Duality Theorem and Drinfeld Double in Braided Tensor Categories \thanks {This work is supported by National Science Foundation (No. 19971074)}
Shouchuan Zhang
Department of Mathematics
Changsha 410082
P.R.China
Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R \# H)\# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *} ) \hbox { \ \ \ as algebras in } {\cal C}.$$ Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.\\ \noindent 2000 Mathematics subject Classification: 16w30.\\ Keywords: Hopf algebra, braided tensor category, duality theorem, Drinfeld double.
math0307255.tex
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     %    This is file    t-manual.tex    ( version 10.09.00 )         %
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     %    Diagrammatic macros for tangles and braided Hopf algebras    %
     %                                                                 %
     %                 (Yu. Bespalov, V. Lyubashenko)                  %
     %                                                                 %
     %       available from      http://www.math.ksu.edu/~lub/         %
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\begin{filecontents*}{t-angles.sty}
% a manual is available from       http://www.math.ksu.edu/~lub/
% \usepackage[emtex]{t-angles}  % for emtex drivers, dviwin and dvips or
% \usepackage{t-angles} = \usepackage[TPIC]{t-angles}
%                       % for TPIC drivers such as dviwin, xdvi, dvips
% \usepackage[kluwer]{t-angles} or \usepackage[emtex,kluwer]{t-angles}
%                       % to use with kluwer.cls

     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     %                                                                 %
     %    This is file    t-angles.sty    ( version 10.09.00 )         %
     %                                                                 %
     %    Diagrammatic macros for tangles and braided Hopf algebras    %
     %                                                                 %
     %                 (Yu. Bespalov, V. Lyubashenko)                  %
     %                                                                 %
     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcount{\hstretch} \hstretch 100
\newcount{\vstretch} \vstretch 100
\newlength{\unitlens} \unitlens 10pt
\def\FillCircDiam{3}

\DeclareOption{emtex}{%
\gdef\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc
  \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth
        \special{em:linewidth \the\@wholewidth}}    %\@wholewidth=0.4pt
\gdef\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
  \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth
        \special{em:linewidth \the\@wholewidth}}    %\@wholewidth=0.8pt
\def\emline#1#2#3#4#5#6{%
       \put(#1,#2){\special{em:moveto}}%
       \put(#4,#5){\special{em:lineto}}}%
\def\drawpath#1#2#3#4{\emline{#1}{#2}1{#3}{#4}2} % graphics for *.lp
}
%
\newcount\@gphlinewidth
%
\DeclareOption{TPIC}{%
\@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736
        \advance\@gphlinewidth\@ne      %\@gphlinewidth=6
\gdef\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc
  \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth
        \@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736
        \advance\@gphlinewidth\@ne}     %\@gphlinewidth=6
\gdef\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
  \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth
        \@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736}
                                        %\@gphlinewidth=11
\def\@paspecial#1#2{%
    \special{pa #1 #2}}
\def\path(#1,#2){%
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    \@path(#1,#2)}
\def\@path(#1,#2){%
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\providecommand{\drawpath}[4]{\path(#1,#2)(#3,#4)} % TPIC graphics for *.lp
\providecommand{\emline}[6]{\path(#1,#2)(#4,#5)}   % TPIC graphics for *.pic
}
\DeclareOption{kluwer}{%
\def\tangle{\begingroup\let\hline\savehline\t@ngle\array{l}}
\def\tanglec{\begingroup\let\hline\savehline\t@ngle\array{c}}
\def\tangler{\begingroup\let\hline\savehline\t@ngle\array{r}}
\def\tangles{\begingroup\let\hline\savehline\t@ngle\array}
} \ExecuteOptions{TPIC}

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\newcount{\v@r}
\newcount{\h@z}
\newcount{\v@t}
\newlength{\unith}
\newlength{\unitv}
\newlength{\textwd}
%
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%
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\divide\unith 1000 \multiply\unith 10 \multiply\unith\hstretch
\@killglue\kern #1\unith}
\def\step{\@ifnextchar[{\@step}{\@step[1]}}
\def\hhstep{\step[-.5]}  \def\hstep{\step[.5]}  \def\Step{\step[2]}
%
\def\object#1{\settowidth{\textwd}{${#1}$}
\hbox{\put(0,0){\kern -.5\textwd${#1}$}\hss}}
%
\def\t@ngl@{%
\gdef\hm@dehalf{1}\gdef\hm@de{2}\gdef\hm@defives{10}\gdef\hm@detens{20}
\gdef\hm@dethree{7}\gdef\hm@deseven{13}\gdef\hm@defiveteen{15}}
%
\def\t@ngle{\t@ngl@
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%
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\@killglue\setbox0=\hbox{\kern -.5\textwd${##1}$}%
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\advance\unith\above@bjectskip\advance\unitv\below@bjectskip%
\ht0=\unith\dp0=\unitv\wd0=0pt\box0}}
%
\unitlength\unitlens\divide\unitlength 1000 \unith\unitlength
\multiply\unith10 \multiply\unith\hstretch \unitv\unitlength
\multiply\unitv10 \multiply\unitv\vstretch
\def\@step[##1]{\@killglue\kern ##1\unith}
%
\def\hbx(##1,##2)##3{%
\@killglue\setbox0=\hbox{##3}\ht0=##2\unitv\dp0=0pt\wd0=##1\unith\box0}
\def\hln##1{\hbx(##1,0){\hbox to ##1\unith{\hrulefill}}}
%
\def\hstr##1{\unith\unitlength \multiply\unith10
\hstretch ##1 \multiply\unith\hstretch}
\def\vstr##1{\unitv\unitlength \multiply\unitv10
\vstretch ##1 \multiply\unitv\vstretch}
%
\def\c@@rds(##1,##2,##3,##4){%
\h@r\hstretch\v@r\vstretch\h@z\hstretch\v@t\vstretch%
\multiply\h@r ##1 \multiply\v@r ##2 \multiply\h@z ##3 \multiply\v@t ##4%
\put(\h@r,\v@r)}
\def\EMl@ne(##1,##2,##3,##4){%
\h@r\hstretch\v@r\vstretch\h@z\hstretch\v@t\vstretch%
\multiply\h@r ##1 \multiply\v@r ##2 \multiply\h@z ##3 \multiply\v@t ##4%
\emline{\h@r}{\v@r}1{\h@z}{\v@t}2}
%
%                      N O D E S   A N D   L I N E S
\def\id{\hbx(0,\hm@de){\c@@rds(0,0,0,\hm@detens){\line(0,1){\v@t}}}}
\def\idash{\hbx(0,2){\c@@rds(0,0,0,3){\line(0,1){\v@t}}%
\c@@rds(0,5,0,2){\line(0,1){\v@t}}\c@@rds(0,9,0,2){\line(0,1){\v@t}}%
\c@@rds(0,13,0,2){\line(0,1){\v@t}}\c@@rds(0,17,0,3){\line(0,1){\v@t}}}}
\def\node{\hbx(0,0){\c@@rds(0,0,\FillCircDiam,0){\circle*{\h@z}}}}
\def\n{\hbx(0,\hm@de){\c@@rds(0,0,0,\hm@detens){\line(0,1){\v@t}}%
\c@@rds(0,0,\FillCircDiam,0){\circle*{\h@z}}}}
\def\s{\hbx(0,\hm@de){\c@@rds(0,0,0,\hm@detens){\line(0,1){\v@t}}%
\c@@rds(0,\hm@detens,\FillCircDiam,0){\circle*{\h@z}}}}
\def\unit{\hbx(0,\hm@de){\c@@rds(0,0,0,10){\line(0,1){\v@t}}%
\c@@rds(0,12,4,0){\circle{\h@z}}}}
\def\counit{\hbx(0,\hm@de){\c@@rds(0,10,0,10){\line(0,1){\v@t}}%
\c@@rds(0,8,4,0){\circle{\h@z}}}}
%
%%%% ARGUMENT OF \ne,\nw,\se,\sw,\ned,\nwd,\sed,\swd IS 0,1,2,3 OR 4;
%%%% 0 PRODUCES EMPTY BOX AND
%%%% OTHER PRODUCE LINES WITH HORIZONTAL PROJECTIONS =##1
\def\@nsew##1##2##3##4##5{\hbx(1,\hm@de){\ifcase##1
\or\EMl@ne(##2)\or\EMl@ne(##3)\or\EMl@ne(##4)\or\EMl@ne(##5)\fi}}
\def\ne##1{\@nsew{##1}{0,0,10,\hm@detens}{-10,0,10,\hm@detens}%
{-20,0,10,\hm@detens}{-30,0,10,\hm@detens}}
\def\nw##1{\@nsew{##1}{0,\hm@detens,10,0}{0,\hm@detens,20,0}%
{0,\hm@detens,30,0}{0,\hm@detens,40,0}}
\def\se##1{\@nsew{##1}{10,0,0,\hm@detens}{10,0,-10,\hm@detens}%
{10,0,-20,\hm@detens}{10,0,-30,\hm@detens}}
\def\sw##1{\@nsew{##1}{0,0,10,\hm@detens}{0,0,20,\hm@detens}%
{0,0,30,\hm@detens}{0,0,40,\hm@detens}}
%
\def\nd{\hbx(0,2){\c@@rds(0,0,0,3){\line(0,1){\v@t}}%
\c@@rds(0,5,0,2){\line(0,1){\v@t}}\c@@rds(0,9,0,2){\line(0,1){\v@t}}%
\c@@rds(0,13,0,2){\line(0,1){\v@t}}\c@@rds(0,17,0,3){\line(0,1){\v@t}}%
\c@@rds(0,0,\FillCircDiam,0){\circle*{\h@z}}}}
\def\ned##1{\hbx(1,2){\ifcase##1\or%
\EMl@ne(0,0,1,2)\EMl@ne(2,4,4,8)\EMl@ne(5,10,7,14)\EMl@ne(8,16,10,20)\or%
\EMl@ne(-10,0,-7,3)\EMl@ne(-5,5,-3,7)\EMl@ne(-1,9,1,11)%
\EMl@ne(3,13,5,15)\EMl@ne(7,17,10,20)\or%
\EMl@ne(-20,0,-14,4)\EMl@ne(-11,6,-8,8)\EMl@ne(-5,10,-2,12)
\EMl@ne(1,14,4,16)\EMl@ne(7,18,10,20)\or%
\EMl@ne(-30,0,-26,2)\EMl@ne(-24,3,-20,5)\EMl@ne(-18,6,-14,8)%
\EMl@ne(-12,9,-8,11)\EMl@ne(-6,12,-2,14)\EMl@ne(0,15,4,17)%
\EMl@ne(6,18,10,20)\fi}}
\def\nwd##1{\hbx(1,2){\ifcase##1\or%
\EMl@ne(0,20,2,16)\EMl@ne(3,14,5,10)\EMl@ne(6,8,8,4)\EMl@ne(9,2,10,0)\or%
\EMl@ne(0,20,3,17)\EMl@ne(5,15,7,13)\EMl@ne(9,11,11,9)%
\EMl@ne(13,7,15,5)\EMl@ne(17,3,20,0)\or%
\EMl@ne(0,20,3,18)\EMl@ne(6,16,9,14)\EMl@ne(12,12,15,10)%
\EMl@ne(18,8,21,6)\EMl@ne(24,4,30,0)\or%
\EMl@ne(0,20,4,18)\EMl@ne(6,17,10,15)\EMl@ne(12,14,16,12)\EMl@ne(18,11,22,9)%
\EMl@ne(24,8,28,6)\EMl@ne(30,5,34,3)\EMl@ne(36,2,40,0)\fi}}
\def\sd{\hbx(0,2){\c@@rds(0,0,0,3){\line(0,1){\v@t}}%
\c@@rds(0,5,0,2){\line(0,1){\v@t}}\c@@rds(0,9,0,2){\line(0,1){\v@t}}%
\c@@rds(0,13,0,2){\line(0,1){\v@t}}\c@@rds(0,17,0,3){\line(0,1){\v@t}}%
\c@@rds(0,20,\FillCircDiam,0){\circle*{\h@z}}}}
\def\sed##1{\hbx(1,2){\ifcase##1\or%
\EMl@ne(10,0,8,4)\EMl@ne(7,6,5,10)\EMl@ne(4,12,2,16)\EMl@ne(1,18,0,20)\or%
\EMl@ne(10,0,7,3)\EMl@ne(5,5,3,7)\EMl@ne(1,9,-1,11)%
\EMl@ne(-3,13,-5,15)\EMl@ne(-7,17,-10,20)\or%
\EMl@ne(10,0,7,2)\EMl@ne(4,4,1,6)\EMl@ne(-2,8,-5,10)%
\EMl@ne(-8,12,-11,14)\EMl@ne(-14,16,-20,20)\or%
\EMl@ne(10,0,6,2)\EMl@ne(4,3,0,5)\EMl@ne(-2,6,-6,8)\EMl@ne(-8,9,-12,11)%
\EMl@ne(-14,12,-18,14)\EMl@ne(-20,15,-24,17)\EMl@ne(-26,18,-30,20)\fi}}
\def\swd##1{\hbx(1,2){\ifcase##1\or%
\EMl@ne(0,0,2,4)\EMl@ne(3,6,5,10)\EMl@ne(6,12,8,16)\EMl@ne(9,18,10,20)\or%
\EMl@ne(0,0,3,3)\EMl@ne(5,5,7,7)\EMl@ne(9,9,11,11)%
\EMl@ne(13,13,15,15)\EMl@ne(17,17,20,20)\or%
\EMl@ne(0,0,3,2)\EMl@ne(6,4,9,6)\EMl@ne(12,8,15,10)%
\EMl@ne(18,12,21,14)\EMl@ne(24,16,30,20)\or%
\EMl@ne(0,0,4,2)\EMl@ne(6,3,10,5)\EMl@ne(12,6,16,8)\EMl@ne(18,9,22,11)%
\EMl@ne(24,12,28,14)\EMl@ne(30,15,34,17)\EMl@ne(36,18,40,20)\fi}}
%
%                      D A S H E D   C R O S S E S
\def\xd{\hbx(2,2){\EMl@ne(0,0,4,4)\EMl@ne(6,6,8,8)\EMl@ne(12,12,14,14)%
\EMl@ne(16,16,20,20)\EMl@ne(0,20,4,16)\EMl@ne(6,14,14,6)\EMl@ne(16,4,20,0)}}
\def\xxd{\hbx(2,2){\EMl@ne(0,0,4,4)\EMl@ne(6,6,14,14)\EMl@ne(16,16,20,20)%
\EMl@ne(0,20,4,16)\EMl@ne(6,14,8,12)\EMl@ne(12,8,14,6)\EMl@ne(16,4,20,0)}}
\def\hxd{\hbx(1,2){\EMl@ne(0,0,2,4)\EMl@ne(3,6,4,8)\EMl@ne(6,12,7,14)%
\EMl@ne(8,16,10,20)\EMl@ne(0,20,2,16)\EMl@ne(3,14,7,6)\EMl@ne(8,4,10,0)}}
\def\hxxd{\hbx(1,2){\EMl@ne(0,0,2,4)\EMl@ne(3,6,7,14)\EMl@ne(8,16,10,20)%
\EMl@ne(0,20,2,16)\EMl@ne(3,14,4,12)\EMl@ne(6,8,7,6)\EMl@ne(8,4,10,0)}}
%
%                        O T H E R   L I N E S
\def\@d##1##2{\hbx(\hm@dehalf,\hm@de){%
\EMl@ne(##1\hm@defives,##2\hm@detens)}}
\def\d{\@d{}{0,0,}}   \def\dd{\@d{0,0,}{}}
\def\@hd##1##2##3{\hbx(.5,\hm@de){\EMl@ne(##1,\hm@defives,##2)%
\c@@rds(##1,\hm@defives,0,\hm@defives){\line(0,##31){\v@t}}}}
\def\hd{\@hd{5}{0,\hm@detens}{-}}     \def\dh{\@hd{0}{5,0}{}}
\def\hdd{\@hd{0}{5,\hm@detens}{-}}    \def\ddh{\@hd{5}{0,0}{}}
%
%             P U T   S O M E T H I N G   A N Y W H E R E
\def\n@de##1##2##3{\hbx(0,0){%
\c@@rds(##1,0,0){\makebox(0,0)[##2]{$\scriptstyle ##3$}}}}
\def\nodeu##1{\hbx(0,0){\n@de{0,3}{cb}{##1}}}
\def\noded##1{\hbx(0,0){\n@de{0,-3}{ct}{##1}}}
\def\nodel##1{\hbx(0,0){\n@de{-2,0}{rc}{##1}}}
\def\noder##1{\hbx(0,0){\n@de{2,0}{lc}{##1}}}
\def\noderu##1{\hbx(0,0){\n@de{5,2}{lb}{##1}}}
\def\noderd##1{\hbx(0,0){\n@de{5,-2}{lt}{##1}}}
\def\nodelu##1{\hbx(0,0){\n@de{-5,2}{rb}{##1}}}
\def\nodeld##1{\hbx(0,0){\n@de{-5,-2}{rt}{##1}}}
\def\Put(##1,##2)[##3]##4{%
\h@r\hstretch \multiply\h@r ##1 \v@r\vstretch \multiply\v@r ##2
\put(\h@r,\v@r){\makebox(\h@z,\v@t)[##3]{\scriptsize$##4$}}}
% \def\negobj##1{\raise -0.2\unitv\hbox{\put(0,0){$##1$}}}
% \def\obj##1{\raise .2\unitv\hbox{\put(0,0){$##1$}}}
% \def\Obj##1{\settowidth{\textwd}{$##1$}%
% \raise 1.1\unitv\hbox{\kern -1\textwd $##1$}}
%
%                            B O X E S
\def\dbox##1##2{\hbx(##1,\hm@de){\multiply\hstretch ##1%
\c@@rds(0,0,10,\hm@detens){\dashbox{100}(\h@z,\v@t){\scriptsize$##2$}}}}
\def\ffbox##1##2{\hbx(##1,\hm@de){\multiply\hstretch ##1%
\c@@rds(0,0,10,\hm@detens){\framebox(\h@z,\v@t){\scriptsize$##2$}}}}
\def\obox##1##2{\hbx(##1,\hm@de){\multiply\hstretch ##1%
\c@@rds(5,\hm@defives,10,\hm@detens){\oval(\h@z,\v@t)}%
\c@@rds(0,0,10,\hm@detens){\makebox(\h@z,\v@t)[cc]{\scriptsize$##2$}}}}
\def\tbox##1##2{\hbx(##1,\hm@de){\multiply\hstretch ##1%
\c@@rds(0,0,10,\hm@detens){\makebox(\h@z,\v@t)[cc]{\scriptsize$##2$}}}}
%
%                       ( C O ) P A I R I N G S
\def\@ev##1##2##3##4{\hbx(##1){\c@@rds(##2,##3,##3){\oval(\h@z,\v@t)[##4]}}}
\def\ev{\@ev{\hm@de,\hm@de}{\hm@defives,\hm@detens}\hm@detens b}
\def\coev{\@ev{\hm@de,\hm@de}{\hm@defives,0}\hm@detens t}
\def\hcoev{\@ev{1,\hm@de}{5,0}{10}t}
\def\hev{\@ev{1,\hm@de}{5,\hm@detens}{10}b}
\def\Coev{\@ev{1,\hm@de}{15,0}{30}t}
\def\Ev{\@ev{1,\hm@de}{15,\hm@detens}{30}b}
\def\COEV{\@ev{2,2}{30,0}{60}t}
\def\EV{\@ev{2,2}{30,20}{60}b}
%
%             ( O V E R / U N D E R ) C R O S S I N G S
\def\X{\hbx(1,\hm@de){\EMl@ne(0,\hm@detens,10,0)\EMl@ne(0,0,10,\hm@detens)}}
\def\XX{\hbx(\hm@de,\hm@de){\EMl@ne(0,\hm@detens,\hm@detens,0)%
\EMl@ne(0,0,\hm@detens,\hm@detens)}}
\def\x{\hbx(\hm@de,\hm@de){\EMl@ne(0,\hm@detens,\hm@detens,0)%
\EMl@ne(0,0,\hm@dethree,\hm@dethree)%
\EMl@ne(\hm@detens,\hm@detens,\hm@deseven,\hm@deseven)}}
\def\xx{\hbx(\hm@de,\hm@de){\EMl@ne(0,\hm@detens,\hm@dethree,\hm@deseven)%
\EMl@ne(0,0,\hm@detens,\hm@detens)%
\EMl@ne(\hm@detens,0,\hm@deseven,\hm@dethree)}}
\def\hx{\hbx(1,2){\EMl@ne(0,20,10,0)\EMl@ne(0,0,3,6)\EMl@ne(10,20,7,14)}}
\def\hxx{\hbx(1,2){\EMl@ne(0,20,3,14)\EMl@ne(0,0,10,20)\EMl@ne(10,0,7,6)}}
%
%                    ( C O ) P R O D U C T S
\def\pr@d##1##2##3{%
\c@@rds(##1){\oval(\h@z,\v@t)[c##2]}\c@@rds(##3){\line(0,1){\v@t}}}%
\def\pr@dd##1##2##3##4{\c@@rds(5,##1,10,10){\oval(\h@z,\v@t)[c##2]}%
\EMl@ne(5,10,##3)\c@@rds(5,##4,0,5){\line(0,1){\v@t}}}
\def\c@cyc##1{\c@@rds(##1,\FillCircDiam,0){\circle*{\h@z}}}
\def\cu{\@ifnextchar*%
{\hbx(\hm@de,\hm@de){\pr@d{\hm@defives,\hm@detens,\hm@detens,\hm@detens}%
{b}{\hm@defives,0,0,\hm@defives}%
\c@cyc{\hm@defives,\hm@defives}}\@gobble}%
{\hbx(\hm@de,\hm@de){\pr@d{\hm@defives,\hm@detens,\hm@detens,\hm@detens}%
{b}{\hm@defives,0,0,\hm@defives}}}}
\def\cd{\@ifnextchar*%
{\hbx(\hm@de,\hm@de){\pr@d{\hm@defives,0,\hm@detens,\hm@detens}%
{t}{\hm@defives,\hm@defives,0,\hm@defives}%
\c@cyc{\hm@defives,\hm@defives}}\@gobble}%
{\hbx(\hm@de,\hm@de){\pr@d{\hm@defives,0,\hm@detens,\hm@detens}%
{t}{\hm@defives,\hm@defives,0,\hm@defives}}}}
\def\hcu{\@ifnextchar*%
{\hbx(1,\hm@de){\pr@d{5,\hm@detens,10,10}{b}{5,0,0,\hm@defiveteen}%
\c@cyc{5,\hm@defiveteen}}\@gobble}%
{\hbx(1,\hm@de){\pr@d{5,\hm@detens,10,10}{b}{5,0,0,\hm@defiveteen}}}}
\def\hcd{\@ifnextchar*%
{\hbx(1,\hm@de){\pr@d{5,0,10,10}{t}{5,5,0,\hm@defiveteen}%
\c@cyc{5,5}}\@gobble}%
{\hbx(1,\hm@de){\pr@d{5,0,10,10}{t}{5,5,0,\hm@defiveteen}}}}
\def\Cu{\@ifnextchar*%
{\hbx(4,2){\pr@d{20,20,40,20}{b}{20,0,0,10}\c@cyc{20,10}}\@gobble}%
{\hbx(4,2){\pr@d{20,20,40,20}{b}{20,0,0,10}}}}
\def\Cd{\@ifnextchar*%
{\hbx(4,2){\pr@d{20,0,40,20}{t}{20,10,0,10}\c@cyc{20,10}}\@gobble}%
{\hbx(4,2){\pr@d{20,0,40,20}{t}{20,10,0,10}}}}
\def\hd@cd##1{\@ifnextchar*%
{\hbx(1,2){\pr@dd{0}{t}{##1,20}{5}\c@cyc{5,5}}\@gobble}%
{\hbx(1,2){\pr@dd{0}{t}{##1,20}{5}}}}
\def\hd@cu##1{\@ifnextchar*%
{\hbx(1,2){\pr@dd{20}{b}{##1,0}{10}\c@cyc{5,15}}\@gobble}%
{\hbx(1,2){\pr@dd{20}{b}{##1,0}{10}}}}
\def\hdcd{\hd@cd 0}   \def\hddcd{\hd@cd{10}}
\def\hdcu{\hd@cu{10}}   \def\hddcu{\hd@cu 0}
%
%                          A C T I O N S
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\def\rd{\@ifnextchar[{\@rd}{\@rd[1]}}
\def\Lu{\lu[2]}\def\Ru{\ru[2]}\def\Ld{\ld[2]}\def\Rd{\rd[2]}
\def\@hlu[##1]{{\divide\unith2\multiply\hstretch##1%
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\def\@hrd[##1]{{\divide\unith2\multiply\hstretch##1%
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\def\hld{\@ifnextchar[{\@hld}{\@hld[1]}}
\def\hrd{\@ifnextchar[{\@hrd}{\@hrd[1]}}
% \def\hlu{\@ct{.5}{5}{\hm@detens}{10}{lb}}
% \def\hru{\@ct{.5}{0}{\hm@detens}{10}{rb}}
% \def\hld{\@ct{.5}{5}{0}{10}{lt}}
% \def\hrd{\@ct{.5}{0}{0}{10}{rt}}
%
%                       C O M P O S I T I O N S
\def\c@mp##1##2##3##4##5##6{\hbx(##1,2){%
\c@@rds(##2,0,##3){\oval(\h@z,\v@t)[##4t]}%
\c@@rds(##2,20,##3){\oval(\h@z,\v@t)[##4b]}%
\c@@rds(##2,##5,0,##6){\line(0,1){\v@t}}}}
\def\cucd{\c@mp{1}{5}{10,10}{}{5}{10}}
\def\k{\c@mp{1}{0}{20,10}{r}{0}{20}}
\def\kk{\c@mp{1}{10}{20,10}{l}{0}{20}}
\def\rurd{\c@mp{.5}{0}{10,10}{r}{0}{20}}
\def\luld{\c@mp{.5}{5}{10,10}{l}{0}{20}}
%
%                M O R P H I S M S   I N   A   C I R C L E
\def\c@rcled##1##2{\c@@rds(##1,12,12){\oval(\h@z,\v@t)}%
\c@@rds(##1,0,0){\makebox(\h@z,\v@t)[cc]{\scriptsize$##2$}}}
%               ( 1, 0 ) -,  ( 0, 1 ) -  T E N S O R S
\def\@Q##1##2{\hbx(0,\hm@de){%
\c@@rds(0,##1,0,4){\line(0,1){\v@t}}\c@rcled{0,10}{##2}}}
\def\Q##1{\@Q{0}{##1}}     \def\QQ##1{\@Q{16}{##1}}
%                    ( 1, 1 ) - T E N S O R S
\def\O##1{\hbx(0,\hm@de){\c@@rds(0,0,0,4){\line(0,1){\v@t}}%
\c@@rds(0,16,0,4){\line(0,1){\v@t}}\c@rcled{0,10}{##1}}}
\def\S{\O{S}}              \def\SS{\O{S^{-1}}}
%\def\circ##1{\hbx(2,2){\c@rcled{10,4}{##1}}}
\def\morph##1{\hbx(2,2){\c@@rds(10,10,10,10){\oval(\h@z,\v@t)}%
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\c@@rds(10,0,0,5){\line(0,1){\v@t}}%
\c@@rds(10,15,0,5){\line(0,1){\v@t}}}}
%               ( 2, 0 ) -,  ( 0, 2 ) -  T E N S O R S
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\c@@rds(0,##1,0,4){\line(0,##41){\v@t}}%
\c@@rds(20,##1,0,4){\line(0,##41){\v@t}}%
\c@@rds(4,##1,8,8){\oval(\h@z,\v@t)[l##2]}%
\c@@rds(16,##1,8,8){\oval(\h@z,\v@t)[r##2]}\c@rcled{##3}{##5}}}
\def\ro##1{\r@{4}{t}{10,8}{-}{##1}}
\def\coro##1{\r@{16}{b}{10,12}{}{##1}}
%
\def\R@##1##2##3{\hbx(4,\hm@de){%
\c@@rds(14,##1,28,20){\oval(\h@z,\v@t)[l##2]}%
\c@@rds(26,##1,28,20){\oval(\h@z,\v@t)[r##2]}\c@rcled{20,10}{##3}}}
\def\Ro##1{\R@{0}{t}{##1}}
\def\coRo##1{\R@{20}{b}{##1}}
%               ( 2, 1 ) -,  ( 1, 2 ) -  T E N S O R S
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\c@@rds(4,##1,8,8){\oval(\h@z,\v@t)[l##2]}%
\c@@rds(16,##1,8,8){\oval(\h@z,\v@t)[r##2]}%
\c@@rds(0,##1,0,5){\line(0,##31){\v@t}}%
\c@@rds(20,##1,0,5){\line(0,##31){\v@t}}%
\c@@rds(10,##4,0,5){\line(0,##31){\v@t}}%
\c@rcled{10,##5}{##6}}}
\def\tu##1{\t@{15}b{}{0}{11}{##1}}  \def\td##1{\t@{5}t{-}{20}{9}{##1}}
%                    ( 2, 2 ) - T E N S O R S
\def\ox##1{\hbx(2,2){\c@@rds(10,10,14,14){\oval(\h@z,\v@t)}%
\c@@rds(10,10,0,0){\makebox(\h@z,\v@t)[cc]{\scriptsize$##1$}}
\EMl@ne(0,0,5,5)\EMl@ne(0,20,5,15)\EMl@ne(20,20,15,15)\EMl@ne(20,0,15,5)}}
%
% HALF HEIGHT ROW IS PRODUCED BY \hh or \HH AT THE BEGINNING OF THE ROW
%   \HH is global for a row, \hh is local
\def\HH{\gdef\hm@dehalf{.5}\gdef\hm@de{1}
\gdef\hm@dethree{3}\gdef\hm@defives{5}\gdef\hm@deseven{7}
\gdef\hm@detens{10}\gdef\hm@defiveteen{5}}
\def\hh{\def\hm@dehalf{.5}\def\hm@de{1}
\def\hm@dethree{3}\def\hm@defives{5}\def\hm@deseven{7}
\def\hm@detens{10}\def\hm@defiveteen{5}}
%
\def\array{\let\@acol\@arrayacol \let\@classz\@arrayclassz
\let\@classiv\@arrayclassiv \def\arraystretch{0}\arraycolsep0pt
\def\\{\gdef\above@bjectskip{4pt}\gdef\below@bjectskip{0pt}\t@ngl@\@arraycr}
\let\@halignto\@empty\@tabarray}
}
%        E N D   \def\t@ngle
%
\def\tangle{\t@ngle\array{l}}    \def\endtangle{\endarray\t@ngl@}
\def\tanglec{\t@ngle\array{c}}   \let\endtanglec\endtangle
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\def\tangles{\t@ngle\array}      \let\endtangles\endtangle
%
\ProcessOptions
\endinput
\end{filecontents*}
\begin{filecontents}{t-manual.bbl}
\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}

\end{filecontents}








\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{t-angles}
%\usepackage[emtex]{t-angles}
%\usepackage{t-erratr}

\setlength{\textwidth}{16 true cm} \setlength{\textheight}{22.5
true cm} \setlength{\topmargin}{-1.4 true cm}
\setlength{\oddsidemargin}{0mm}
\title{
Duality Theorem  and Drinfeld Double  in Braided Tensor Categories
 \thanks {This work is supported by National Science Foundation  (No. 19971074)}}
\author{
Shouchuan Zhang \\ Department  of Mathematics, Hunan  University,
Changsha  410082, P.R.China\\
E-mail:z9491@yahoo.com.cn }
\date{}
\begin{document}
% \tableofcontents
\newtheorem{Theorem}{\quad Theorem}[section]
\newtheorem{Proposition}[Theorem]{\quad Proposition}
\newtheorem{Definition}[Theorem]{\quad Definition}
\newtheorem{Corollary}[Theorem]{\quad Corollary}
\newtheorem{Lemma}[Theorem]{\quad Lemma}
\newtheorem{Example}[Theorem]{\quad Example}
\maketitle \addtocounter{section}{0}






 \begin {abstract} Let $H$ be a finite Hopf algebra with  $C_{H,H} = C_{H,H}^{-1}.$
  The duality  theorem is shown for $H$, i.e., $$   (R \# H)\# H^{\hat *}   \cong R \otimes (H \bar
  \otimes H^{\hat *} ) \hbox { \ \ \  as algebras in } {\cal C}.$$
Also, it is proved that the Drinfeld double $(D(H),[b])$ is a
quasi-triangular Hopf
algebra in ${\cal C}$.\\
\noindent 2000 Mathematics subject Classification: 16w30.\\
Keywords: Hopf algebra, braided tensor category, duality theorem,
Drinfeld double.
 \end {abstract}


\section {Introduction and Preliminaries}

               It is well-known that in the work of $C^*$-algebras and
               von Neumann algebras, for an abelian group $G$,the product of $R$ crossed
by $G$ crossed by $\hat G$  is isomorphic to the tensor product of
$R$ and  the  compact operator.
 Its generalization to Hopf-von Neumann algebras was known again.( see,  for example,
  Stratila \cite {St81}). Blattner and Montgomery
strip off the functional analysis and  duplicate the result at the
level of Hopf algebras (see \cite {BM85} and \cite {Mo93}). They
proved that for an ordinary Hopf algebra $H$ and some subalgebra
$U$ of $H^*$,
\begin {eqnarray*} \label {e1}
(R \# H)\# U   \cong R \otimes (H \# U ) \hbox { \ \ \ as algebras
\ . }
 \end {eqnarray*}


The basic construction of the Drinfeld double is due to Drinfeld
\cite {Dr86}. S.Majid  \cite {Ma90a} and D.E.Radiford \cite
{Ra93a} modified the treatment.

In this paper, we generalize the duality theorem and Drinfeld
double into the braided case, i.e., for a finite  Hopf algebra $H$
with  $C_{H,H} = C_{H,H}^{-1},$ we show that
  \begin {eqnarray*} \label {e2}
(R \# H)\# H^{\hat *}   \cong R \otimes (H \bar
  \otimes H^{\hat *} )  \hbox { \ \ \ as
algebras in } {\cal C}
 \end {eqnarray*}
We also show that the  Drinfeld double (D(H),[b]) is a
quasi-triangular Hopf algebra in ${\cal C}$.





In this paper, $({\cal C}, \otimes, I,  C )$  is always   a
braided tensor category, where $I$ is the identity object and $C$
is the braiding.


By \cite [Theorem 0.1] {ZC},
 we can view every braided tensor category as
 a strict braided tensor category.



  For an object $V $ in  ${\cal C}$, if there exists
an object $U$ and morphisms : $d_V : U\otimes V \rightarrow I
\hbox { \ \ \ and \ \ \ } b_V : I \rightarrow V \otimes U$ in
${\cal C}$ such that $ (d_V \otimes id _U) (id _U \otimes b_V)
=id_ U \hbox {\ \ \ and \ \ \ } (id_V \otimes d _V)(b_V \otimes id
_V)= id_ V,$ then $U$  is called  a  left dual of $V,$ written as
$V^{ *}.$ In this case, $d_V$ and $b_V$ are called the evaluation
morphism and coevaluation morphism of $V$, respectively. In
general, we use $d$ and $b$ instead of $d_V$ and $b_V.$
Furthermore, $V$ is said to be finite if $V$ has a left dual (see
\cite {Ta} ).


Let us  define the transpose $f^* =(d \otimes id _{U^*} )(id_{V^*}
\otimes f \otimes id _{U^*}) (id _{V^*}\otimes b): V^* \rightarrow
U^*$ of
 a morphism $f : U \rightarrow V$.






Let $\Delta ^{cop} = C_{H,H}\Delta$  and $m^{op} = m C_{H,H}.$ We
denote $(H, \Delta ^{cop}, \epsilon)$  by $H^{cop}$ and  $(H,
m^{op}, \eta)$  by $H^{op}$. Furthermore we denote  $((H^{ * })^{\
op }) ^ {  cop }$ by $H^{\hat *}$.


A bialgebra $(H, R, \bar \Delta )$ with convolution-invertible $R$
in $Hom _{\cal C} (I, H\otimes H)$ is called a quasi-triangular
bialgebra
 in ${\cal C}$ if
$(H, \bar \Delta , \epsilon )$  is a coalgebra and the following
conditions are satisfied:

 (QT1)  $(\bar \Delta \otimes id ) R = (id \otimes id  \otimes m )
(id \otimes R  \otimes id)R$;



 (QT2)  $(id   \otimes \Delta ) R = (m  \otimes id  \otimes id)
(id \otimes R  \otimes id)R$;


 (QT3)  $(m \otimes m )(id \otimes C_{H,H} \otimes id  )(\bar \Delta \otimes R )
=  (m \otimes m )(id \otimes C_{H,H} \otimes id  )(R \otimes
\Delta ).$


 \noindent In this case, we also
say that $(H,R, \bar \Delta )$ is a braided quasi-triangular
bialgebra.



Dually, we can define a coquasi-triangular bialgebra $(H, r, \bar
m)$  in the  braided tensor category  ${\cal C}$ .



In particular, we  say that  $(H,R)$  is quasi-triangular if $(H,
R, \Delta ^{cop})$ is quasi-triangular. Dually, we  say that
$(H,r)$  is coquasi-triangular if $(H, r, m ^{op})$ is
coquasi-triangular.


A morphism $\tau$ from $H\otimes A$ to $I$ in ${\cal C}$ is called
a skew pairing on $H\otimes A$ if the following conditions are
satisfied:

(SP1): $\tau (m \otimes id ) = \tau (id \otimes \tau \otimes id )
(id \otimes id  \otimes \Delta )$ ;

(SP2): $\tau ( id \otimes m ) = (\tau  \otimes \tau )(id \otimes
C_{H,A} \otimes id ) (\Delta \otimes id \otimes id)$;

(SP3):$\tau (id \otimes \eta ) = \epsilon _H$;

(SP4):$\tau (\eta \otimes id ) = \epsilon _A$.











 If a morphism $\tau$ from $V\otimes W$ to $I$ in ${\cal C}$
 satisfies
 $$(id _U\otimes \tau ) (C_{V,U}\otimes id_W )= (\tau \otimes id _U  ) (id_V\otimes C_{U,W} )$$
for $U = V, W,$ then $\tau $ is said  to be symmetric with respect
to the braiding $C$ \ .

\begin {Lemma} \label {01''} If $H$ has a left dual $H^*$, then the
 following conditions are equivalent:

(i) The evaluation of $H$ is symmetric with respect to the
braiding $C$.

(ii) $C_{U,V} = C_{V,U}^{-1}$  for $U, V = H $ or $H^*$.

(iii)  $C_{H,H} = C_{H,H}^{-1}$.

(iv)  $C_{H^*, H^*} = C_{H^*, H^*}^{-1}$.

(v) $(id _H\otimes d ) (C_{H^{ *},H}\otimes id_H )= (d \otimes id
_H  ) (id_{H^{ *}}\otimes C_{H,H} )$.

(vi) $(id _{H ^{ *}} \otimes d ) (C_{H^{ *},H^{ * }}\otimes id_H
)= (d \otimes id _{H^{ * }}  ) (id_{H^{ *}}\otimes C_{H^{ *},H}
)$.

 \end {Lemma}

{\bf Proof.} It is straightforward.\ \
\begin{picture}(5,5)
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\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
%$\Box$
\section {   The Duality Theorem }

                            In this section, we obtain the duality
theorem
 for Hopf algebras living in the braided tensor category ${\cal C}.$

Throughout this section, $H$  is a finite Hopf algebra  with
$C_{H,H}= C_{H,H}^{-1}$ living in ${\cal C}$.
 $(R, \alpha )$  is a left $H$-module algebra in ${\cal C}$ and
 $R\# H$  is the smash product in ${\cal C}$. Let $H^{\hat * }= ((H^ {  * })^{ op })^{  cop }. $

The proof of   Lemmas \ref {1.1}
---\ref {1.7} is very similar to that of correspending results in \cite {Mo93}.
\begin {Lemma} \label {1.1}
If $(R, \alpha )$ is an $H$-module algebra, let $\phi = (id
\otimes \alpha )(b' \otimes id ):$ \ $R \rightarrow  H^{*\ op }
\otimes R,$ where $b' = C_{H, H^{* \ op }}b$. Then
$$\phi m =
(m\otimes m )(id \otimes C_{R, H^{*\ op}}\otimes id)(\phi \otimes
\phi ). $$
\end {Lemma}

\begin {Lemma} \label {1.2}
(1)  $(H, \rightharpoonup )$  is a left  $H^{ \hat *}$-module
algebra under the module operation  $\rightharpoonup \  = (id
\otimes d) (C_{H^{\hat *}, H} \otimes id  ) (id \otimes \Delta )
$.

(2)  $(H^{\hat *}, \rightharpoonup )$  is a left $H$-module
algebra under the module operation  $\rightharpoonup \ = (id
\otimes d)(id \otimes C_{H, H^{\hat *}}) (C_{ H,H^{\hat *}}
\otimes id  ) (id \otimes \Delta )$.

(3)  $(H, \leftharpoonup )$  is a right $H^{ \hat *}$-module
algebra   under the module operation  $\leftharpoonup \ = (
d\otimes id) (C_{H,H^{\hat *} } \otimes id  )(id \otimes C_{H,
H^{\hat *}}) (\Delta \otimes id )$.

(4)  $(H^{\hat *}, \leftharpoonup )$  is a right $H$-module
algebra under the module operation  $\leftharpoonup \  = (
d\otimes id) (id \otimes C_{ H^{\hat *},H}) (\Delta \otimes id )$.


\end {Lemma}

\begin {Lemma}  \label {1.3} The object $H \otimes H^{\hat *}$ becomes an algebra, written as   $H
\bar \otimes H^{\hat *}$, under the  multiplication $m_{H \bar
\otimes H^{\hat *}} = (id _ H \otimes d \otimes id _{H^{\hat *}}
)$  and unity $\eta _{H \bar \otimes H^{\hat *}} = b. $
          \end {Lemma}

          In fact, if ${\cal C}$  is a braided  tensor category
          determined  by the (co)quasi-triangular structure of
          a (co)quasi-triangular Hopf algebra over a field $k$,
           then $H\bar \otimes
          H^{\hat *}$  can be viewed as  $End _k(H)$ or $M_n (k)= \{A \mid A \hbox { is an  } n\times n
\hbox { matrix  over the field } k\}$.


















\begin {Lemma} \label {1.4}
Let $\lambda = (m \otimes d \otimes id ) (id \otimes C_{H^{\hat
* }, H}\otimes id \otimes id )(id \otimes id \otimes \Delta \otimes id )(id \otimes id \otimes
b)$  and $\rho  =  ( d \otimes m\otimes id ) (id \otimes id
\otimes C_{ H, H}\otimes id  )(id \otimes C_{H, H}\otimes id
\otimes id )(id \otimes id \otimes \Delta \otimes id )(id \otimes
id \otimes b)$. Then $\lambda $ is an algebra morphism from $H\#H
^{\hat *}$ to $H\bar \otimes H ^{\hat *}$ and $\rho $ is an
anti-algebra morphism from $H ^{\hat *}\#H$ to $H\bar \otimes H
^{\hat *}$.
\end {Lemma}


\begin {Lemma} \label {1.5} The following relation holds:
 $ m (\lambda
\otimes \rho ) =  m (\rho \otimes \lambda )(id \otimes
\rightharpoonup \otimes \leftharpoonup \otimes id )(id \otimes
C_{H, H^{\hat *}} \otimes C_{H^{\hat *},H} \otimes id) (id \otimes
id \otimes C_{H^{\hat *}, H^{\hat *}} \otimes id \otimes id )(id
\otimes C_{H^{\hat *},H} \otimes C_{H, H^{\hat *}} \otimes id)(
 C_{H^{\hat *},H^{\hat *}}\otimes  id \otimes id \otimes C_{H^{\hat *}, H^{\hat *}}
 )(S \otimes id  \otimes id  \otimes id  \otimes id  \otimes id )(\Delta  \otimes id
  \otimes id  \otimes \Delta  ) (id \otimes C_{H,H}\otimes id ) (C_{H, H^{\hat *} } \otimes
  C_{ H^{\hat *},H })(id \otimes C_{H ^{\hat *},H ^{\hat *}}\otimes id
  )$.
\end {Lemma}
\begin {Lemma} \label {1.6}
If the antipode of $H$ is  invertible, then  $\lambda $ is invertible.
\end {Lemma}

\begin {Lemma} \label {1.7}
$R \# H$ becomes    an  $H^{ *}$-module algebra under the
 module operation  $\rightharpoonup ' \ =  (id \otimes
\rightharpoonup )(C_{H^{\hat *}, R} \otimes id ).$
\end {Lemma}


          \begin {Theorem} \label {1.8}
 If $H$ is a  finite Hopf algebra with  $C_{H,H}=C_{H,H}^{-1}$,
then
  $$   (R \# H)\# H^{\hat *}   \cong R \otimes (H \bar
  \otimes H^{\hat *} ) \hbox { \ \ \  as algebras in }  {\cal C},$$

Where $H \bar
  \otimes H^{\hat *} $ is defined in Lemma \ref {1.3}.



\end {Theorem}






{\bf Proof. }   We first define a morphism  $w$ from $H^{\hat *} $
to $H \# H^{\hat *}$ such that  $w = \lambda ^{-1}\rho (S^{-1}
\otimes \eta _H )$; this can be done by Lemma \ref  {1.6} and
\cite [Theorem 4.1]{Ta} . Since $\rho $ and $S^{-1}$ are
anti-algebra morphisms, $w$ is an algebra morphism.  Set $ \phi =
(id \otimes \alpha  ) (C_{H, H^{\hat *}} \otimes id )(b \otimes id
)$.

We now define a morphism  $\Phi = (id \otimes  m_{H \# H^{\hat
*}})(id  \otimes w \otimes id \otimes id )(C_{ H^{\hat *}, R
}\otimes id \otimes id ) (\phi \otimes id \otimes id )$ from
$(R\#H)\# H^{\hat *}$ to $R \otimes (H \bar \otimes H^{\hat *})$
and a morphism
 $\Psi = (id \otimes  m_{H \#
H^{\hat *}})(id  \otimes w \otimes id \otimes id )(C_{ H^{\hat *
},R}\otimes id \otimes id ) (S \otimes id \otimes id \otimes id
)(\phi \otimes id \otimes id )$ from $R \otimes (H \bar \otimes
H^{\hat *})$ to $(R\#H)\# H^{\hat *}.$
 It is straightforward to verify  that $\Phi \Psi = id  $ and $\Psi \Phi =
id$.

To see that $\Phi$ is an algebra morphism, we only need to show
that $\Phi ' = (id \otimes \lambda )\Phi $ is an algebra morphism.
Set  $\xi = (id \otimes \rho  )(S^{-1} \otimes \eta _H) C_{H^{\hat
*}, R} \phi $ , which is a morphism from $R$ to $R \otimes (H \bar
\otimes H^{\hat *} )$. We have that $\xi $ is an algebra morphism
and  $\Phi ' = (id \otimes m) (\xi \otimes \lambda ).$ Using Lemma
\ref {1.5}, we can show that $ (id \otimes m)(C_{H \bar \otimes
H^{\hat *}, R}\otimes id ) (\lambda \otimes \xi) = (id \otimes m)
( \xi \otimes\lambda ) (\alpha \otimes id \otimes id  ) (id
\otimes C_{H, R} \otimes id )(id \otimes id \otimes C_{H^{\hat *},
R} )(\Delta \otimes id \otimes id ).$ Applying this, we see that
$\Phi'$ is an algebra morphism.\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
%$\Box$




 \section {Drinfeld Double}

In this section, we  construct the
 Drinfeld double $D(H)$ for  a finite Hopf algebra $H$ with  $C_{H,H} = C_{H,H} ^{-1}$
in the braided tensor category ${\cal C}$. We show that
$(D(H),[b])$ is quasi-triangular.



\begin {Theorem} \label {2.1}
Let $H$
and $A$ be two bialgebras  in ${\cal C}$. Assume that  $\tau$  is  an invertible
skew pairing on $H \otimes A$ and  symmetric  with respect to the braiding. If we
 define
$\alpha = (\tau \otimes id \otimes \bar \tau ) ( id \otimes id
\otimes  C_{H, A} \otimes id )( id \otimes   C_{H, A} \otimes
\Delta  )  (\Delta \otimes \Delta )$ and $\beta = (\tau \otimes id
\otimes \bar \tau ) ( id \otimes  C_{H, A} \otimes id \otimes id
)(\Delta \otimes   C_{H, A} \otimes   id )  (\Delta \otimes \Delta
),$
 then  the double cross product $A _\alpha \bowtie _\beta H$, defined in \cite [P36]{ZC}),
  of $A$ and $H$  is an algebra and a
coalgebra.
 If $A$ and $H$ are Hopf algebras, then $A _\alpha \bowtie _\beta H$ has an antipode.
 Furthermore, $A _\alpha \bowtie _\beta H$ is a bialgebra if and only
 if
$ C_{A, H}C_{H, A} = id .$
\end {Theorem}
{\bf Proof.} We can check that $(A, \alpha )$  is an $H$-module
coalgebra and $(H, \beta )$ is an $A$-module coalgebra step by
step. We can also  check that (M1)--(M3) in \cite [P36--37] {ZC}
hold  step by step. Consequently,  it follows from \cite
[Corollary 1.8 , Theorem 1.5]{ZC} that  $A _\alpha \bowtie _\beta
H$ is an algebra and a coalgebra. Observing  the proof of \cite
[Theorem 1.5] {ZC}, we know that the condition (M4) is not needed
in the proof.  Consequently, $A _\alpha \bowtie _\beta H$ has an
antipode.

From \cite [Proposition 3.6] {BD}, we obtain our last assertion.\
\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
% $\Box$



In this case, $A _\alpha \bowtie _\beta H$ can be
written as $A  \bowtie _\tau H$ and  called a double cross
product.

\begin {Theorem} \label {2.2}
Let $H$ be a finite  Hopf algebra with $C_{H,H}= C_{H,H}^{-1}$.
Set $A= (H^*)^{op}$ and $ \tau = d_H C_{H,A}$. Then $(D(H), [b])$
is a quasi-triangular
 Hopf algebra in ${\cal C}$ with  $[b]= \eta _A \otimes b \otimes \eta
_H$ and $D(H) =A \bowtie _{\tau} H,$ called the Drinfeld double of
$H$.
\end {Theorem}
{\bf Proof.} Using \cite [Proposition 2.4] {Ma95a} or the
definition of the evaluation and coevaluation on tensor product,
we can obtain that
 $\tau $ is a skew pairing on $H \otimes A$ and
 $[b]$ satisfies (QT1) and  (QT2). For  (QT3), see that
$(m \otimes m )(id \otimes C_{D(H),D(H)} \otimes id  )( \Delta
^{cop}\otimes [b] ) =(id \otimes id  \otimes m \otimes id ) (id
\otimes C_{A,H} \otimes id \otimes id )(id \otimes id  \otimes m
\times id  \otimes id )(id \otimes id  \otimes id \otimes m \times
id \otimes id )(id \otimes id \otimes id \otimes id  \otimes m
\times id \otimes id ) (C_{A,A} \otimes id \otimes id \otimes
C_{H,H} \times C_{H,A} )(id \otimes id \otimes id \otimes C_{H,H}
\otimes C_{H,H}\otimes  id ) (id \otimes id
 \otimes C_{H,H} \otimes C_{H,H}\otimes b )
(id \otimes id \otimes id
 \otimes C_{H,H} \otimes S^{-1} )
 (\Delta  \otimes \Delta
 \otimes C_{H,H}   )
(id   \otimes \Delta
 \otimes id )
(id   \otimes \Delta ) = (id \otimes C_{A,H} \otimes id  ) (id
\otimes m  \otimes id \otimes id  )
 (C_{A,A} \otimes C_{H,A}  \otimes id   )
  (id \otimes id \otimes  m  \otimes id  \otimes id )
 (id \otimes id   \otimes C_{H,H} \otimes C_{H,A}  )
 (id \otimes id   \otimes id \otimes C_{H,H} \otimes id  )
 (\Delta \otimes \Delta   \otimes b )$
 and $ (m \otimes m )(id \otimes C_{D(H),D(H)} \otimes id  )([b]
\otimes \Delta )= (id \otimes m  \otimes id \otimes id ) (C_{H,A}
\otimes C_{A,H}  \otimes id  ) (id \otimes id \otimes  m  \otimes
id \otimes id ) (id \otimes id \otimes m  \otimes id \otimes id
\otimes id  )(id \otimes id \otimes m  \otimes id \otimes id
\otimes id  \otimes id)(id \otimes C_{A,A} \otimes id \otimes  S
\otimes id \otimes id \otimes id  ) (id \otimes id \otimes C_{A,A}
\otimes id \otimes id \otimes id \otimes id  ) (id \otimes C_{A,A}
\otimes \Delta \otimes id \otimes id \otimes id  )(b \otimes
\Delta \otimes id \otimes id \otimes id  )(\Delta \otimes \Delta
)= (id \otimes C_{A,H} \otimes id  ) (id \otimes m  \otimes id
\otimes id )
 (C_{A,A} \otimes C_{H,A}  \otimes id   )
  (id \otimes id \otimes  m  \otimes id \otimes id )
 (id \otimes id   \otimes C_{H,H} \otimes C_{H,A}  )
 (id \otimes id   \otimes id \otimes C_{H,H} \otimes id  )
 (\Delta \otimes \Delta   \otimes b )$. Thus
 (QT3) holds.\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
 %$\Box$

 In fact, there exists a very closed relation between the
Drinfeld double $D(H)$ defined in Theorem \ref {2.2} and the
Drinfeld double  ${\cal D} (H)$ defined in \cite {Ch}. Since
$C_{H^*, H} : \ \ D(H)= H^{*op} \bowtie _{\tau} H \longrightarrow
{\cal D} (H^{op})= H^{op} \bowtie _d H^* $ is an anti-algebra
isomorphism  and a coalgebra isomorphism, we have  $D(H) \cong
{\cal D}(H^{op})^{op} \ \  \hbox{ as Hopf algebras in } {\cal C}.
$

\section { Example }

In this section, using preceding conclusions, we give some
examples for the duality theorem and Drinfeld double in a braided
tensor category ${\cal C}.$


\begin {Proposition} \label {3.1}(See \cite [Definition 2.8 (R2)] {LZ})
  A $\chi$-Hopf algebra $H$ is a Hopf algebra living in the braided tensor
category $({}^{kG}{\cal M}, C^r)$, where $r(g,h) = v ^ {\chi (g,
h)}$ for any $g, h \in G$; $\chi$ is a map from $G \times G$ to
${\bf Z}$ with $\chi (a,b) = \chi (b,a)$ and $\chi (a +b, c) =
\chi(a, c)+ \chi (b, c)$ for any $a, b , c \in G$.
\end {Proposition}


It follows from the preceding proposition and \cite [Ex. 2.12,
Section 3] {LZ} that
 Lusztig's algebra  ${\ }'f$, the twisted Ringel-Hall algebra and Ringel's
 composition algebra  are all
Hopf algebras in braided tensor categories.



\begin {Example} \label {3.2}  The bilinear map $\tau$, defined in
\cite [Pro. 1.2.3]{L}, of
 Lusztig's algebra ${\ }'f$
is  symmetric with respect to the braiding.
\end {Example}
{\bf Proof.} For any homogeneous elements  $x, y, z \in { }'f$, we
have that $(\tau \otimes id)(id \otimes C_{{\ }'f,{\ } 'f} ) (x
\otimes y \otimes z ) = \tau (x,z)v^{\mid y \mid \cdot \mid z\mid
} \delta _{\mid x\mid , \mid z \mid} = \tau (x,z)v^{\mid y \mid
\cdot \mid x\mid } \delta _{\mid x\mid , \mid z \mid} = (id\otimes
\tau )( C_{{\ }'f, {\ }'f} \otimes id ) (x \otimes y \otimes z ).$
\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
%$\Box$

\begin {Example} \label {3.3} (see \cite [Example 9.4.9]{Ma95b}) The evaluation of the braided group analogue
 $\underline H$ of an ordinary coquasi-triangular cocommutative Hopf algebra $(H,r)$ is
 symmetric  with respect to braiding $C^r$. In particular, the above conclusion holds for  $H =kG$.
\end {Example}
{\bf Proof. } It is straightforward.\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
%$\Box$


\begin {Example} \label {3.4} Let $H$ denote  Lusztig's algebra ${ }'f$.
If  $A = {}'f ^{op}$,  then the bilinear map $\tau $ as in Example
\ref {3.2} is a skew pairing on $H \otimes A$ and symmetric with
respect to the braiding. Thus, by Theorem \ref {2.1}, $A  \bowtie
_\tau H$ is an algebra and a coalgebra with an antipode in $({}^{k
{\bf Z}[I]}{\cal M}, C^r) $, but it  is never a bialgebra in
$({}^{k {\bf Z}[I]}{\cal M},C^r) $ since $ C_{A, H}C_{H, A} \neq
id .$
\end {Example}

It has been known that the category of comodules of  every
ordinary coquasi-triangular Hopf algebra is a braided tensor
category. For example, let $H={\bf C Z}_n$ and $r(a,b) = (e
^{\frac {2 \pi i}{n}})^{ab}$  for any $a, b \in {\bf Z}_n$, where
${\bf C}$ is the complex field. It is clear that $({\bf CZ}_n, r
)$  is a coquasi-triangular Hopf algebra. Thus, $ ({} ^{{\bf CZ}
_n} {\cal M} , C^r)$ is a braided tensor category, usually written
as ${\cal C}_n.$  Every algebra or Hopf algebra living in ${\cal
C}_n $ is called an anyonic algebra or anyonic Hopf algebra (see
\cite [Example 9.2.4] {Ma95b}).
 Every algebra or Hopf algebra living   in ${\cal C}_2 $
is called a superalgebra or super-Hopf algebra. In particular,
${\cal C}_n$ is a strictly braided tensor category when $n> 2$.



It follows from Theorem \ref {2.2} that

\begin {Corollary} \label {3.5} (Duality Theorem) Let $H$ be a
finite dimensional Hopf algebra with $C_{H,H} = C_{H,H}^{-1}$ in
$({\cal C},C)$. Then

   $$
(R \# H)\# H^{\hat *}   \cong M_n(R)  \hbox { \ \ \  as algebras
in } ({\cal C}, C) $$ in the following three cases:

(i)  $({\cal C}, C)$ is the braided tensor category $({}_B{\cal
M}, C^R)$ determined by the quasi-triangular structure $R$;

(ii) $({\cal C}, C)$  is  the braided tensor category $({}^B{\cal
M}, C^r)$ determined by the coquasi-triangular structure $r$

(iii)  $({\cal C}, C)$ is the braided tensor category ${}^B_B
{\cal YD}$ or  ${}_B {\cal YD} ^B$ of Yetter-Drinfeld modules.
\end {Corollary}



\begin {Example} \label {3.6}
(see \cite  {MR94} )   Let $A$  denote  the anyonic line algebra,
  i.e., $A= {\bf C}\{x \}/\langle x^n \rangle$, where  ${\bf C} \{x\}$
 is a free algebra
  over the complex
 field {\bf C}  and $\langle x^n\rangle$  is an ideal generalized by $x^n$ of
  ${\bf C} \{x\}$.
Set ${\bf C} [\xi]={\bf C} \{x\} / \langle x^n\rangle  $  with
$\xi ^n=0$.
 Its comultiplication, counit,  and antipode are
 $$ \Delta (\xi) = \xi \otimes 1 + 1 \otimes \xi , \hbox { \ \ \  \ \ \ }
 \epsilon (\xi ) =0 , \hbox { \ \ \ and \ \ \ } S(\xi ) =- \xi , $$
 respectively.
 It is straightforward to check that  $H$  is an anyonic Hopf algebra.
 Let $H$ denote the braided group analogue  $\underline {{\bf CZ} _n }$ of   ${\bf CZ} _n $.
Since  $  {\bf CZ} _n $ is commutative, we have  $C_{ {\bf CZ} _n
, {\bf CZ} _n }=C_{ {\bf CZ} _n ,   {\bf CZ} _n }^{-1}$. Let $H$
act on $A$ trivially. By Corollary \ref {3.5}, we have
 $$  ({\bf C} [\xi] \# \underline {{\bf CZ} _n })\#(\underline {{\bf CZ} _n })^{\hat *}
 \cong  M_n({\bf C}[\xi]) \hbox { \ \ \  as algebras in } ({} ^{{\bf CZ} _n}
{\cal M} , C^r).$$


By the way, we also have that the Drinfeld double of  $\underline
{{\bf CZ} _n }$ is  quasi-triangular in
 $ ({} ^{{\bf CZ} _n}
{\cal M} , C^r)$ with the quasi-triangular structure  $[b]$.

 \end {Example}





\vskip 1cm

{\bf Acknowledgement }  The author thanks the referee and the
editor for valuable suggestions.


\begin{thebibliography}{150}


\bibitem {BD} Y. Bespalov and B.Drabant, Cross product bialgebras, I,
 J. algebra, {\bf 219} (1999), 466--505.




\bibitem {BM85}  R. J. Blattner and   S. Montgomery, A duality theorem for Hopf
module algebras,  J. algebra, {\bf 95} (1985), 153--172.



\bibitem {Ch}  H.X.Chen, Quantum double in monoidal categories,
Comm.Algebra,
 {\bf 28} (2000)5, 2303--2328.



\bibitem {Dr86} V. G. Drinfeld,  Quantum groups, in ``Proceedings International Congress of
Mathematicians, August 3-11, 1986, Berkeley, CA" pp. 798--820,
Amer. Math. Soc., Providence, RI, 1987.




\bibitem {L} G.Lusztig, Introduction to    Quantum  Groups,
Progress on Math., Vol.110, Birkhauser, Berlin, 1993.

 \bibitem {LZ} L.Li and P.Zhang, Twisted Hopf algebras, Ringel-Hall algebras and Green's ca
tegories, J.Algebra, {\bf 231}(2000), P713--743.



\bibitem {Ma90a} {  S. Majid,
Physics for algebraists: Non-commutative and non-cocommutative
Hopf algebras by a bicrossproduct construction, J.Algebra }  {\bf
130}  (1990), 17--64.


  \bibitem {Ma95a} S. Majid, Algebras and Hopf algebras
  in braided categories,
Lecture notes in pure and applied mathematics advances in Hopf
algebras, Vol. 158, edited by J. Bergen and S. Montgomery, Marcel
Dekker, New York, 1994, 55--105.

  \bibitem {Ma95b} S. Majid, Foundations of Quantum Group Theory,  Cambridge University Press, Cambridge, 1995.
\bibitem {MR94} S. Majid and M.J. Rodriguez Piaza, Random walk and the heat
equation on superspace and anyspace, J. Math. Phys., {\bf
35}(1994) 7, 3753--3760.



\bibitem {Mo93}  S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS
  Number 82, AMS, Providence, RI, 1993.



 \bibitem{Ra93a}   D. E. Radford,
 Minimal quasi-triangular Hopf algebras,
 J. algebra,
 {\bf 157} (1993), 281--315.


\bibitem {St81} S. Stratila, Modular Theory in Operator Algebras,
Editura Academiei and Abacus Press, Bucuresti, 1981.


\bibitem {Sw}   M.E.Sweedler, Hopf Algebras, Benjamin, New York, 1969.

 \bibitem {Ta}   M. Takeuchi, Finite Hopf algebras in braided tensor categories ,
J.Pure and Applied Algebra, {\bf 138}(1999), 59-82.


\bibitem {ZC} Shouchuan Zhang, Hui-Xiang Chen, The double bicrossproducts
in braided tensor categories,
    Communications in Algebra,   {\bf 29}(2001)1, P31--66.




\end {thebibliography}
\newpage
\section {Appendix}
We denote the multiplication, comutiplication, antipode , braiding
and   inverse braiding  by \[
\begin{tangle}
\cu \step[1],\step[2] \cd \step[2], \step[2]\S \step[2],\step[2]\x
\step[2] \hbox { and } \step[2] \xx \step[2], \hbox {respectively
. In particular, }\\
 \hbox {we denote }\step [2] \object {C_{U,V}  }\step
[3]  \hbox { by } \step [2] \X \ ,\step [3] \hbox {for any } \step
[3] \object {U, V = H } \step [3] \hbox { or } \step [1] \object
{H^*}.\end{tangle}
\]





The proof of Lemma2.4\\
Proof.
\[
\begin{tangle}
\object{H\# {H^{\hat{*}}}}\step[6]\object{H\#
{H^{\hat{*}}}}\\
\step\Cu\\
\step\Step\O \lambda\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[2]\object{H^{\hat{*}}}\Step\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\step\cd\step\id\Step\id\Step\coev\\
\id\step\id\Step\X\Step\id\step\cd\step\id\\
\id\step\tu \rightharpoonup\step\cu\step\id\Step\id\step\id\\
\cu\step[3]\XX\Step\id\step\id\\
\step\id\step[3]\ne2\Step\ev\step\id\\
\step\cu\step[7]\id\\
\Step\object{H}\step[8]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[4]\object{H^{\hat{*}}}\step[4]\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\cd\Step\cd\Step\id\Step\coev\\
\id\Step\id\Step\XX\Step\id\Step\id\step\cd\step\id\\
\id\Step\XX\Step\XX\Step\id\step\id\Step\id\step\id\\
\cu\Step\ev\Step\cu\step\id\Step\id\step\id\\
\step\id\step[8]\XX\Step\id\step\id\\
\step\id\step[7]\ne3\Step\ev\step\id\\
\step\id\step[4]\ne3\step[8]\id\\
\step\cu\step[11]\id\\
\Step\object{H}\step[12]\object{H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\object{H}\step[4]\object{H^{\hat{*}}}\step[4]\object{H}\step[2]\object{H^{\hat{*}}}\\
\id\Step\cd\Step\cd\step\id\step[3]\Coev\\
\id\Step\id\Step\XX\Step\id\step\id\Step\cd\step[2]\id\\
\id\Step\XX\Step\XX\step\XX\step\cd\step\id\\
\cu\Step\ev\Step\X\Step\X\Step\id\step\id\\
\step\id\step[6]\ne3\step\ev\step\ev\step\id\\
\step\id\step[3]\ne2\step[10]\id\\
\step\cu\step[12]\id\\
\Step\object{H}\step[13]\object{H^{\hat{*}}}\\
\end{tangle} \ \ \ \ \ \ \ \
and \]
\[
\begin{tangle}
\object{H\# {H^{\hat{*}}}}\step[6]\object{H\#
{H^{\hat{*}}}}\\
\O \lambda\step[4]\O \lambda\\
\Cu\\
\step[2]\object{H\bar{\otimes}{H^{\hat{*}}}}
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\Step\object{H^{\hat{*}}}\Step\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\id\Step\id\Step\id\Step\coev\\
\id\Step\id\Step\id\Step\id\step\cd\step\id\\
\id\Step\id\Step\id\Step\X\Step\id\step\id\\
\id\Step\id\Step\cu\step\ev\step\id\\
\id\Step\id\Step\cd\step[4]\id\\
\id\Step\XX\Step\id\step[4]\id\\
\cu\Step\ev\step[4]\id\\
\step\object{H}\step[8]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\Step\object{H^{\hat{*}}}\Step\object{H}\step[4]\object{H^{\hat{*}}}\\
\id\Step\id\Step\id\step[3]\id\Step\coev\\
\id\Step\id\Step\id\step[3]\id\step\cd\step\id\\
\id\Step\id\Step\id\step[3]\X\Step\id\step\id\\
\id\Step\id\step\cd\step\cd\ev\step\id\\
\id\Step\id\step\id\Step\X\Step\id\step[3]\id\\
\id\Step\id\step\cu\step\cu\step[3]\id\\
\id\Step\XX\Step\dd\step[4]\id\\
\cu\Step\ev\step[5]\id\\
\step\object{H}\step[10]\object{H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\Step\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\cd\Step\id\step[3]\id\Step\coev\\
\id\Step\XX\Step\id\step[3]\id\step\cd\step\id\\
\id\Step\id\Step\id\Step\id\step[3]\X\Step\id\step\id\\
\id\Step\id\Step\id\step\cd\step\cd\ev\step\id\\
\id\Step\id\Step\id\step\id\Step\X\Step\id\step[3]\id\\
\id\Step\id\Step\id\step\cu\step\id\Step\id\step[3]\id\\
\id\Step\id\Step\XX\Step\id\Step\id\step[3]\id\\
\id\Step\XX\Step\ev\Step\id\step[3]\id\\
\id\Step\id\Step\nw2\step[4]\ne2\step[3]\id\\
\cu\step[4]\ev\step[5]\id\\
\step\object{H}\step[12]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[4]\object{H^{\hat{*}}}\step[3]\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\cd\Step\cd\step\id\step[3]\Coev\\
\id\Step\id\Step\XX\Step\id\step\id\Step\cd\step[2]\id\\
\id\Step\XX\Step\XX\step\XX\step\cd\step\id\\
\cu\Step\ev\Step\X\Step\X\Step\id\step\id\\
\step\nw2\step[5]\ne3\step\ev\step\ev\step\id\\
\step[3]\cu\step[10]\id\\
\step[4]\object{H}\step[11]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H\# {H^{\hat{*}}}}\step[6]\object{H\#
{H^{\hat{*}}}}\\
\step\Cu\\
\step\Step\O \lambda\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle} .
\]
Thus $\lambda $ is an algebraic morphism.

Similarly, we can show that $\rho $ is an anti-algebraic
morphism.\ \ \ \ \begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\

The proof of Theorem 2.5.


\[
\begin{tangle}
\object{H\# {H^{\hat{*}}}}\step[6]\object{{H^{\hat{*}}}\#
H}\\
\O \lambda\step[4]\O \rho\\
\Cu\\
\step[2]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{H}\Step\object{H^{\hat{*}}}\step[3]\object{H^{\hat{*}}}\Step\object{H}\\
\step\id\Step\XX\Step\id\\
\step\XX\Step\XX\\
\cd\step\XX\Step\d\\
\O S\Step\id\step\id\Step\d\step\cd\\
\XX\step\id\step[3]\id\step\XX\\
\id\Step\X\step[3]\X\Step\id\\
\id\Step\id\step\d\step\dd\step\id\Step\id\\
\id\Step\id\Step\X\Step\id\Step\id\\
\id\Step\XX\step\XX\Step\id\\
\d\step\tu \rightharpoonup\step\tu \leftharpoonup\step\dd\\
\step\tu \rho\step[3]\tu \lambda\\
\Step\d\step[4]\id\\
\step[3]\Cu\\
\step[5]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\Step\Step ......(1)
\]

We show (1) by following five steps. It is easy to check the
following (i) and (ii).

(i)
\[
\begin{tangle}
\object{H\#
{\eta_{H^{\hat{*}}}}}\step[7]\object{{\eta_{H^{\hat{*}}}}\#
H}\\
\Step\O \lambda\step[4]\O \rho\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\object{H\#
{\eta_{H^{\hat{*}}}}}\step[7]\object{{\eta_{H^{\hat{*}}}}\#
H}\\
\step\d\Step\dd\\
\Step\XX\\
\step\dd\Step\d\\
\step\O \rho\step[4]\O \lambda\\
\step\Cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\]

(ii)
\[
\begin{tangle}
\object{\eta_H\# {H^{\hat{*}}}}\step[7]\object{H^{\hat{*}}\#
\eta_H}\\
\Step\O \lambda\step[4]\O \rho\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\object{\eta_H\# {H^{\hat{*}}}}\step[7]\object{H^{\hat{*}}\#
\eta_H}\\
\step\d\Step\dd\\
\Step\XX\\
\step\dd\Step\d\\
\step\O \rho\step[4]\O \lambda\\
\step\Cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\]


(iii)
\[
\begin{tangle}
\object{{H^{\hat{*}}}\#\eta_H}\step[7]\object{{H^{\hat{*}}}\#\eta_H
}\\
\Step\O \rho\step[4]\O \lambda\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{\eta_H}\Step\object{H}\step[3]\object{\eta_{H^{\hat{*}}}}\\
\id\Step\XX\Step\id\\
\XX\Step\XX\\
\id\Step\XX\Step\id\\
\id\Step\id\step\cd\step\id\\
\id\Step\X\step\dd\step\id\\
\tu \leftharpoonup\step\id\step\tu \rho\\
\step\tu \lambda\step\dd\\
\Step\cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\]


\[
\hbox {In fact,the right side} \step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{\eta_H}\step[2]\object{H}\step[3]\object{\eta_{H^{\hat{*}}}}\\
\step\id\Step\XX\Step\id\step[3]\coev\\
\step\XX\Step\XX\Step\cd\step\id\\
\cd\step\XX\Step\XX\Step\id\step\id\\
\id\Step\id\step\id\step\cd\step\d\step\XX\step\id\\
\id\Step\id\step\X\Step\ev\step\cu\step\id\\
\id\Step\X\step\nw3\step[4]\cd\step\id\\
\XX\step\nw2\step[3]\XX\Step\id\step\id\\
\ev\step[3]\cu\Step\ev\step\id\\
\step[6]\object{H}\step[6]\object{H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\step\object{H^{\hat{*}}}\step[4]\object{H}\\
\cd\step\cd\Step\Coev\\
\id\Step\X\Step\id\step\cd\Step\id\\
\ev\step\id\Step\X\Step\id\Step\id\\
\step[3]\ev\step\cu\Step\id\\
\step[7]\object{H}\step[3]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step \hbox{the left side.}
\]
Thus (iii) holds.


(iv)
\[
\begin{tangle}
\object{H\# {\eta_{H^{\hat{*}}}}}\step[7]\object{{H^{\hat{*}}}\#
{\eta_H}}\\
\Step\O \lambda\step[4]\O \rho\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H^{\hat{*}}}\step[2]\object{H}\\
\step\id\Step\XX\Step\id\\
\step\XX\Step\XX\\
\cd\step\XX\Step\id\\
\O S\Step\id\step\id\Step\id\Step\id\\
\XX\step\id\Step\id\Step\id\\
\id\Step\X\Step\id\Step\id\\
\tu \rho\step\XX\step\dd\\
\step\d\step\tu \leftharpoonup\step\id\\
\Step\d\step\tu \lambda\\
\step[3]\cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\]




\[
\hbox{The right side }\stackrel{by\ (iii)}{=}
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H^{\hat{*}}}\step[2]\object{\eta_{H}}\\
\step\id\Step\XX\Step\id\\
\step\XX\Step\XX\\
\cd\step\XX\Step\id\\
\O S\Step\id\step\id\Step\id\Step\id\\
\XX\step\id\Step\id\Step\id\\
\id\Step\X\Step\id\Step\id\\
\id\Step\id\step\XX\Step\id\\
\id\Step\id\step\tu \leftharpoonup\Step\id\\
\id\Step\XX\Step\dd\\
\XX\Step\XX\\
\id\Step\XX\Step\d\\
\id\Step\id\step\cd\Step\id\\
\id\Step\X\Step\tu \rho\\
\tu \leftharpoonup\step\id\Step\dd\\
\step\tu \lambda\step\dd\\
\Step\cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\Step\object{H^{\hat{*}}}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H^{\hat{*}}}\step[2]\object{\eta_{H}}\\
\step[3]\id\Step\XX\Step\id\Step\Coev\\
\step[3]\XX\Step\XX\step\cd\Step\id\\
\Step\cd\step\XX\Step\id\step\id\Step\id\Step\id\\
\Step\O S\step\dd\step\XX\Step\id\step\id\Step\id\Step\id\\
\step\dd\step\XX\Step\XX\step\id\Step\id\Step\id\\
\dd\step\dd\step\cd\step\id\Step\X\Step\id\Step\id\\
\XX\Step\id\Step\id\step\XX\step\XX\Step\id\\
\d\step\cu\Step\hev\Step\id\step\cu\Step\id\\
\step\tu \leftharpoonup\step[6]\id\step\cd\Step\id\\
\Step\nw3\step[6]\X\Step\id\Step\id\\
\step[5]\Cu\step\ev\Step\id\\
\step[7]\object{H}\step[7]\object{H^{\hat{*}}}\\
\end{tangle}\]
\[
\step=\step
\begin{tangle}
\step\object{H}\step[3]\object{H^{\hat{*}}}\\
\cd\step\cd\step[4]\Coev\\
\id\Step\id\step\O S\step\cd\Step\cd\Step\id\\
\id\Step\id\step\id\step\d\step\id\Step\id\Step\id\Step\id\\
\id\Step\id\step\cu\step\ev\step\dd\Step\id\\
\id\Step\XX\step[4]\dd\step[3]\id\\
\XX\Step\Cu\step[4]\id\\
\ev\step[4]\id\step[6]\id\\
\step[6]\object{H}\step[6]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\id\step[3]\Coev\\
\id\Step\id\Step\cd\Step\id\\
\d\step\ev\step\dd\Step\id\\
\step\Cu\step[3]\id\\
\step[3]\object{H}\step[5]\object{H^{\hat{*}}}\\
\end{tangle}
\]


and
\[
\hbox{the left side}\step=\step
\begin{tangle}
\object{H}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H^{\hat{*}}}\step[2]\object{\eta_H}\\
\id\Step\id\step[3]\id\Step\id\Step\Coev\\
\id\Step\id\step[3]\id\Step\id\step\cd\Step\id\\
\d\step\d\step[2]\id\Step\X\Step\id\Step\id\\
\step\d\step\d\step\ev\step\XX\Step\id\\
\Step\d\step\d\step[3]\cu\Step\id\\
\step[3]\d\step\d\Step\cd\Step\id\\
\step[4]\id\Step\XX\Step\id\Step\id\\
\step[4]\cu\Step\ev\Step\id\\
\step[5]\object{H}\step[7]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\Step\id\step[3]\Coev\\
\id\Step\id\Step\cd\Step\id\\
\d\step\ev\step\dd\Step\id\\
\step\Cu\step[3]\id\\
\step[3]\object{H}\step[5]\object{H^{\hat{*}}}\\
\end{tangle}\ \ .
\]

Thus (iv) holds.

(v)
\[
\begin{tangle}
\object{\eta_H\#
{H^{\hat{*}}}}\step[7]\object{\eta_{H^{\hat{*}}}\#
H}\\
\Step\O \lambda\step[4]\O \rho\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{\eta_H}\Step\object{H^{\hat{*}}}\step[3]\object{\eta_{H^{\hat{*}}}}\step[2]\object{H}\\
\id\Step\XX\Step\id\\
\XX\Step\XX\\
\id\Step\XX\step\cd\\
\id\Step\id\Step\id\step\XX\\
\id\Step\id\Step\X\Step\id\\
\id\Step\XX\step\tu \lambda\\
\d\step\tu \rightharpoonup\step\dd\\
\step\tu \rho\step\dd\\
\Step\cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}\ \ \ .
\]




\[
\hbox{In fact, the right side}\step=\step
\begin{tangle}
\object{\eta_H}\Step\object{H^{\hat{*}}}\step[3]\object{\eta_{H^{\hat{*}}}}\step[2]\object{H}\\
\id\Step\XX\Step\id\step[3]\Coev\\
\XX\Step\XX\Step\cd\Step\id\\
\id\Step\XX\step\cd\step\id\Step\id\Step\id\\
\id\step\cd\step\id\step\XX\step\id\Step\id\Step\id\\
\id\step\id\Step\id\step\X\Step\X\Step\id\Step\id\\
\id\step\d\step\X\step\cu\step\ev\Step\id\\
\d\step\d\hev\step\cd\step[5]\id\\
\step\id\Step\XX\Step\id\step[5]\id\\
\step\ev\Step\XX\step[5]\id\\
\step[5]\cu\step[5]\id\\
\step[6]\object{H}\step[6]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{H^{\hat{*}}}\step[3]\object{H}\\
\cd\step\cd\Step\Coev\\
\XX\step\XX\step\cd\Step\id\\
\id\Step\X\Step\id\step\id\Step\id\Step\id\\
\ev\step\XX\step\id\Step\id\Step\id\\
\step[3]\id\Step\X\Step\id\Step\id\\
\step[3]\XX\step\ev\step[2]\id\\
\step[3]\cu\step[5]\id\\
\step[4]\object{H}\step[6]\object{H^{\hat{*}}}\\
\end{tangle}
\]


and
\[
\hbox{the left side}\step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{H}\\
\id\Step\id\Step\coev\\
\id\Step\XX\Step\id\\
\id\Step\cu\Step\id\\
\id\Step\cd\Step\id\\
\XX\Step\id\Step\id\\
\id\Step\ev\Step\id\\
\object{H}\step[6]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{H}\\
\id\Step\id\Step\Coev\\
\id\Step\XX\step[3]\id\\
\id\Step\id\Step\d\step[2]\id\\
\id\step\cd\step\cd\step\id\\
\id\step\id\Step\X\Step\id\step\id\\
\id\step\cu\step\cu\step\id\\
\XX\Step\dd\step[2]\id\\
\id\Step\ev\step[3]\id\\
\object{H}\step[7]\object{H^{\hat{*}}}\\
\end{tangle}\]
\[
\step=\step
\begin{tangle}
\step\object{H^{\hat{*}}}\step[3]\object{H}\\
\cd\step[2]\id\Step\Coev\\
\id\Step\id\step[2]\XX\step[3]\id\\
\id\step[2]\id\Step\id\Step\d\Step\id\\
\id\step[2]\id\step\cd\step\cd\step\id\\
\id\Step\id\step\id\Step\X\Step\id\step\id\\
\id\Step\id\step\cu\step\id\Step\id\step\id\\
\id\Step\XX\Step\id\Step\id\step\id\\
\XX\Step\XX\Step\id\step\id\\
\id\Step\ev\Step\ev\step\id\\
\object{H}\step[9]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{H^{\hat{*}}}\step[3]\object{H}\\
\cd\step\cd\Step\Coev\\
\id\Step\id\step\id\Step\id\step\cd\Step\id\\
\id\Step\id\step\id\Step\X\Step\id\Step\id\\
\id\Step\id\step\XX\step\id\Step\id\Step\id\\
\id\Step\id\step\cu\step\id\Step\id\Step\id\\
\id\Step\XX\Step\id\step\dd\Step\id\\
\XX\Step\ev\dd\step[3]\id\\
\id\Step\d\step[3]\id\step[4]\id\\
\id\step[3]\Ev\step[6]\id\\
\object{H}\step[10]\object{H^{\hat{*}}}\\
\end{tangle}
\step=\step\hbox{the right side.}
\]
Thus (v) holds.

Now we show that the relation (1) holds.
\[
\hbox{the left side of (1)}\stackrel{\hbox{by
Lemma1.4}}{\step=\step[5]}
\begin{tangle}
\object{H\# {\eta_{H^{\hat{*}}}}}\step[8]\object{{\eta_H}\#
H^{\hat{*}}}\step[9]\object{H^{\hat{*}}\#\eta_H}\step[6]\object{\eta_{H^{\hat{*}}}\# H}\\
\O \lambda\step[6]\Step\O \lambda\step[11]\XX \\
\nw3\step[7]\nw3\step[10]\O \rho\step[2]\O \rho\\
\step[3]\nw3\step[7]\nw3\step[7]\cu\\
\step[6]\nw3\step[7]\d\step[4]\dd\\
\step[9]\nw3\step[5]\Cu\\
\step[12]\d\step[4]\id\\
\step[13]\Cu\\
\step[15]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle} \]
\[
\step\stackrel{\hbox{by (v)}}{=}\step
\begin{tangle}
\object{H\#
{\eta_{H^{\hat{*}}}}}\step[5]\object{\eta_H}\step[3]\object{H^{\hat{*}}}\step[5]
\object{H^{\hat{*}}\#\eta_H}\step[5]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H}\\
\O \lambda\step[5]\nw3\step[2]\nw3\step[4]\d\Step\dd\Step\dd\\
\d\step[7]\nw2\Step\d\Step\XX\Step\dd\\
\step\d\step[8]\id\Step\XX\Step\XX\\
\Step\d\step[7]\XX\Step\XX\Step\O \rho\\
\step[3]\d\step[6]\id\Step\XX\step\cd\step\id\\
\step[4]\d\step[5]\id\Step\id\Step\id\step\XX\step\id\\
\step[5]\d\step[4]\id\Step\id\Step\X\Step\id\step\id\\
\step[6]\d\step[3]\id\Step\XX\step\tu \lambda\step\id\\
\step[7]\d\Step\d\step\tu \rightharpoonup\Step\cu\\
\step[8]\d\Step\tu \rho\step[3]\dd\\
\step[9]\d\Step\Cu\\
\step[10]\Cu\\
\step[12]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\]

\[
\step\stackrel{\hbox{by (i)(ii)}}{=}\step
\begin{tangle}
\object{H}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{\eta_H}\step[3]\object{H^{\hat{*}}}
\step[3]\object{H^{\hat{*}}}\step[3]\object{\eta_H}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H}\\
\d\Step\d\Step\d\Step\d\Step\d\Step\d\Step\id\Step\dd\\
\step\d\Step\d\Step\d\Step\d\Step\d\Step\XX\Step\id\\
\Step\d\Step\d\Step\d\Step\d\Step\XX\Step\XX\\
\step[3]\d\Step\d\Step\d\Step\XX\Step\XX\Step\id\\
\step[4]\d\Step\d\Step\XX\Step\XX\Step\id\Step\id\\
\step[5]\d\Step\XX\Step\XX\step\cd\step\id\Step\id\\
\step[6]\XX\Step\id\Step\id\Step\id\step\XX\step\id\Step\id\\
\step[6]\id\Step\id\Step\id\Step\id\Step\X\Step\X\Step\id\\
\step[6]\id\Step\id\Step\id\Step\XX\step\XX\step\XX\\
\step[6]\id\Step\id\Step\d\step\tu
\rightharpoonup\step\id\Step\X\Step\id\\
\step[6]\id\Step\d\Step\XX\Step\tu \rho\step\tu \lambda\\
\step[6]\d\Step\XX\Step\id\step[3]\id\Step\dd\\
\step[7]\tu \rho\Step\tu \lambda\step[3]\id\Step\id\\
\step[8]\nw2\step[3]\Cu\Step\id\\
\step[10]\nw2\step[3]\Cu\\
\step[12]\Cu\\
\step[14]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\]
\[
\step\stackrel{\hbox{by (iv)}}{=}\step
\begin{tangle}
\object{H}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{\eta_H}\step[3]\object{H^{\hat{*}}}
\step[3]\object{H^{\hat{*}}}\step[3]\object{\eta_H}\step[3]\object{\eta_{H^{\hat{*}}}}\step[3]\object{H}\\
\d\Step\d\Step\d\Step\d\Step\d\Step\d\Step\id\Step\dd\\
\step\d\Step\d\Step\d\Step\d\Step\d\Step\XX\Step\id\\
\Step\d\Step\d\Step\d\Step\d\Step\XX\Step\XX\\
\step[3]\d\Step\d\Step\d\Step\XX\Step\XX\Step\id\\
\step[4]\d\Step\d\Step\XX\Step\XX\Step\id\Step\id\\
\step[5]\d\Step\XX\Step\XX\step\cd\step\id\Step\id\\
\step[6]\XX\Step\id\Step\id\Step\id\step\XX\step\id\Step\id\\
\step[6]\id\Step\id\Step\id\Step\id\Step\X\Step\X\Step\id\\
\step[6]\id\Step\id\Step\id\Step\XX\step\XX\step\XX\\
\step[6]\id\Step\id\Step\d\step\tu
\rightharpoonup\step\id\Step\X\Step\id\\
\step[6]\id\Step\d\Step\XX\Step\id\Step\id\step\tu \lambda\\
\step[6]\d\Step\XX\Step\XX\Step\id\Step\id\\
\step[7]\tu \rho\Step\XX\Step\XX\Step\id\\
\step[8]\id\Step\cd\step\XX\Step\id\Step\id\\
\step[8]\id\Step\O S\Step\id\step\id\Step\id\Step\id\Step\id\\
\step[8]\id\Step\XX\step\id\Step\id\Step\id\Step\id\\
\step[8]\id\Step\id\Step\X\Step\id\Step\id\Step\id\\
\step[8]\d\step\tu \rho\step\XX\Step\id\Step\id\\
\step[9]\cu\Step\tu \leftharpoonup\step\dd\Step\id\\
\step[10]\d\step[3]\tu \lambda\step[2]\dd\\
\step[11]\Cu\step[2]\dd\\
\step[13]\Cu\\
\step[15]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step\stackrel{\hbox{by Lemma1.4}}{=}\step[3]
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\step[3]\object{H^{\hat{*}}}\Step\object{H}\\
\id\step[3]\id\Step\XX\\
\id\step[3]\XX\Step\id\\
\id\step[3]\id\step\cd\step\id\\
\id\Step\dd\step\XX\step\id\\
\id\Step\XX\Step\X\\
\d\step\tu \rightharpoonup\Step\id\step\id\\
\step\XX\Step\dd\step\id\\
\dd\Step\XX\Step\id\\
\id\Step\cd\step\d\step\id\\
\id\Step\O S\Step\id\Step\id\step\id\\
\id\Step\XX\Step\id\step\id\\
\XX\Step\XX\step\id\\
\tu \rho\Step\tu \leftharpoonup\step\id\\
\step\d\step[3]\tu \lambda\\
\Step\Cu\\
\step[4]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\]
\[
\step=\step
\begin{tangle}
\step\object{H}\step[2]\object{H^{\hat{*}}}\step[3]\object{H^{\hat{*}}}\Step\object{H}\\
\step\id\Step\XX\Step\id\\
\step\XX\Step\XX\\
\cd\step\XX\step\cd\\
\O S\Step\id\step\id\Step\id\step\XX\\
\XX\step\id\Step\X\Step\id\\
\id\Step\X\step\dd\step\id\Step\id\\
\id\step\dd\step\X\Step\id\Step\id\\
\id\step\XX\step\XX\Step\id\\
\id\step\tu \rightharpoonup\step\tu \leftharpoonup\step\dd\\
\tu \rho\step[3]\tu \lambda\\
\step\id\step[4]\dd\\
\step\Cu\\
\step[3]\object{H\bar{\otimes}{H^{\hat{*}}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\hbox{the right side of (1). }
\end{tangle}
\ \ \ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\
\]

The proof of Theorem 2.8 .\\
(i)
\[
\step Let\step[3]
\begin{tangle}
\object{H^{\hat{*}}}\\
\id\\
\O w\\
\id\\
\object{H\# H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H^{\hat{*}}}\step[3]\object{\eta_H}\\
\step[0.5]\O { \bar S}\Step\id\\
\step[0.5]\tu \rho\\
\obox 3{\lambda^{-1}}\\
\step[1.5]\id\\
\step[1.5]\object{H\# H^{\hat{*}}}\\
\end{tangle}
\Step and\Step
\begin{tangle}
\step\object{R}\\
\td \phi\\
\object{H^{\hat{*}}}\Step\object{R}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[4]\object{R}\\
\ro {b'} \Step\id\\
\id\Step\tu \alpha\\
\object{H^{\hat{*}}}\step[3]\object{R}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[4]\object{R}\\
\coev\Step\id\\
\XX\Step\id\\
\id\Step\tu \alpha\\
\object{H^{\hat{*}}}\step[3]\object{R}\\
\end{tangle}\ \ \ , \hbox {where } \step [4] \object { b' = C_{H,
H^*}b_H .}
\]


We define

\[
\begin{tangle}
\object{R\otimes(H\# H^{\hat{*}})}\\
\id\\
\O \Psi\\
\id\\
\object{(R\# H)\# H^{\hat{*}}}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\step\object{R}\step[4]\object{H}\Step\object{H^{\hat{*}}}\\
\td \phi\step[3]\id\Step\id\\
\O S\Step\id\step[3]\id\Step\id\\
\x\step[3]\id\Step\id\\
\id\step\td w\Step\id\Step\id\\
\id\step\id\step\cd\step\id\Step\id\\
\id\step\id\step\id\Step\X\Step\id\\
\id\step\id\step\tu \rightharpoonup\step\cu\\
\id\step\cu\step[3]\id\\
\object{R}\Step\object{H}\step[4]\object{H^{\hat{*}}}\\
\end{tangle}
\step[3]and\step[5]
\begin{tangle}
\object{(R\# H)\# H^{\hat{*}}}\\
\id\\
\O \Phi\\
\id\\
\object{R\otimes(H\# H^{\hat{*}})}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\step\object{R}\step[4]\object{H}\Step\object{H^{\hat{*}}}\\
\td \phi\step[3]\id\Step\id\\
\x\step[3]\id\Step\id\\
\id\step\td w\Step\id\Step\id\\
\id\step\id\step\cd\step\id\Step\id\\
\id\step\id\step\id\Step\X\Step\id\\
\id\step\id\step\tu \rightharpoonup\step\cu\\
\id\step\cu\step[3]\id\\
\object{R}\Step\object{H}\step[4]\object{H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\]

We see that
\[
\Psi\Phi\step=\step
\begin{tangle}
\step[7]\object{R}\step[5]\object{H\# H^{\hat{*}}}\\
\ro {b'} \step\ro {b'}\Step\id\step[4]\dd\\
\O S\Step\id\step\d\step\tu \alpha\step[3]\dd\\
\O w\Step\id\Step\x\step[3]\dd\\
\d\step\tu \alpha\Step\O w\Step\dd\\
\step\x\Step\dd\step\dd\\
\step\id\Step\cu\step\dd\\
\step\id\step[3]\cu\\
\step\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\end{tangle}
\step \stackrel{\hbox{ since } w \hbox{ is algebraic }}{=}\step
\begin{tangle}
\step[7]\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\step[3]\coev\Step\id\step[4]\id\\
\coev\step\d\step\tu \alpha\step[4]\id\\
\O S\Step\id\Step\x\step[4]\dd\\
\d\step\tu \alpha\Step\id\step[3]\dd\\
\step\x\Step\dd\step[2]\dd\\
\step\id\Step\cu\step[2]\dd\\
\step\id\step[3]\O w\Step\dd\\
\step\id\step[3]\cu\\
\step\object{R}\step[5]\object{H\# H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\step[6]\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\coev\step\coev\step\id\step[4]\id\\
\O S\Step\X\Step\id\step\id\step[4]\id\\
\cu\step\cu\step\id\step[3]\dd\\
\step\O w\step[3]\tu \alpha\Step\dd\\
\step\d\Step\dd\Step\dd\\
\Step\x\Step\dd\\
\Step\id\step[2]\cu\\
\Step\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[5]\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\step\coev\Step\id\step[4]\id\\
\cd\step\tu \alpha\step[4]\id\\
\O S\Step\id\Step\id\step[5]\id\\
\cu\Step\id\step[5]\id\\
\step\O w\Step\dd\step[4]\dd\\
\step\x\step[4]\dd\\
\step\id\Step\Cu\\
\step\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\id\step[4]\id\\
\id\step[4]\id\\
\object{R}\step[4]\object{H\# H^{\hat{*}}}\\
\end{tangle} \ \ \ \ \ .
\]
Similarly, we have  $\Phi \Psi = id $. Thus $\Phi$ is invertible.

Now we show that $\Phi$ is algebraic.
\[
Let\step[12]
\begin{tangle}
\step\object{(R\# H)\# H^{\hat{*}}}\\
\step\id\\
\obox 2{\Phi'}\\
\step\id\\
\step\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\step[4]\object{R}\step[3]\object{\eta_H}\step[4]\object{H\# H^{\hat{*}}}\\
\ro {b'}\Step\id\step[3]\id\step[4]\id\\
\O {\bar{S}}\Step\tu \alpha\step[3]\id\step[4]\O \lambda\\
\d\Step\id\step[3]\dd\step[3]\dd\\
\step\x\Step\dd\step[3]\dd\\
\step\id\Step\tu \rho\step[3]\dd\\
\step\id\step[3]\Cu\\
\step\object{R}\step[5]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\]

It is clear that $\Phi = (id \otimes \lambda ^{-1})\Phi '$.
Consequently, we only need show that $\Phi '$ is alegebraic.

\[
Let\step[12]
\begin{tangle}
\object{R}\\
\id\\
\O \xi\\
\id\\
\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[3]=\step[3]
\begin{tangle}
\step[4]\object{R}\step[3]\object{\eta_H}\\
\ro {b'}\Step\id\step[3]\id\\
\O {\bar{S}}\Step\tu \alpha\Step\dd\\
\d\Step\id\Step\dd\\
\step\x\step[2]\id\\
\step\id\Step\tu \rho\\
\step\object{R}\step[4]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\]

We have that

\[
\begin{tangle}
\step\object{(R\# H)\# H^{\hat{*}}}\\
\step\id\\
\obox 2{\Phi'}\\
\step\id\\
\step\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step=\step[3]
\begin{tangle}
\step\object{R}\step[5]\object{H\# H^{\hat{*}}}\\
\td \xi\step[4]\O \lambda\\
\id\Step\Cu\\
\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\]

We claim that

\[
\begin{tangle}
\object{H}\step[2]\object{H^{\hat{*}}}\Step\object{R}\\
\tu \lambda\step\td \xi\\
\step\x\Step\id\\
\step\id\Step\cu\\
\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{H}\step[2]\object{H^{\hat{*}}}\step[3]\object{R}\\
\cd\Step\x\\
\id\Step\x\Step\id\\
\cu\Step\tu \lambda\\
\td \xi\Step\dd\\
\id\Step\cu\\
\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\step[5] ......(*)
\]


\[
\hbox{the left side}\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\step[4]\object{R}\step[3]\object{\eta_H}\\
\tu \lambda\step\ro {b'}\Step\id\step[3]\id\\
\step\id\Step\O {\bar{S}}\Step\tu \alpha\step[3]\id\\
\step\id\Step\d\Step\id\step[3]\dd\\
\step\d\Step\x\Step\dd\\
\Step\x\Step\tu \rho\\
\Step\id\Step\d\Step\id\\
\Step\id\step[3]\cu\\
\Step\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step\stackrel{\hbox{by Lemma1.5(1)}}{\step[3]=\step[3]}\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\step[5]\object{R}\step[3]\object{\eta_H}\\
\id\Step\id\Step\ro {b'}\Step\id\step[3]\id\\
\id\Step\id\Step\O {\bar{S}}\Step\tu \alpha\step[3]\id\\
\id\Step\id\Step\id\Step\dd\step[4]\id\\
\id\Step\id\Step\x\step[4]\dd\\
\id\Step\x\Step\id\step[3]\dd\\
\x\Step\XX\step[2]\dd\\
\id\Step\XX\Step\XX\\
\id\step\cd\step\XX\step\cd\\
\id\step\O S\Step\id\step\id\Step\id\step\XX\\
\id\step\XX\step\id\Step\X\Step\id\\
\id\step\id\Step\X\Step\id\step\id\Step\id\\
\id\step\id\step\dd\step\XX\step\d\step\id\\
\id\step\id\step\XX\Step\XX\step\id\\
\id\step\id\step\tu \rightharpoonup\Step\tu
\leftharpoonup\step\id\\
\id\step\tu \rho\step[4]\tu \lambda\\
\id\Step\d\step[4]\dd\\
\id\step[3]\Cu\\
\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\step[5]\object{R}\step[3]\object{\eta_H}\\
\id\Step\id\Step\ro {b'}\Step\id\step[3]\id\\
\id\Step\id\Step\O {\bar{S}}\Step\tu \alpha\step[3]\id\\
\id\Step\id\Step\id\Step\dd\step[4]\id\\
\id\Step\id\Step\x\step[4]\dd\\
\id\Step\x\Step\id\step[3]\dd\\
\x\Step\XX\step[2]\dd\\
\id\Step\XX\Step\XX\\
\id\step\cd\step\XX\Step\id\\
\id\step\O {\bar S}\Step\id\step\id\Step\id\Step\id\\
\id\step\XX\step\id\Step\id\Step\id\\
\id\step\id\Step\X\Step\id\Step\id\\
\id\step\tu \rho\step\XX\Step\id\\
\id\Step\id\Step\tu \leftharpoonup\step\dd\\
\id\Step\id\step[3]\tu \lambda\\
\id\Step\Cu\\
\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{H}\step[3]\object{H^{\hat{*}}}\step[5]\object{R}\\
\id\Step\id\Step\ro {b'}\Step\id\\
\id\Step\XX\Step\tu \alpha\\
\XX\Step\id\Step\dd\\
\id\Step\id\Step\x\\
\id\Step\x\Step\id\\
\x\Step\id\Step\id\\
\id\step\cd\step\d\step\id\\
\id\step\O {\bar S}\Step\XX\step\id\\
\id\step\id\Step\tu \leftharpoonup\step\id\\
\id\step\id\step[3]\tu \lambda\\
\id\step\id\step\obox 2{\eta_H}\step\id\\
\id\step\tu \rho\step\dd\\
\id\Step\cu\\
\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\]
\[
\step=\step
\begin{tangle}
\step\object{H}\step[5]\object{H^{\hat{*}}}\step[4]\object{R}\\
\cd\step\ro {b'}\step\id\step\ro {b'}\step\id\\
\id\Step\X\Step\X\step\id\Step\id\step\id\\
\XX\step\id\Step\id\step\X\Step\id\step\id\\
\id\Step\id\step\id\Step\X\step\cu\step\id\\
\id\Step\id\step\XX\step\d\step\tu \alpha\\
\d\step\X\Step\id\Step\x\\
\step\d\hev\Step\x\Step\id\\
\Step\d\Step\id\Step\tu \lambda\\
\step[3]\x\step[3]\d\\
\step[3]\id\Step\O {\bar{S}}\step\obox 2{\eta_H}\step\id\\
\step[3]\id\Step\tu \rho\step\dd\\
\step[3]\id\step[3]\cu\\
\step[3]\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[4]\object{H}\step[3]\object{H^{\hat{*}}}\step[3]\object{R}\\
\ro {b'}\step\cd\step[2]\id\Step\id\\
\id\Step\id\step\XX\Step\id\Step\id\\
\id\Step\X\Step\id\Step\id\Step\id\\
\id\Step\id\step\cu\step\dd\step\dd\\
\id\Step\id\step [2]\XX\step [2]\id\\
\id\Step\id\step [2]\d\step\tu \alpha\\
\id\Step\d\Step\x\\
\d\Step\x\Step\id\\
\step\x\Step\tu \lambda\\
\step\id\Step\id\step[3]\d\\
\step\id\Step\O {\bar{S}}\step\obox 2{\eta_H}\step\id\\
\step\id\Step\tu \rho\step\dd\\
\step\id\step[3]\cu\\
\step\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[3]\object{H}\step[3]\object{H^{\hat{*}}}\step[3]\object{R}\\
\Step\cd\Step\id\Step\id\\
\Step\XX\Step\id\Step\id\\
\step [2]\id \step[2]\XX  \Step\id\\
\step [2]\id \step[2]\id \step [2]\tu \alpha\\
\step \dd \step \dd\ro {b'}\step [1]\d\\
\step \id \step[2]\id \step \id\step [2]\tu \alpha\\
\step \d \step \X\step[3]\id\\
\step[2]\id \step[1]\id \step \d \step [2] \id \\
\step[2]\X\step[2]\x \\
\step [2] \id \step [1] \x \step [2]\id \\
\step[2]\hx\step[2]\id\Step\id\\
\step [2]\id \step\O {\bar S}\step \step \tu \lambda\\
\step [2]\id\step\id\step\obox 2{\eta_H}\d\\
\step [2]\id\step\tu \rho\step\dd\\
\step [2]\id\step[2]\cu\\
\step [2]\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}\]
\[
\step=\step
\begin{tangle}
\step\object{H}\step[3]\object{H^{\hat{*}}}\step[3]\object{R}\\
\cd\Step\id\Step\id\\
\XX\Step\id\Step\id\\
\id\Step\XX\Step\id\\
\id\Step\d\step\tu \alpha\\
\d\Step\x\\
\step\x\Step\id\\
\td \xi\step\tu \lambda\\
\id\Step\cu\\
\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{H}\step[3]\object{H^{\hat{*}}}\step[3]\object{R}\\
\cd\Step\x\\
\id\Step\x\Step\id\\
\tu \alpha\Step\tu \lambda\\
\td \xi\Step\dd\\
\id\Step\cu\\
\object{R}\step[4]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}\ \ \ .
\]
Thus relation (*) holds.

Next, we check that $\xi$ is algebraic. We see that
\[
\begin{tangle}
\object{R}\Step\object{R}\\
\O \xi\Step\O \xi\\
\cu\\
\step\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[3]=\step
\begin{tangle}
\step\object{R}\step[6]\object{R}\\
\td \phi\step[4]\td \phi\\
\O {\bar{S}}\Step\id\step[4]\O {\bar{S}}\Step\id\\
\x\step\obox 2{\eta_H}\step\x\step\obox 2{\eta_H}\\
\id\Step\tu \rho\step\dd\Step\tu \rho\\
\d\Step\x\step[4]\id\\
\step\cu\Step\Cu\\
\Step\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{R}\step[3]\object{R}\\
\td \phi\step\td \phi\\
\O {\bar{S}}\Step\id\step\O {\bar{S}}\Step\id\\
\x\step\id\Step\id\\
\id\Step\X\Step\id\\
\id\Step\id\step\x\\
\id\Step\hx\Step\id\\
\cu\step\cu\\\
\step\id\step[3]\d\step\obox 2{\eta_H}\\
\step\id\step[4]\tu \rho\\
\step\object{R}\step[5]\object{H\bar{\otimes} H^{\hat{*}}}\\
\end{tangle}
\step=\step[4]
\begin{tangle}
\object{R}\Step\object{R}\\
\cu\\
\step\O \xi\\
\step\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\]
and obviously
\[
\begin{tangle}
\object{\eta_R}\\
\id\\
\O \xi\\
\id\\
\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[6]=\step[6]
\begin{tangle}
\object{\eta_{R\otimes(H\bar{\otimes}H^{\hat{*}})}}\\
\id\\
\id\\
\id\\
\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}\ \ \ \ \ \ \ \ \ .
\]
Thus $\xi$ is algebraic.

Now we show that $\Phi'$ is algebraic.
\[
\begin{tangle}
\object{(R\# H)\# H^{\hat{*}}}\step[11]\object{(R\# H)\# H^{\hat{*}}}\\
\step\nw2\step[6]\ne2\\
\step[2]\obox 2{\Phi'}\step[2]\obox 2{\Phi'}\\
\step[3]\Cu\\
\step[6]\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[3]=\Step
\begin{tangle}
\step\object{R}\step[4]\object{H}\step[3]\object{H^{\hat{*}}}
\step[3]\object{R}\step[4]\object{H}\step[3]\object{H^{\hat{*}}}\\
\td \xi\step[3]\id\Step\id\step[3]\td \xi\step[3]\id\Step\id\\
\id\Step\id\step[3]\tu \lambda\Step\dd\Step\id\step[3]\tu \lambda\\
\id\Step\Cu\Step\dd\step[3]\Cu\\
\id\step[4]\d\Step\dd\step[5]\ne2\\
\d\step[4]\x\step[4]\dd\\
\step\Cu\Step\Cu\\
\step[3]\object{R}\step[6]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}\]
\[
\step=\step
\begin{tangle}
\step\object{R}\step[3]\object{H}\step[2]\object{H^{\hat{*}}}
\step[3]\object{R}\step[3]\object{H}\step[2]\object{H^{\hat{*}}}\\
\td \xi\step[2]\id\Step\id\step[2]\td \xi\step[2]\id\Step\id\\
\id\Step\id\Step\tu \lambda\step\dd\step\dd\Step\tu \lambda\\
\id\Step\id\step[3]\x\Step\id\step[4]\id\\
\id\Step\id\Step\dd\Step\cu\step[3]\dd\\
\id\Step\x\step[4]\Cu\\
\cu\Step\d\step[4]\dd\\
\step\id\step[4]\Cu\\
\step\object{R}\step[6]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}
\stackrel{\hbox{by (*)}}{=}
\begin{tangle}
\step\object{R}\step[3]\object{H}\step[2]\object{H^{\hat{*}}}
\step[2]\object{R}\step[2]\object{H}\step[3]\object{H^{\hat{*}}}\\
\td \xi\step\cd\step\x\Step\id\Step\id\\
\id\Step\id\step\id\Step\hx\Step\id\Step\tu \lambda\\
\id\Step\id\step\tu \alpha\step\tu \lambda\step[3]\id\\
\id\Step\id\step\td \xi\Step\id\step[4]\id\\
\id\Step\hx\Step\cu\step[3]\dd\\
\cu\step\d\Step\Cu\\
\step\id\step[3]\Cu\\
\step\object{R}\step[5]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}
\]
\[
\step\stackrel{\hbox{by Lemma1.4}}{=}\step
\begin{tangle}
\step\object{R}\step[3]\object{H}\step[2]\object{H^{\hat{*}}}
\step[2]\object{R}\step[2]\object{H}\step[3]\object{H^{\hat{*}}}\\
\td \xi\step\cd\step\x\Step\id\Step\id\\
\id\Step\id\step\id\Step\hx\step\cd\step\id\Step\id\\
\id\Step\id\step\tu \alpha\step\id\step\id\Step\X\Step\id\\
\id\Step\id\step\td \xi\step\id\step\tu \rightharpoonup\step\cu\\
\id\Step\hx\Step\id\step\cu\step[2]\ne2\\
\cu\step\id\Step\d\step\tu \lambda\\
\step\id\Step\id\step[3]\cu\\
\step\id\Step\Cu\\
\step\object{R}\step[5]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle}
\step\stackrel{\hbox{ since } \xi \hbox{ is algebraic }}{=}\step
\begin{tangle}
\object{R}\step[2]\object{H}\step[2]\object{H^{\hat{*}}}
\step[2]\object{R}\step[2]\object{H}\step[3]\object{H^{\hat{*}}}\\
\id\step\cd\step\x\Step\id\Step\id\\
\id\step\id\Step\hx\step\cd\step\id\Step\id\\
\id\step\tu \alpha\step\id\step\id\Step\X\Step\id\\
\cu\Step\id\step\tu \rightharpoonup\step\cu\\
\td \xi\Step\cu\Step\ne2\\
\id\Step\id\step[3]\tu \lambda\\
\id\Step\Cu\\
\object{R}\step[5]\object{H\bar{\otimes}H^{\hat{*}}}\\
\end{tangle} \]
\[
\step=\step[6]
\begin{tangle}
\object{(R\# H)\# H^{\hat{*}}}\step[11]\object{(R\# H)\# H^{\hat{*}}}\\
\step\nw2\step[6]\ne2\\
\step[3]\Cu\\
\step[4]\obox 2{\Phi'}\\
\step[5]\id\\
\step[4]\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}\ \ \ \ \ \ \ \ \ \ .
\]

It is clear that
\[
\begin{tangle}
\object{\eta_{(R\# H)\# H^{\hat{*}}}}\\
\step\id\\
\obox 2{\Phi'}\\
\step\id\\
\step\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}
\step[6]=\step[6]
\begin{tangle}
\object{\eta_{R\otimes(H\bar{\otimes}H^{\hat{*}})}}\\
\id\\
\id\\
\id\\
\object{R\otimes(H\bar{\otimes}H^{\hat{*}})}\\
\end{tangle}\ \ \ \ \ \ \ \ \ \ \ \ \ .\] Thus $\Phi '$ is algebraic. \ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\


The proof of Lemma3.2.\\
\[
\begin{tangle}
\step[2]\object{D(H)}\\
\step[2]\id\\
\obox 4{\Delta^{cop}}\step\obox 3{[b]}\\
\Step\id\step[2]\XX\step\d\\
\Step\cu\Step\cu\\
\step\object{D(H)}\step[6]\object{D(H)}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{A}\step[3]\object{H}\Step\object{\eta}\step[9]\object{\eta}\\
\cd\step\cd\step\id\Step\Coev\step[5]\id\\
\id\Step\X\Step\id\step\id\Step\id\step[2]\cd\Step\id\\
\id\Step\X\Step\id\step\id\Step\id\Step\id\step\cd\step\id\\
\XX\step\XX\step\id\Step\id\Step\id\step\id\Step\id\step\id\\
\id\Step\X\Step\X\Step\id\Step\id\step\id\Step\id\step\id\\
\id\Step\id\step\XX\step\XX\Step\d\d\step\d\d\\
\id\Step\X\Step\X\Step\d\Step\id\step\id\Step\id\step\id\\
\cu\step\cu\step\id\step[2]\cd\step\id\step\id\Step\id\step\id\\
\step\id\step[3]\id\Step\id\step\cd\step\X\step\id\Step\id\step\id\\
\step\id\step[3]\id\Step\id\step\id\Step\X\step\X\Step\id\step\id\\
\step\id\step[3]\id\Step\id\step\coro \tau \step\X\step\coro {\bar \tau }\step\id\\
\step\id\step[3]\id\Step\Cu\step\Cu\\
\step\object{A}\step[3]\object{H}\step[4]\object{A}\step[5]\object{H}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{A}\step[3]\object{H}\\
\cd\step\cd\Step\COEV\\
\XX\step\id\Step\id\step\dd\Step\Coev\step[3]\id\\
\id\Step\id\step\id\Step\id\step\id\Step\ne2\coev\step\id\step\id\\
\id\Step\id\step\id\Step\id\step\id\step\cu\Step\id\step\id\step\id\\
\id\Step\id\step\id\Step\id\step\cu\step[3]\id\step\id\step\id\\
\id\Step\id\step\d\step\cu\step[4]\id\step\id\step\id\\
\id\Step\id\Step\XX\step[4]\dd\step\id\step\id\\
\id\Step\id\step\dd\step\cd\Step\dd\Step\id\step\id\\
\id\Step\X\step\cd\step\XX\Step\dd\step\id\\
\id\Step\id\step\id\step\id\Step\X\Step\XX\Step\id\\
\id\Step\id\step\id\step\coro \tau \step\XX\Step\coro {\bar \tau }\\
\id\Step\id\step\Cu\Step\id\\
\object{A}\Step\object{H}\step[3]\object{A}\step[4]\object{H}\\
\end{tangle} \]
\[
\step=\step
\begin{tangle}
\step\object{A}\step[5]\object{H}\\
\cd\step[3]\cd\step\coev\\
\XX\Step\cd\step\id\step\id\Step\id\\
\id\Step\id\step\cd\step\X\step\id\Step\id\\
\id\Step\id\step\id\Step\X\step\O {\bar{S}}\step\id\Step\id\\
\id\Step\id\step\XX\step\X\step\id\Step\id\\
\id\Step\id\step\id\Step\X\step\X\Step\id\\
\id\Step\id\step\id\step\dd\step\X\step\XX\\
\id\Step\id\step\id\step\id\step\dd\step\id\step\id\Step\id\\
\id\Step\id\step\id\step\id\step\cu\step\id\Step\id\\
\id\Step\id\step\id\step\cu\Step\id\Step\id\\
\id\Step\id\step\cu\step[3]\id\Step\id\\
\id\Step\XX\step[4]\id\Step\id\\
\id\Step\id\Step\Cu\Step\id\\
\object{A}\Step\object{H}\step[4]\object{A}\step[4]\object{H}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{A}\step[3]\object{H}\\
\cd\step\cd\step\coev\\
\XX\step\id\Step\X\Step\id\\
\id\Step\id\step\XX\step\XX\\
\id\Step\id\step\cu\step\id\Step\id\\
\id\Step\id\Step\XX\Step\id\\
\id\Step\cu\step\dd\Step\id\\
\id\step[3]\XX\step[3]\id\\
\object{A}\step[3]\object{H}\Step\object{A}\step[3]\object{H}\\
\end{tangle}
\]

and
\[
\begin{tangle}
\step[5]\object{D(H)}\\
\step[5]\id\\
\obox 3{[b]}\step\cd\\
\step\id\step[2]\X\Step\id\\
\step\cu\step\cu\\
\step[0.5]\object{D(H)}\step[5.5]\object{D(H)}\\
\end{tangle}
\step=\step
\begin{tangle}
\object{\eta}\step[8]\object{\eta}\step[3]\object{A}\step[3]\object{H}\\
\id\step[2.5]\obox 3{b}\step[2.5]\id\Step\cd\step\cd\\
\id\Step\cd\step\d\Step\XX\Step\X\Step\id\\
\id\step\cd\step\id\Step\XX\Step\XX\step\id\Step\id\\
\id\step\id\Step\id\step\id\step\cd\step\XX\Step\X\Step\id\\
\id\step\id\Step\id\step\X\step\cd\d\step\cu\step\cu\\
\id\step\id\Step\X\step\X\Step\id\step\id\Step\id\step[3]\id\\
\id\step\coro \tau \step\X\step\coro {\bar \tau }\step\id\Step\id\step[3]\id\\
\Cu\step\Cu\Step\id\step[3]\id\\
\Step\object{A}\step[5]\object{H}\step[4]\object{A}\step[3]\object{H}\\
\end{tangle}
\step=\step
\begin{tangle}
\step \step[9]\object{A}\step[3]\object{H}\\
\step \COEV\step[6]\cd\step\cd\\
\step \id\step\coev\step\coev\d\step\id\Step\id\step\id\Step\id\\
\step \id\step\id\Step\X\Step\id\step\id\step\id\Step\id\step\id\Step\id\\
\step \id\step\XX\step\cu\step\id\step\id\Step\id\step\id\Step\id\\
\step \X\Step\id\Step\cu\step\id\Step\id\step\id\Step\id\\
\step \id\step\XX\step[3]\XX\Step\id\step\id\Step\id\\
\step \id\step\id\Step\id\Step\cd\step\cu\step\id\Step\id\\
\step \id\step\id\Step\XX\step\cd\step\XX\Step\id\\
\dd\step\XX\Step\X\Step\id\step\id\Step\id\Step\id\\
\coro \tau \Step\XX\step\coro {\bar \tau }\step\id\Step\id\Step\id\\
\step \step[3]\id\Step\Cu\Step\id\Step\id\\
\step \step[3]\object{A}\step[4]\object{H}\step[4]\object{A}\Step\object{H}\\
\end{tangle}\]
\[
\step=\step
\begin{tangle}
\step[5]\object{A}\step[4]\object{H}\\
\step[4]\cd\Step\cd\\
\coev\step\cd\step\d\step\id\Step\id\\
\id\Step\X\step\cd\step\id\step\id\Step\id\\
\id\Step\id\step\X\Step\O S\step\id\step\id\Step\id\\
\id\Step\X\step\d\step\id\step\id\step\id\Step\id\\
\XX\step\cu\step\id\step\id\step\id\Step\id\\
\id\Step\id\Step\cu\step\id\step\id\Step\id\\
\id\Step\id\step[3]\cu\step\id\Step\id\\
\id\Step\id\step[4]\XX\Step\id\\
\id\Step\Cu\Step\id\Step\id\\
\object{A}\step[4]\object{H}\step[4]\object{A}\Step\object{H}\\
\end{tangle}
\step=\step
\begin{tangle}
\step\object{A}\step[6]\object{H}\\
\cd\step\coev\step\cd\\
\XX\step\id\Step\id\step\id\Step\id\\
\id\Step\X\Step\id\step\id\Step\id\\
\id\Step\id\step\cu\step\id\Step\id\\
\id\Step\id\Step\XX\Step\id\\
\id\Step\cu\Step\id\Step\id\\
\object{A}\step[3]\object{H}\step[3]\object{A}\step[2]\object{H}\\
\end{tangle}\ \ \ .
\]

Thus
\[
\begin{tangle}
\step[2]\object{D(H)}\\
\step[2]\id\\
\obox 4{\Delta^{cop}}\step\obox 3{[b]}\\
\Step\id\step[2]\XX\step\d\\
\Step\cu\Step\cu\\
\step\object{D(H)}\step[6]\object{D(H)}\\
\end{tangle}
\step=\step
\begin{tangle}
\step[5]\object{D(H)}\\
\step[5]\id\\
\obox 3{[b]}\step\cd\\
\step\id\step[2]\X\Step\id\\
\step\cu\step\cu\\
\step[0.5]\object{D(H)}\step[5.5]\object{D(H)}\\
\end{tangle}\ \ \ .
\]
We complete the proof.\ \
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}\put(5,5){\line(0,-1){5}}
\put(0,0){\line(1,0){5}}\put(5,5){\line(-1,0){5}}
\end{picture}\\

{\bf Remark:} If $U$ and $V$ have left dual $U^*$ and $V^*$,
respectively, then $U^*\otimes V^*$ and $V^* \otimes U^*$ both are
the left duals of $U\otimes V$. Their evaluations and
coevaluations are $$ d_{U\otimes V}=(d_{U} \otimes d_V) (id _{U^*}
\otimes C_{V^*, U}\otimes  id _V), \ \ \ \ b_{U\otimes V}=(id
_U\otimes (C_{V, U^*})^{-1}\otimes id_ {V^*})(b_U \otimes b_V);$$\
$$ d_{U\otimes V}=d_V (id _{V^*}\otimes d_U \otimes id _V ) ,
\ \ \ \ b_{U\otimes V}=(id _{U^*}\otimes b_V \otimes id_U)b_U,
$$ respectively. In this paper, we use the second.  $H^{\hat * }$ is the left dual of $H$ under the first.





\end{document}

\end {document}

math0307255.extracted_bib
\begin{thebibliography}{150}


\bibitem {BD} Y. Bespalov and B.Drabant, Cross product bialgebras, I,
 J. algebra, {\bf 219} (1999), 466--505.




\bibitem {BM85}  R. J. Blattner and   S. Montgomery, A duality theorem for Hopf
module algebras,  J. algebra, {\bf 95} (1985), 153--172.



\bibitem {Ch}  H.X.Chen, Quantum double in monoidal categories,
Comm.Algebra,
 {\bf 28} (2000)5, 2303--2328.



\bibitem {Dr86} V. G. Drinfeld,  Quantum groups, in ``Proceedings International Congress of
Mathematicians, August 3-11, 1986, Berkeley, CA" pp. 798--820,
Amer. Math. Soc., Providence, RI, 1987.




\bibitem {L} G.Lusztig, Introduction to    Quantum  Groups,
Progress on Math., Vol.110, Birkhauser, Berlin, 1993.

 \bibitem {LZ} L.Li and P.Zhang, Twisted Hopf algebras, Ringel-Hall algebras and Green's ca
tegories, J.Algebra, {\bf 231}(2000), P713--743.



\bibitem {Ma90a} {  S. Majid,
Physics for algebraists: Non-commutative and non-cocommutative
Hopf algebras by a bicrossproduct construction, J.Algebra }  {\bf
130}  (1990), 17--64.


  \bibitem {Ma95a} S. Majid, Algebras and Hopf algebras
  in braided categories,
Lecture notes in pure and applied mathematics advances in Hopf
algebras, Vol. 158, edited by J. Bergen and S. Montgomery, Marcel
Dekker, New York, 1994, 55--105.

  \bibitem {Ma95b} S. Majid, Foundations of Quantum Group Theory,  Cambridge University Press, Cambridge, 1995.
\bibitem {MR94} S. Majid and M.J. Rodriguez Piaza, Random walk and the heat
equation on superspace and anyspace, J. Math. Phys., {\bf
35}(1994) 7, 3753--3760.



\bibitem {Mo93}  S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS
  Number 82, AMS, Providence, RI, 1993.



 \bibitem{Ra93a}   D. E. Radford,
 Minimal quasi-triangular Hopf algebras,
 J. algebra,
 {\bf 157} (1993), 281--315.


\bibitem {St81} S. Stratila, Modular Theory in Operator Algebras,
Editura Academiei and Abacus Press, Bucuresti, 1981.


\bibitem {Sw}   M.E.Sweedler, Hopf Algebras, Benjamin, New York, 1969.

 \bibitem {Ta}   M. Takeuchi, Finite Hopf algebras in braided tensor categories ,
J.Pure and Applied Algebra, {\bf 138}(1999), 59-82.


\bibitem {ZC} Shouchuan Zhang, Hui-Xiang Chen, The double bicrossproducts
in braided tensor categories,
    Communications in Algebra,   {\bf 29}(2001)1, P31--66.




\end {thebibliography}