On unconventional integrations and cross ratio on supermanifolds
The conventional integration theory on supermanifolds had been constructed so as to possess (an analog of) Stokes' formula. In it, the exterior differential $d$ is vital and the integrand is a section of a fiber bundle of finite rank. Other, not so popular, but, nevertheless, known integrations are analogs of Berezin integral associated with infinite dimensional fibers. Here I offer other unconventional integrations that appear thanks to existence of several versions of traces and determinants and do not allow Stokes formula. Such unconventional integrations have no counterpart on manifolds except in characteristic $p$. Another type of invariants considered are analogs of the cross ratio for ``classical superspaces''. As a digression, homological fields corresponding to simple Lie algebras and superalgebras are described.\vspace{1pc}
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\newcommand {\Cee}    {{\mathbb  C}}
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\newcommand {\Pee}    {{\mathbb  P}}
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\rmname{id}
\rmname{Hom}
\rmname{Mat}
\rmname{qet}
\rmname{st}
\rmname{tr}
\rmname{Span}
\rmname{str}
\rmname{Vol}
\rmname{vol}

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\title{On unconventional integrations and cross ratio on
supermanifolds}

\author{Dimitry Leites\address[SU]{Department of Mathematics,
University of Stockholm, \\
Kr\"aftriket hus 6, SU-106 91, Stockholm,
Sweden; mleites@matematik.su.se}%
\thanks{I am thankful to NFR and TBSS for financial support, to
J.~Lukiersky for hospitality, to B.~Zupnik, O.~Hudaverdyan and
A.~Vaintrob for inspiring discussions, V.~Serganova and P.~Grozman for
help.}}

\begin{document}

\begin{abstract}
The conventional integration theory on supermanifolds had been
constructed so as to possess (an analog of) Stokes' formula.  In it,
the exterior differential $d$ is vital and the integrand is a section
of a fiber bundle of finite rank.  Other, not so popular, but,
nevertheless, known integrations are analogs of Berezin integral
associated with infinite dimensional fibers.  Here I offer other
unconventional integrations that appear thanks to existence of several
versions of traces and determinants and do not allow Stokes formula. 
Such unconventional integrations have no counterpart on manifolds
except in characteristic $p$.

Another type of invariants considered are analogs of the cross
ratio for ``classical superspaces''.  

As a digression, homological fields corresponding to simple Lie
algebras and superalgebras are described.\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle


For the basics on Linear Algebra in Superspaces and Supermanifold
theory see \cite{D}; for notations and useful facts see \cite{Sch},
\cite{Ser}.  At the talk I also considered related issues partly
collected in \cite{GLS1}.  As compared with the talk, \S\S2, 3 are
new; they are a part of the talk given 10 years earlier \cite{LSV} but
yet unpublished.  Encouraged by Manin's selected examples \cite{Mn}
and recent results in classification of simple Lie superalgebras
\cite{LSh1} I decided to draw attention to these issues.



\section{INTEGRATION}


\subsection{Integration with Stokes' formula} In mid 1970's
J.~Bernstein and I discussed how to construct an analog of integration
theory on supermanifolds.  We had at our disposal (1) the differential
forms, i.e., functions polynomial in differentials of the coordinates,
the coefficients of these polynomials being usual functions and (2)
volume forms, the latter constituted a rank one module $\Vol$ over the
algebra of functions and under the change of coordinates the generator
$\vol(x(y))$ of $\Vol$ accrued the Berezinian (superdeterminant) of the
Jacobi matrix as the factor.  Each of the above notions had to be
carefully reconsidered in super setting because even the most
innocent-looking notions and theorems (e.g., the Foubini theorem)
displayed, in supersetting, funny signs at unexpected places, see
\cite{L}, v.  31.

On manifolds, one {\it can} integrate differential forms; on
supermanifolds, one can {\it not}: their transformation rule yields no
analog of determinant, except in the absence of odd parameters.  On
the other hand, one, clearly, can integrate elements of $\Vol$,
provided they are with compact support, of course.  But we wanted to
have some analog of Stokes' formula, and, therefore, needed (1)
elements of ``degrees'' lesser than that of volume forms, and (2) the
notion of the supermanifold with boundary to overcome the puzzle
demonstrated by ``Rudakov's example''; for solution see \cite{BL1}.

To have integration theory, one needs not only what to
integrate (the integrand), but over what (cycle), and orientation. 
The latter two notions turned out to be more involved than we
originally thought; Shander clarified this in his development of
integration theory, see \cite{Sh} and the details in \cite{L}.

Actually, what we had had was sufficient to construct the integration
theory desired: by setting $\Sigma_{-i}=\Hom_{\tt F}(\Omega^i, \Vol)$,
where ${\tt F}$ is the superspace of functions, we obtain a complex
dual to the de Rham one with $\Sigma_{0}= \Vol$ as forms of the
highest degree.  We called the elements of $\Sigma_{\bcdot}$ {\it
integrable} forms (the ones one can integrate) and described how to
integrate such forms in \cite{BL1}.

\subsection{Veblen's problem and Rudakov} We wondered for a while if
there is another integration theory with Stokes' formula, and to
investigate the options, considered the following problem: {\sl
describe all differential operators acting in the spaces of tensor
fields and invariant with respect to any changes of variables}. 
Indeed, the exterior differential (instrumental in Stokes' formula)
is, evidently, an invariant and, as is proven in \cite{R}, this is the
only invariant unary differential operator between spaces of tensor
fields whose fibers are irreducible $\mathfrak{gl}(n)$-modules with
vacuum vector.  So, in order to scan the options on supermanifolds,
it was necessary to list all such operators.

This problem (to list all invariant differential operators) goes back
to O.~Veblen (see \cite{GLS1} for a review).  For unary operators on
manifolds it was solved by Rudakov \cite{R} as a part of another
problem (description of irreducible vacuum vector modules over simple
Lie algebras of formal or polynomial vector fields).

\subsection{Unconventional integrations} 
A.~Shwarts and his students \cite{AS} attempted to integrate densities
and objects depending on higher jets of the diffeomorphism but all
their examples boil down to either pseudodifferential (\cite{BL2})
or integral forms.  Having obtained an analog of Rudakov's result for
the general vectorial Lie {\it super}algebra \cite{BL3}, we can be
sure that there is only one integration theory on supermanifolds {\it
provided the integration involves tensors}
$$
\hbox{{\it with irreducible finite
dimensional fibers}.}\eqno{(*)}
$$

The result of \cite{BL3} do not preclude, however, unconventional
integrations.  For tensors other than $(*)$ constructions \`a la
Shwarts may lead to an integration theory (perhaps, \cite{H} leads to
it).  Such a theory does exist and in \cite{LKV} the calculations from
\cite{BL3} are used to consider infinite dimensional fibers snubbed at
in \cite{BL3} for no reason except tradition.  It turns out that in
the spaces of such tensors there act invariant operators similar to
Berezin integral.  Next, observe that having stated that it is
impossible to integrate differential forms on supermanifolds, we
almost immediately published a paper \cite{BL2} showing, nevertheless,
how to do it if one is very eager to.  More exactly, one has to
consider {\it pseudodifferential forms}, i.e., functions nonpolynomial
in differentials.  Of course, there are no such functions on
manifolds.  Certain types of pseudoforms lead to new invariants ---
semi-infinite cohomology of supermanifolds; quite criminally, no
examples are calculated yet.

Here I consider still another type of ``integrations''.


\subsection{Supertraces and superdeterminants} From the very beginning
I wondered what if we stop insisting on having an analog of the
Stokes' formula?  What remains of the integration then?  Only the
Jacobian, one can say.  Since it is easier to deal with Lie
algebras than with groups, let me list analogs of trace for Lie
superalgebras.  Then, if the Lie superalgebra $\mathfrak{g}$ can be
exponentiated to a Lie supergroup, we can consider the analog of the
determinate defined via the formula
$$
\det \exp X= e^{\tr (X)} \quad\hbox{  for any } \; X\in\mathfrak{
g}.\eqno{(1)}
$$
In other words, I mean: 

1) Let us consider the Lie superalgebras $\mathfrak{g}$ with a trace
also denoted by $\tr$ (i.e., $\tr([x, y])=0$ for any
$x,y\in\mathfrak{g}$); then for the role of $\Vol$ we can take tensor
fields of type $\tr$, its infinitesimal transformations being the
Cartan prolongation (see \cite{GLS1}) of the pair $(\id,
\mathfrak{g})$, where $\id$ is the ``standard'' or ``identity''
representation of $\mathfrak{g}$.

The prime example is provided by the Poisson Lie superalgebra
$\mathfrak{g}=\mathfrak{po}(0|2n)$.  Indeed, there is a parametric family
(quantization) of Lie superalgebras $\mathfrak{g}_{t}$ which at $t=0$
coincides with $\mathfrak{po}(0|2n)$ and $\mathfrak{
g}_{t}\simeq\mathfrak{gl}(2^{n-1}|2^{n-1})$ for $t\neq 0$, see 
\cite{LSh2}.

2) From various points of view it is clear that $\mathfrak{gl}(n)$ has
at least two superanalogs: the ``simple-minded'' one,
$\mathfrak{gl}(n|m)$, and the ``queer'' one, $\mathfrak{q}(n)$.  On
$\mathfrak{q}(n)$, the supertrace vanishes identically but there are
specially designed for it its particular, queer, trace and
determinant.  Regrettably, the queer trace is odd and, therefore, to
describe the corresponding representation, we need odd parameters,
which causes extra difficulties.

So, still another versions of integration theory, if exist, are
related with the queertrace and its ``quasiclassical limit'' as
$t\longrightarrow 0$: the restriction of the above quantization is a
parametric family $\mathfrak{g}_{t}$ which at $t=0$ coinsides with
$\mathfrak{po}(0|2n-1)$ and $\mathfrak{g}_{t}\simeq\mathfrak{
q}(2^{n-1})$ for $t\neq 0$. In 1) and 2) the identity representation 
of $\mathfrak{po}(0|m)$ is the adjoint one and the Berezin integral 
serves as $\tr$.

3) The analog of trace on the general vectorial algebra $\mathfrak{
vect}(m|n)$ is the divergence.  I do not know how to generalize
formula $(1)$ with divergence serves as $\tr$, so let me mention two
other, more obvious, analogs of $\tr$: (a) in characteristic $p$ such
analog exists, e.g., for Lie algebras of contact vector fields (but
not only), see \cite{St}; another one is provided by (b)
``superconformal'' algebras of divergence-free series $\mathfrak{
svect}(1|N)$and the exceptions related with $N=4$ and $N=5$ extended
Neveu-Schwarz algebras, see \cite{GLS2}; these traces are of the same
parity as $N$.

\section{Jordan superalgebras}

I consider here certain algebraic structures associated with certain
selected ``classical superdomains''.  So far, nobody knows yet (as far
as I know) even a ``right'' definition of this basic notion, to start
with: for finite dimensional manifolds all is clear, for
supermanifolds we just consider the most easy to handle simple Lie
supergroups and their Lie superalgebras whereas the elusive ``right''
definition requires, perhaps, semisimple or almost simple Lie
superalgebras.  So we take the road of least resistance:

Unless otherwise mentioned the ground field is $\Cee$, the classical
superdomains are considered as quotients of {\it simple} or close to
them ``classical'' Lie supergroups modulo {\it certain} maximal
parabolic subsupergroups; for the list see \cite{S2}.

This paper is an attempt to tackle the following questions: What are
the criteria for selecting the above-mentioned subgroups among other
maximal ones?  Why cosets modulo other parabolic subsupergroups are
seldom considered in Differential Geometry whereas only these
``other'' cosets are the main topic of study in analytical mechanics
of nonholonomic dynamical systems and in supergravity (\cite{Ma},
\cite{GL})?

In these questions ``super'' is beside the point, so we can very well
begin with manifolds.  The classical domains are distinguished among
symmetric spaces by the fact that the Lie algebra of the symmetry
group of any classical domain (Hermitian symmetric space) $M$ is a
simple complex Lie algebra of the form 
$$
\mathfrak{g} = \oplus_{|i|\leq 1}\; \mathfrak{g}_i;\eqno{(d=1)}
$$ 
the tangent space to $M$ at a fixed point can be identified with the
$\mathfrak{g}_{-1}$ and, on it, one can always define a Jordan algebra
structure, by fixing any element $p\in\mathfrak{g}_{1}$ and setting
$$
x\circ y= [[p, x], y]\quad\hbox{  for any } x, y\in \mathfrak{g}_{-1}. 
\eqno{(2)}
$$
Recall that a {\it Jordan algebra} is a commutative
algebra $J$ with product $\circ$ satisfying, instead of
associativity, the identity
$$
(x^2\circ y)\circ x = x^2\circ (y\circ x). \eqno{(JI)}
$$
In a very inspiring paper \cite{Mc} McCrimmon gave an account of some
applications of Jordan algebras from antiquity to nowadays, see also
refs.  in \cite{U}, \cite{Koe}.  The paper and books strengthen my
prejudice that {\it general} Jordan algebras are, bluntly speaking,
useless.  Contrariwise, {\it simple} Jordan algebras give rise to
several notions important in various problems.  For simple Jordan
algebras the so-called {\it general norm} \cite{K4} should be
nondegenerate which imposes additional constraints on the parabolic
subalgebra.  This answers the above questions but tempts one to make
use of the other coset spaces as well.

I wish to make similar use of simple Jordan {\it super}algebras,
especially infinite dimensional ones, associated with infinite
dimensional classical superdomains listed in \cite{LSV}; for
convenience I reproduce these tables.


In the '60s a remarkable correspondence between Jordan algebras and
certain $\Zee$-graded Lie algebras became explicit, cf.  \cite{T},
\cite{K1}--\cite{K2} and \cite{Koe}.  Kantor used this correspondence
to list simple Jordan algebras (over $\Cee$ and $\Ree$) by the, so
far, simplest known method.  He clarified the mysterious relation of
Jordan algebras with classical domains and actively studied certain
generalizations of Jordan algebras associated with $\Zee$-graded
simple Lie algebras of finite depth $d$ (since all of them are of the
form $\mathfrak{g}=\mathop{\oplus}\limits_{|i|\leq d}\mathfrak{g}_i$,
their {\it length} is equal to $d$).  Supersymmetry, supertwistors
etc.  are related with gradings $d>1$ almost without exceptions; so it
is interesting to find generalizations of Jordan algebras (or, rather,
useful related structures).

Following Freudental and Springer, Kantor generalized products $(2)$
to several arguments which is natural for $d>1$.  I suggest,
contrariwise, to stick to formula $(2)$, even for $d>1$, with the
minimal modification: fix $p\in\mathfrak{g}_{1}$ and for $\mathfrak{
g}_{-}=\mathop{\oplus}\limits_{i\leq 0}\mathfrak{g}_i$ set
$$
x\circ y= [[p, x], y]\quad\hbox{  for any } x, y\in
\mathfrak{g}_{-}.  \eqno{(3)}
$$
In this way we obtain noncommutative generalizations of Jordan
algebras (with unknown relations instead of (JI)) and it is
interesting to investigate what type of integrable systems are
associated with them under the Sokolov-Svinolupov's approach, cf. 
\cite{SS}, \cite{HSY}.

Kac \cite{Ka} has already applied this correspondence to 
list simple {\it finite} dimensional Jordan superalgebras. 

Tables (borrowed from \cite{LSV}) provide with a list of simple Jordan
superalgebras associated with the known in 1991 simple $\Zee$-graded
Lie superalgebras of polynomial growth (SZGLSAPGs for short), cf. 
\cite{Ka}, \cite{LSh2}, \cite{KMZ}, namely, with $\Zee$-gradings of
depth 1 of SZGLSAPGs {\it including} finite dimensional ones.  (This
is the place where \cite{Ka} contains an omission --- cf.  Kac'
exceptional Jordan superalgebra $K$ with our series $\mathfrak{sh}$
discovered by Serganova in 1983, see \cite{L}, and later rediscovered
several times.)

\subsubsection{Tits--Kantor--K\"ocher's functor $\mathfrak{kan}$ }

Let $J$ be a Jordan superalgebra and $p$ the tensor that determines
the product in $J$, i.e., $p(x, y) = x \circ y$.  To $J$, Kantor
assigns (see \cite{Ka}) a $\Zee$-graded Lie superalgebra
$\mathfrak{kan}(J)=\mathop{\oplus}\limits_{|i|\leq 1}
\mathfrak{kan}(J)_{i}$, a $\Zee$-graded Lie subalgebra in
$\mathfrak{vect} (J)$ such that (here $L_a(x)= a\circ x$)
$$
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
\mathfrak{kan}(J)_{-1}=\mathfrak{vect} (J)_{-1}, \\
\mathfrak{kan}(J)_{0}= \Span( L_{a}, [L_a, L_b]\mid  a, b \in J),\\ 
\mathfrak{kan}(J)_{1}= \Span(p, [L_a, p]\mid  \;a\in J).
\end{array}
$$

Conversely, for any $\Zee$-graded Lie superalgebra of the form
$\mathfrak{g}=\mathop{\oplus}\limits_{i\geq -1}\mathfrak{g}_i$ we
define a Jordan superalgebra structure on $\mathfrak{g}_{-1}$ if
$(\mathfrak{g}_1)_{\bar{0}} \not=0$ in the following way.  Take $p \in
(\mathfrak{g}_1)_{\bar{0}}$ and for $x, y \in \mathfrak{g}_{-1}$ set
$$
x \circ y = [[p, x], y] . \eqno{(J)}
$$

\subsubsection{Digressioin: on homological fields}
I do not know what structure is related with an arbitrary odd $p$ but
if $p$ is {\it homologic}, i.e., $[p, p]=0$, then the formula
$$
[x, y]' =[[p, x], y] \eqno{(L)}
$$
determines a Lie superalgebra structure on $\Pi(\mathfrak{g}_{-1})$. 
This structure had been first noticed, perhaps, by M.~Gerstenhaber in
'60s and rediscovered many times since then.  It seemed interesting to
describe in intrinsic terms the $p$'s which determine {\it simple} Lie
(super)algebras.  Homological vector fields were first introduced, in
connection with the problem of integration of differential equations
on supermanifolds, by V.~Shander \cite{Sha}, who gave a normal form
for the nonsingular fields.  However, Shander did not consider
singularities of the fields in that work; this was recently done by
Vaintrob in a series of articles (e.g.,\cite{V}) in which he showed
that the study of singularities of homological fields, and their
classification, turns out to be rather similar to the case of
singularities of smooth functions.  Regrettably, the answer for $p$
corresponding to the simple algebras is more trivial than expected, as
we have recently established with Grozman:

Let, first, $\mathfrak{g}=\mathfrak{gl}(n)$; denote the matrix units
by $\partial_{i}^j$, let $(\partial_{i}^j)^*=x_{j}^i$ be the dual
basis.  Then $p=\sum \xi_{j}^i\xi_{k}^j\delta_{i}^k$, where 
$\xi_{j}^i$ is the odd copy of $x_{j}^i$ and $\delta_{i}^k=
\frac{\partial}{\partial\xi_{k}^i}$.  Similarly, if
the $X_{i}$ form a basis of $\mathfrak{g}$ and $[X_{i}, X_{j}]=\sum
c_{ij}^kX_{k}$, then the operator $p\in \mathfrak{vect}(0|\dim
\mathfrak{g})$ is of the form $\frac12\sum c_{ij}^kX_{i}^*X_{j}^*
X_{k}$, where $X_{i}^*$ is the dual of $X_{i}$.

Having observed that every simple finite dimensional Lie algebra (over
$\Cee$) possesses a nondegenerate symmetric bilinear form, we see that
for such algebras $p$ is a hamiltonian vector field; to find the
corresponding generating function is easy: it is the sum of all
elements of degree 1 and weight $0$ with respect to the Cartan
subalgebra of $\mathfrak{po}(0|\dim \mathfrak{g})$.  

Generalization to Lie superalgebras is straightforward.  Still,
observe that some simple Lie superalgebras have no form at all, some
(e.g., $\mathfrak{q}(n)$) possess an {\it odd} nondegenerate symmetric
bilinear form in which case $p$ belongs to the antibracket algebra,
not to the Poisson one.

\subsection{SZGLSAPGs of depth 1 and length 1} 
All possible $\Zee$-gradings of SZGLSAPGs $\mathfrak{g}$ are listed in
\cite{Ka} for $\dim \mathfrak{g} < \infty$ and in \cite{L} for most of
the other cases.  Our job is to pick those of them which are of depth
1, in particularly, of the form $\mathop{\oplus}\limits_{|i| \leq 1}\;
\mathfrak{g}_i$, see Tables.

Albert's notation for Jordan algebras were given in accordance with
Cartan's notations for the corresponding Lie algebras.  As follows
from Serganova's classification of systems of simple roots of simple
Lie superalgebras \cite{S}, Cartan's notations are highly
inappropriate for Lie superalgebras.

\subsubsection{Matrix Jordan superalgebras}
Let $B_{m, 2n}=\left(\matrix{1_{m}& 0\cr 
0&J_{2n}}\right)$, where 
$J_{2n}=\left(\matrix{0&1_{n}\cr -1_{n}&0}\right)$. Set 
$$
\renewcommand{\arraystretch}{1.4}
\begin{array}{l}
{\tt Mat}(m|n)=\{X \in \Mat(m|n)\}, \\
{\tt Q}(n|n)=
\{X \mid [X,J_{2n}]=0\},\\ 
{\tt OSp}(m|2n)=
\{X \mid X^{st}B_{m, 2n}=B_{m, 2n}X\}, \\ 
{\tt Pe}(n|n)=\{X \mid
X^{st}J_{2n}=(-1)^{p(X)}J_{2n}X \}. 
\end{array}
$$
In the first two of these spaces the Jordan product is given by the
formula
$$
X \circ Y = XY +(-1)^{p(X)p(Y)}YX.
$$
I leave it as an excersise to figure out the formula in the other two 
cases; for the answer see \cite{Ka}.

\subsubsection{Jordan algebras from bilinear forms} Set
$$
{\tt Q}_{m|2n}={\Cee}^{m|2n}
$$ 
with a nondegenerate even symmetric 
bilinear form
$(\cdot, \cdot)$ and the product
$$
x\circ y =(e, x)y + x(e, y) -(x, y)e\eqno{(Q)}
$$
where $e \in ({\tt Q}_{m|2n})_{\bar{0}}$ satisfies $(e, e)=1$.

Set ${\tt HQ}_{m|2n}= \Pi(\Cee [p, q, \Theta])$, where $m>0$, $p=(p_1,
\ldots, p_n)$, $q=(q_1, \ldots, q_n)$, $\Theta=(\xi_1, \ldots, \xi_r,
\eta_1, \ldots, \eta_r)$ for $m=2r$, of $\Theta=(\xi, \eta, \theta)$
for $m=2r+1$ with the Jordan product defined with the help of the
symplectic form $\omega$ on the supermanifold with coordinates $p, q,
\Theta$:
$$
x\circ y=\omega (e, x)y + x \omega (e, y) -\omega (x, y)e,\eqno{(H')}
$$
where $e\in ({\tt HQ}_{m|2n})_{\bar{0}}$ satisfies $\omega (e, e)=1$.

To explicitly give the product, consider the space $\Cee [p, q,
\Theta, \alpha, \beta]$ with two extra odd indeterminates and the
Poisson bracket such that $p$ and $q$, $\xi$ and $\eta$, and $\alpha,
\beta$ are dual.  Setting $\deg \alpha=-\deg \beta=-1$ the degrees of
the other indeterminates being $0$, we obtain the $\Zee$-grading of
the Poisson algebra, and its quotient modulo center, of the form
$(d=1)$.  On $\mathfrak{g}_{-1}= \Cee [p, q, \Theta]\alpha$, define
the product \footnotesize
$$
H_{f\alpha}\circ H_{g\alpha} = \{\{H_{\beta},\,  H_{f\alpha}\}, \, 
H_{g\alpha}\}\\
= (-1)^{p(f)}H_{\{f, g\}\alpha}.
$$ \normalsize
In other words, on the superspace of functions with shifted parity, we
set
$$
f\circ g = (-1)^{p(f)+1} \{f, g\}. \eqno{(H,K)}
$$

\subsubsection{Exceptional Jordan superalgebras} There are two of them
associated with the gradings of $\mathfrak{osp}(4|2; \alpha)$ and
$\mathfrak{ab}_3$ from Table 1 and the corresponding loops.

\subsubsection{Stringy Jordan superalgebras} These are obtained from
$\mathfrak{k}^L(1|n)$ for $n>2$ and $\mathfrak{k}^M(1|n)$ (see
\cite{GLS1}) for $n>3$ by formula $(J)$ with the grading from Table 1. 
They will be denoted, respectively, by
$$
{\tt K{}^LQ}_{1|n} \cong \Pi(\Cee [t^{-1}, t, \theta_1, \dots ,
\theta_n])
$$
$$
{\tt K{}^MQ}_{1|n} \cong
\Pi(\Cee[t^{-1}, t, \theta_1, \dots , \sqrt{t} \theta_n]).
$$
The product is given by formula $(H,K)$.

 
\subsubsection{Loop Jordan superalgebras} For a finite-dimensional
Jordan superalgebra $J$ denote by $J^{(1)}= J \otimes
\Cee[t^{-1}, t]$ the loops with values in $J$ and point-wise product.

\subsubsection{Twisted loop Jordan superalgebras} These are associated 
by formula $(J)$ with the Lie superalgebras from Table 2.

	
\section{Cross ratios}	
In \cite{K3} Kantor generalized the cross ratio of four points on
$\Pee^1$ to most of the quotients $G/P$, where $G$ is a simple Lie
group and $P$ is its parabolic subgroup.  The Lie algebra
$\mathfrak{g}=\hbox{ Lie}(G)$ in these cases is of the form
$\mathfrak{g}=\mathop{\oplus}\limits_{|i| \leq d}\; \mathfrak{g}_i$. 
I do not know any paper referring to \cite{K3}, so Kantor's studies
drew no attention at all.  His constructions, however, naturally
appear in supersetting \cite{Mn}; this prompts me to try to decipher a
part of \cite{K3}.  I will consider here the simplest case, when $d=1$
and the corresponding Jordan algebra is simple.  In this case one can
generalize the cross ratio from $\Pee^1={\tt Gr}_{1}^2$ to a
collection of ${\tt Gl}(2m|2n)$-invariants of four points on ${\tt
Gr}_{m|n}^{2m|2n}$. First, consider \footnotesize
$$
(A, B, C, D)=(A-B)(C-B)^{-1}(C-D)(A-D)^{-1}.\eqno{(CR)} 
$$
\normalsize Let $mn=0$.  Now, replace the rhs of (CR) --- call it $X$
--- with $\det(X-\lambda E)$. The collection of all coefficients of
the powers of $\lambda$ is the analog of the cross ratio.  

By dimension considerations these are all the invariants of four
points for the general, orthogonal and Lagrangian grassmannians.

For their super counterparts we take the Berezinian (superdeterminant)
and the amount of {\it polynomially independent} invariants is
infinite, cf. \cite{Ser}. If, however, we consider rational 
dependence, which is natural in super setting, the coefficients of 
the first $n+m$ powers of $\lambda$ generate the algebra of 
invariants and is a natural candidate for the cross ratio.

On ${\tt Q}(n|n)$, we should take the queerdeterminant, $\qet$, instead of $\det$; the 
collection obtained is finite.

For loop Jordan superalgebras we consider matrix-valued functions and $\det(X-\lambda
E)$ returns a collection of functions, rather than numbers.  

I do not know the complete cross ratio for Jordan superalgebras
related to quadrics and do not know at all what are they for twisted
loops and in stringy cases, most interesting to me.  One invariant is
obvious (but there should be several if $\dim {\tt Q}>1$): given the form
$(\cdot, \cdot)$, or the symplectic form $\omega$ in the curved case,
set (for the $\Lambda$-points (see \cite{D}) of the Jordan algebra
$J$, i.e., for $A, B, C, D\in (J\otimes \Lambda)_{\bar 0}$
\footnotesize
$$
(A, B, C, D)=\frac{(A-B, A-B)}{(C-B, C-B)}\frac{(C-D, C-D)}{(A-D,
A-D)}.\eqno{(CRQ)}
$$
\normalsize
For curved quadrics, take $\omega(H_{A-B}, H_{A-B})$ instead of $(A-B,
A-B)$, etc.

Perhaps, other invariants (in non-super case) can be dug out from
Reichstein's results.

I almost forgot to add refs.  \cite{EHPSW} that studies four-point
functions in $N=2$ superconformal field theories and \cite{M}, where
matrix cross ratio is applied to Riccati equation; together with
\cite{Mn} they provide a wide setting for applications of our cross
ratios.

\section{Tables} Everywhere we assume the notational
conventions of \cite{Ser} and definitions adopted there.

In Table 1 we say that the homogeneous superspace $G/P$, where $G$ is a
simple Lie supergroup, $P$ its parabolic subsupergroup corresponding to
several omitted generators of a Borel subalgebra (description of these
generators can be found in \cite{GLP}), of {\it depth} $d$ and {\it
length} $l$ if such are the depth and length of $\mathfrak{g}={\rm Lie}(G)$ in the
$\Zee$-grading compatible with that of ${\rm Lie}(P)$.  Note that all
superspaces of Table 1 possess an hermitian structure (hence are of
depth 1) except $PeGr$ (no hermitian structure), $PeQ$ (no hermitian structure,
length 2), $CGr_{0, k}^{0, n}$ and $SCGr_{0, k}^{0, n}$ (no hermitian structure,
lengths $n-k$ and, resp.,  $n-k-1$).  

Let $\mathfrak{s}(\mathfrak{g})$ be the traceless part of
$\mathfrak{g}$ and $\mathfrak{p}(\mathfrak{g})=\mathfrak{g}/{\rm
center}$; let $\mathfrak{g}_{\varphi}^{(m)}$ be the stationary
subalgebra of the loop algebra with values in $\mathfrak{g}$ singled
out by the degree $m$ automorphism $\varphi$ of $\mathfrak{g}$;for
$G=\mathfrak{g}_{\varphi}^{(m)}$ with the $\Zee$-grading of type
$(d=1)$ the last column of Table 1 contains $G_{0}$; the map $-\st$,
``minus supertransposition'', sends $X$ to $-X^{st}$, the map $\Pi$
sends $\left(\matrix{a& b\cr c&d}\right)$ to $\left(\matrix{d& c\cr
b&a}\right)$, and $\delta_{x}$ sends $\left(\matrix{a& b\cr c&d}\right)$
to $\left(\matrix{a& xb\cr xc&d}\right)$; the automorphism $A$ of 
$\mathfrak{po}$ is defined on monomials $f(\theta)$ as $\id$ if 
$\frac{\partial f}{\partial \theta_1}=0$ 
and otherwise; $irr(\ldots)$ is any of the
two irrducible components; $LGr$ and $OGr$ stand for the Lagrangian
and orthogonal Grassmannian, respectively; the dual domain is endowed
with an asterisk as a left superscript.

Table 2: for the lack of space I give an interpretation of the
supergrassmannians here, linewise: the
supergrassmannian of $p|q$-dimensional subsuperspaces in $\Cee^{m|n}$
and same for $n=m$, $p=q$; superquadric of $1|0$-dimensional isotropic
(wrt a non-degenerate even form) lines in $\Cee^{m|n}$; ortolagrangian
supergrassmannian; queergrassmannian; ``odd'' superquadric (wrt a
non-degenerate even form) of $1|0$-lines in $\Cee^{n|n}$;
odd-lagrangian supergrassmannian; curved supergrassmannian of
$0|1$-dimensional subsupermanifolds in $\Cee^{0|n}$; curved
superquadric; two exceptions.

\begin{table*}[htb]
\caption{Gradings of twisted loop 
(super)algebras corresponding to hermitian superdomains.}
\label{table:1}
\tiny
$$
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{cccc}
\hline
$\mathfrak{g}_{\varphi}^{(m)}$ &$\varphi$ & grading elements from
$\mathfrak{h}$& $(\mathfrak{g}_{\varphi}^{(m)})_0$ \cr
\hline
$\mathfrak{sl}(2m|2n)^{(2)}$&$(-st)\circ \; {\rm Ad}\; {\rm diag}(\Pi
_{2m}, J_{2n})$ &${\rm diag}(1_m, -1_m, 1_n,
-1_n)$&$\mathfrak{sl}(m|n)^{(1)}$\cr
$\mathfrak{sl}(2m)^{(2)}$&$(-t)\circ {\rm Ad}
(\Pi_{2m})$&&$\mathfrak{sl}(m)^{(1)}$\cr $\mathfrak{sl}(2n)^{(2)}
$&$(t)\circ {\rm Ad} (J _{2n})$&&$\mathfrak{sl}(n)^{(1)}$\cr
$\mathfrak{sl}(n|n)^{(2)}$&$\Pi$&${\rm diag}(1_p, 0_{n-p}, 1_p,
0_{n-p})$& $\mathfrak{s}(\mathfrak{gl}(p|p)^{(2)}_{\Pi}\oplus
\mathfrak{gl}(n-p|n-p)^{(2)}_{\Pi})$\cr
$\mathfrak{sl}(n|n)^{(2)}$&$\Pi\circ (-st)$&${\rm diag}(1_p, -1_{n-p},
-1_p, 1_{n-p})$& $\mathfrak{s}(\mathfrak{gl}(p|p)^{(2)}_{\Pi\circ
(-st)}\oplus \mathfrak{gl}(n-p|n-p)^{(2)}_{\Pi\circ (-st)})$\cr
\hline
$\mathfrak{osp}(2m|2n)^{(2)}$&$\varphi_{m, n}{\rm Ad}\; {\rm
diag}(1_{2m-1}, 1, 1_{2n})$ &${\rm diag} (J_2, O_{2(m+n-1)}) $&$
(\mathfrak{cosp}(2m-2|2n))^{(1)}_{\varphi_{m-1, n}}$\cr
$\mathfrak{o}(2m)^{(2)}$& &&$(\mathfrak{co}(2m-2))^{(1)}$\cr
\hline
$\mathfrak{psq}(2n)^{(4)}$&$ (-st)\circ \delta_{\sqrt{-1}}$&${\rm diag}(J_{2n},
J_{2n})$& $\mathfrak{psq}(n)^{(2)}_{\delta_{-1}}$\cr
$\mathfrak{sh}(2n)^{(2)}$&$ A$&& $ (\mathfrak{sh}
(2n-2)\oplus \Lambda(2n-2))^{(2)}_A $\cr 
$\mathfrak{psq}(n)^{(2)}$&$
\delta_{-1}$&${\rm diag}(1_p, 0_{n-p}, 1_p, 0_{n-p})$& $\mathfrak{ps}
(\mathfrak{q}(p)^{(2)}_{\delta _{-1}}\oplus
\mathfrak{q}(n-p)^{(2)}_{\delta _{-1}})$\cr
\hline
\end{tabular}
$$

\end{table*}

\begin{table*}[htb]
\caption{Classical superspaces of depth 1.}
\label{table:2}
\tiny
$$
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{ccccc}
\hline
$\mathfrak{g} $&$ \mathfrak{g}_0 $&$
\mathfrak{g}_{-1}$&Underlying&Name of the\cr 
&&&domain&superdomain\cr\hline
$\mathfrak{sl}(m|n)$&$ \mathfrak{s}(\mathfrak{gl}(p|q)\oplus
\mathfrak{gl}(m-p|n-q))$&$id\otimes id^*$& $Gr_p^m\times
Gr_q^n$&$Gr_{p, q}^{m, n}$\cr
$\mathfrak{psl}(m|m)$&$\mathfrak{ps}(\mathfrak{gl}(p| p)\oplus
\mathfrak{gl}(m-p|m-q))$&$ id\otimes id^*$&$G_p^m\times Gr_p^m
$&$Gr_{p, p}^{m, m}$\cr 
\hline
$\mathfrak{osp}(m| 2n)$&$ \mathfrak{cosp}(m-2| 2n)$& $id$& $Q_{m-2}$&
$Q_{m-2, n}$\cr 
\hline
$\mathfrak{osp}(2m| 2n)$& $\mathfrak{gl}(m|n)$ & $\Lambda^2$&
 ${}^*OGr_m\times LGr_n
$&$OLGr_{m, n}$ \cr 
\hline
$\mathfrak{sq}(n)$&$ \mathfrak{s}(\mathfrak{q}(p) \oplus
\mathfrak{q}(n-p))$&$irr(id\otimes id*)$& $Gr_p^n$&$ QGr_p^n$\cr
$\mathfrak{psq}(n)$&$\mathfrak{ps}(\mathfrak{q}(p) \oplus
\mathfrak{q}(n-p))$& &&\cr
\hline
$\mathfrak{pe}(n)$&$\mathfrak{cpe}(n-1)$&$ id$& $\Cee P^{n-1}$&
$PeQ_{n-1}$ \cr 
$\mathfrak{spe}(n)$&$\mathfrak{cspe}(n-1)$& &&\cr 
\hline
$\mathfrak{pe}(n)$&$\mathfrak{gl}(p|n-p)$&$\Pi(S^2 (id))$& $Gr_p^n$&
$PeGr_p^n$\cr 
$(\mathfrak{spe}(n))$&$ (\mathfrak{sl}(p|n-p))$& or $\Pi
(\Lambda^2 (id))$&&\cr
\hline
\hline
$\mathfrak{vect}(0| n)$&$\mathfrak{vect}(0| n-k)\oplus
\mathfrak{gl}(k; \Lambda (n-k))$ &$\Lambda (k)\otimes \Pi
(id)$&&$CGr_{0, k}^{0, n}$\cr 
$\mathfrak{svect}(0|n)$&$\mathfrak{vect}(0| n-k)\oplus \mathfrak{sl}(k; \Lambda (n-k))$&
$\Pi (Vol)$ if $k=1$ &--&$SCGr_{0,
k}^{0, n}$\cr 
\hline
$\mathfrak{h}(0| m)$&$\mathfrak{h}(0|m-2)\oplus \Lambda (m-2)\cdot
z$&& --&$CQ_{m-2, 0}$ \cr $\mathfrak{sh}(m)$&$\mathfrak{sh}(m-2)\oplus
\Lambda (m-2)\cdot z$ &$ \Pi (id)$&&\cr 
\hline
\hline
$\mathfrak{osp}(4|2;\alpha)$&$\mathfrak{cosp}(2|2)\simeq
(\mathfrak{gl}(2|1))$ & $id$& $\Cee \Pee^1\times \Cee \Pee^1$ 
&$E_{\alpha}$\cr
\hline
$\mathfrak{ab}(3)$&$\mathfrak{cosp}(2|4)$&$ L_{3\varepsilon_{1}}$ & 
$\Cee \Pee^1\times Q_5$ &$AB(3)$\cr \hline
\end{tabular}
$$\\[2pt]
The curved superquadric has infinite dimensional ``stringy'' 
counterpart with $\mathfrak{h}(0| m)$ replaced with the centerless 
$N$-extended Neveu-Schwarz algebra
$\mathfrak{k}^L(1|N)$ or Ramond algebra $\mathfrak{k}^M(1|N)$.
\end{table*}



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D.~Leites, V.~Serganova and G.~Vinel. In: C. Bartocci, U. Bruzzo and R.
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Yu.~Manin, {\em Topics in noncommutative geometry}, Princeton Univ. 
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\bibitem{L}
D.~Leites (ed.), {\em Seminar on supermanifolds}, vv.  1--34,
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J.~Bernstein and D.~Leites, Functional Anal. 
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\bibitem{Sh}
V.~Shander, Funct.  Anal.  Appl.  22 (1988) no.  1, 80

\bibitem{R}
A.~Rudakov, Math.  USSR Izvestiya, v.  38 (1974) n.  4, 835

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A.~Shvarts, Nuclear Phys.  B 171 (1980), no.  1-2, 154; A.~Gaiduk,
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\bibitem{BL3}
J.~Bernstein and D.~Leites, Selecta Math.  Soviet.  1 (1981) no.  2,
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\bibitem{BL2}
J.~Bernstein and D.~Leites, Functional Analysis and Its Applications
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O.~Khudaverdian, Comm.  Math.  Phys.  198 (1998), no.  3, 591

\bibitem{LKV}
D.~Leites, Yu.~Kochetkov and A.~Vaintrob, In: S.~Andima et.  al (eds.) 
{\em General topology and its applications}, LN in pure and applied
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\bibitem{LSh2}
D.~Leites and I.~Shchepochkina, Theor.  and Math.  Physics, v.  126
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\bibitem{St} 
H.~Strade and R.~Farnsteiner, Modular Lie algebras and their
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P.~Grozman, D.~Leites and I.~Shchepochkina, hep-th 9702120; Acta
Mathematica Vietnamica, v.  26 (2001) no.  1, 27

\bibitem{S2}
V.~Serganova, Functional Anal. Appl. 17 (1983), no. 3, 200

\bibitem{Ma}
Yu.~Manin, Gauge field theory and complex geometry.  Second edition. 
Springer-Verlag, Berlin, 1997

\bibitem{GL}
P.~Grozman and D.~Leites In: J.~Wess, E.~Ivanov (eds.), {\em
Supersymmetries and quantum symmetries}, (SQS'97, 22--26 July, 1997),
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\bibitem{Mc}
K.~McCrimmon, Bull.  Amer.  Math.  Soc., v.84 (1978) no.  4, 612

\bibitem{U}
H.~Upmeier,Jordan algebras in analysis, operator theory, and quantum
mechanics.  CBMS Regional Conference Series in Mathematics, 67.  AMS,
Providence, RI, 1987.

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M.~Koecher (edited by A.~Krieg and S.~Walcher) {\em The Minnesota
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I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
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\bibitem{T}
J.~Tits, Indag. Math. v. 24 (1962) 530

\bibitem{K1}
I.~Kantor, Soviet. Math. Doklady v. 5 (1964) 1404

\bibitem{K2}
I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
analysis.  v. 13 (1966), 310 (in Russian)

\bibitem{SS}
S.~Svinolupov and V.~Sokolov, Theoret.  and Math.  Phys.  108 (1996)
no.  3, 1160; id., Acta Appl.  Math.  41 (1995) no.  1--3, 323; id.,
Math.  Notes 53 (1993) no.  1--2, 201

\bibitem{HSY}
I.~Habibullin, V.~Sokolov and R.~Yamilov.  In: Nonlinear physics:
theory and experiment (Lecce, 1995) World Sci., River Edge, NJ, 1996,
139

\bibitem{Ka}
V.~Kac, Commun.  Alg. 5 (13) (1977) 1375; L.~Hogben and V.~Kac,
Erratum Comm.  Algebra 11 (1983) no.  10, 1155

\bibitem{KMZ}
V.~Kac, C.~Martinez and E.~Zelmanov, Graded simple Jordan
superalgebras of growth one.  Mem.  Amer.  Math.  Soc.  150 (2001)
no.  711

\bibitem{Sha}
V.~Shander, Functional Anal. Appl. 14 (1980), no. 2, 160

\bibitem{V}
A.~Vaintrob, J. Math.  Sci.  82 (1996), no.  6, 3865

\bibitem{S}
V.~Serganova, Comm.  Algebra 24 (1996) no.  13, 4281

\bibitem{K3}
I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
analysis.  v. 17 (1974) 250; v. 18 (1975) 234 (in Russian)


\bibitem{GLP}
P.~Grozman, D.~Leites and E.~Poletaeva,
In: E.~Ivanov et.  al.  (eds.)  {\em Supersymmetries and Quantum
Symmetries} (SQS'99, 27--31 July, 1999), Dubna, JINR, 2000, 387

\bibitem{EHPSW}
B.~Eden, P.~Howe, A.~Pickering,
E.~Sokatchev, P.~West, Nuclear Phys.  B 581 (2000), no.  1-2, 523--558

\bibitem{M}
M.~Zelikin,{\em Control theory and optimization.  I.
Homogeneous spaces and the Riccati equation in the calculus of
variations}. Encyclopaedia of Mathematical
Sciences, 86.  Springer-Verlag, Berlin, 2000.  xii+284 pp.

\end{thebibliography}

\end{document}

math0202194.extracted_bib
\begin{thebibliography}{99}

\bibitem{D}
Deligne P. et al (eds.)  {\em Quantum fields and strings: a course for
mathematicians}.  Vol.  1, 2.  AMS, Providence, RI, 1999

\bibitem{Sch}
I.~Shchepochkina,  Represent.  Theory (electronic
journal of AMS), v.  3, 1999, 3 (1999) 373

\bibitem{Ser}
A.~Sergeev, Michigan Math.  J. 49 (2001) no.  1, 113 (math.RT/9810113,
math.RT/9904079)


\bibitem{GLS1}
P.~Grozman,  D.~Leites and  I.~Shchepochkina, In: M.~Olshanetsky and 
A.~Vainshtein (eds.) Multiple facets of quantization and supersymmetry
Michael Marinov Memorial Volume, World Sci., to appear [ESI preprint 
1111 (2001)]

\bibitem{LSV} 
D.~Leites, V.~Serganova and G.~Vinel. In: C. Bartocci, U. Bruzzo and R.
Cianci (eds.)  {\em Differential Geometric Methods in Theoretical
Physics} Proc.  DGM-XIX, 1990, Springer, LN Phys.  375 (1991) 286


\bibitem{Mn}
Yu.~Manin, {\em Topics in noncommutative geometry}, Princeton Univ. 
Press, Princeton, NJ, 1991 

\bibitem{LSh1} 
D.~Leites and I.~Shchepochkina  Classification of simple Lie
superalgebras of vector fields, to appear

\bibitem{L}
D.~Leites (ed.), {\em Seminar on supermanifolds}, vv.  1--34,
1986--90, Reports of Dept.  of Math.  University of Stockholm, 2100
pp.

\bibitem{BL1}
J.~Bernstein and D.~Leites, Functional Anal. 
Appl.  11 (1977) no.  1, 45

\bibitem{Sh}
V.~Shander, Funct.  Anal.  Appl.  22 (1988) no.  1, 80

\bibitem{R}
A.~Rudakov, Math.  USSR Izvestiya, v.  38 (1974) n.  4, 835

\bibitem{AS}
A.~Shvarts, Nuclear Phys.  B 171 (1980), no.  1-2, 154; A.~Gaiduk,
V.~Romanov and A.~Shvarts, Comm.  Math.  Phys.  79 (1981), no.  4,
507; A.~Gaiduk, O.~Khudaverdyan and A.~ Shvarts, Theoret.  and Math. 
Phys.  52 (1982), no.  3, 862

\bibitem{BL3}
J.~Bernstein and D.~Leites, Selecta Math.  Soviet.  1 (1981) no.  2,
143

\bibitem{BL2}
J.~Bernstein and D.~Leites, Functional Analysis and Its Applications
11 (1977) 219

\bibitem{H}
O.~Khudaverdian, Comm.  Math.  Phys.  198 (1998), no.  3, 591

\bibitem{LKV}
D.~Leites, Yu.~Kochetkov and A.~Vaintrob, In: S.~Andima et.  al (eds.) 
{\em General topology and its applications}, LN in pure and applied
math, v.  134, Marcel Decker, NY (1991) 217

\bibitem{LSh2}
D.~Leites and I.~Shchepochkina, Theor.  and Math.  Physics, v.  126
(2000) no.  3, 339

\bibitem{St} 
H.~Strade and R.~Farnsteiner, Modular Lie algebras and their
representations.  Marcel Dekker, NY, 1988

\bibitem{GLS2}
P.~Grozman, D.~Leites and I.~Shchepochkina, hep-th 9702120; Acta
Mathematica Vietnamica, v.  26 (2001) no.  1, 27

\bibitem{S2}
V.~Serganova, Functional Anal. Appl. 17 (1983), no. 3, 200

\bibitem{Ma}
Yu.~Manin, Gauge field theory and complex geometry.  Second edition. 
Springer-Verlag, Berlin, 1997

\bibitem{GL}
P.~Grozman and D.~Leites In: J.~Wess, E.~Ivanov (eds.), {\em
Supersymmetries and quantum symmetries}, (SQS'97, 22--26 July, 1997),
Lecture Notes in Phys., 524, 1999, 58

\bibitem{Mc}
K.~McCrimmon, Bull.  Amer.  Math.  Soc., v.84 (1978) no.  4, 612

\bibitem{U}
H.~Upmeier,Jordan algebras in analysis, operator theory, and quantum
mechanics.  CBMS Regional Conference Series in Mathematics, 67.  AMS,
Providence, RI, 1987.

\bibitem{Koe}
M.~Koecher (edited by A.~Krieg and S.~Walcher) {\em The Minnesota
Notes on Jordan Algebras and Their Applications}, Lect.  Notes Math.,
1710, Springer, 1999

\bibitem{K4}
I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
analysis.  v. 14 (1968), 114 (in Russian)

\bibitem{T}
J.~Tits, Indag. Math. v. 24 (1962) 530

\bibitem{K1}
I.~Kantor, Soviet. Math. Doklady v. 5 (1964) 1404

\bibitem{K2}
I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
analysis.  v. 13 (1966), 310 (in Russian)

\bibitem{SS}
S.~Svinolupov and V.~Sokolov, Theoret.  and Math.  Phys.  108 (1996)
no.  3, 1160; id., Acta Appl.  Math.  41 (1995) no.  1--3, 323; id.,
Math.  Notes 53 (1993) no.  1--2, 201

\bibitem{HSY}
I.~Habibullin, V.~Sokolov and R.~Yamilov.  In: Nonlinear physics:
theory and experiment (Lecce, 1995) World Sci., River Edge, NJ, 1996,
139

\bibitem{Ka}
V.~Kac, Commun.  Alg. 5 (13) (1977) 1375; L.~Hogben and V.~Kac,
Erratum Comm.  Algebra 11 (1983) no.  10, 1155

\bibitem{KMZ}
V.~Kac, C.~Martinez and E.~Zelmanov, Graded simple Jordan
superalgebras of growth one.  Mem.  Amer.  Math.  Soc.  150 (2001)
no.  711

\bibitem{Sha}
V.~Shander, Functional Anal. Appl. 14 (1980), no. 2, 160

\bibitem{V}
A.~Vaintrob, J. Math.  Sci.  82 (1996), no.  6, 3865

\bibitem{S}
V.~Serganova, Comm.  Algebra 24 (1996) no.  13, 4281

\bibitem{K3}
I.~Kantor, Proc.  of P.~K.~Rashevsky's seminar on vector and tensor
analysis.  v. 17 (1974) 250; v. 18 (1975) 234 (in Russian)


\bibitem{GLP}
P.~Grozman, D.~Leites and E.~Poletaeva,
In: E.~Ivanov et.  al.  (eds.)  {\em Supersymmetries and Quantum
Symmetries} (SQS'99, 27--31 July, 1999), Dubna, JINR, 2000, 387

\bibitem{EHPSW}
B.~Eden, P.~Howe, A.~Pickering,
E.~Sokatchev, P.~West, Nuclear Phys.  B 581 (2000), no.  1-2, 523--558

\bibitem{M}
M.~Zelikin,{\em Control theory and optimization.  I.
Homogeneous spaces and the Riccati equation in the calculus of
variations}. Encyclopaedia of Mathematical
Sciences, 86.  Springer-Verlag, Berlin, 2000.  xii+284 pp.

\end{thebibliography}