On the Cohomology of the Lie Algebra Arising from the Lower Central Series of a $p$-Group.
Justin Mauger
We study the cohomology $H^*(A)=\Ext_A^*(k,k)$ of a locally finite, connected, cocommutative Hopf algebra $A$ over $k=\f$. Specifically, we are interested in those algebras $A$ for which $H^*(A)$ is generated as an algebra by $H^1(A)$ and $H^2(A)$. We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras $F\ra A\ra B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be semi-Koszul. Special attention is given to the case in which $A$ is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central series of a $p$-group. We show that the algebras arising in this way from extensions by $\z$ of an abelian $p$-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 $p$-groups, and it is shown that these are all semi-Koszul for $p\geq 5$.
math0303327.tex
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\begin{document}

\author{Justin Mauger}
\address{Whittier College\\
Whittier, CA 90608}
 \email{jmauger@whittier.edu}
\keywords{Koszul algebras, cohomology of algebras, Hopf algebras}
\subjclass[2000]{Primary 16E40 ; Secondary 16S30, 16S37}
%16W30 Coalgebras, bialgebras, Hopf algebras
%16S30 Universal enveloping algebras of Lie algebras
%16S37 Quadratic and Koszul algebras
%16E40 (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)

\date{\today}

\title{On the Cohomology of the Lie Algebra Arising from the Lower Central Series of a $p$-Group.}
%}arising from a filtration on p-groups.

\begin{abstract}
We study the cohomology $H^*(A)=\Ext_A^*(k,k)$ of a locally finite, connected,
cocommutative Hopf algebra $A$ over $k=\f$. Specifically, we are interested in those
algebras $A$ for which $H^*(A)$ is generated as an algebra by $H^1(A)$ and $H^2(A)$.  We
shall call such algebras \emph{semi-Koszul}.  Given a central extension of Hopf algebras
$F\ra A\ra B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg
spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be
semi-Koszul.  Special attention is given to the case in which $A$ is the restricted
universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central
series of a $p$-group.  We show that the algebras arising in this way from extensions by
$\z$ of an abelian $p$-group are semi-Koszul.  Explicit calculations are carried out for
algebras arising from rank 2 $p$-groups, and it is shown that these are all semi-Koszul
for $p\geq 5$.
\end{abstract}


\maketitle

\section{Introduction}

A \emph{quadratic} algebra over a field $k$ is an associative algebra of the form
$A=T(V)\big/(R)$ where $(R)$ is the two-sided ideal of relations generated by some
$R\subset V\ot V$, and $V$ is a locally finite graded vector space.
%Priddy calls this a homogeneous pre-Koszul algebra.%He constructed a Koszul resolution.
In \cite{Priddy}, Priddy defines a quadratic algebra $A$ to be a \emph{Koszul algebra} if
$H^*(A)=Ext_A^*(k,k)$ is generated as an algebra by $Ext_A^1(k,k)$. Basic examples
include tensor, polynomial, and exterior algebras.
%The restricted universal enveloping
%algebra $V(L)$ of a graded Lie algebra $L$ over $\mathbb{F}_2$ is another example. NOT QUADRATIC

When $A$ is Koszul, Priddy constructs projective resolutions over $A$ which are ``small''
subcomplexes of the bar resolution. He also shows that if $H^*(A)$ is itself Koszul, then
$A\simeq H^*(H^*(A))$ (Koszul duality). Equivalent definitions of Koszulity include:
\begin{enumerate}
\item A certain complex associated to $A$, the \emph{Koszul} complex, is acyclic
\cite[Theorem 1.2]{Lofwall}.
%\item A satisfies the ``Fr\"oberg formula'' $P_A(x,y)H_A(-x,y)=1$, where $P$ is
%the Poincare polynomial of a graded or bigraded algebra \cite{Lofwall}
%\item A has a distributive lattice of ideals, in the sense of Backelin \cite{Backelin}.
\item There exists a graded projective resolution
$$\dots\rightarrow P^2\rightarrow P^1\rightarrow P^0\rightarrow k\rightarrow 0$$
of $k$ such that $P^i$ is generated by its component of degree $i$, $P^i=A\cdot P^i_i$
\cite[Definition 1.2.1]{BGS}. Equivalently, $\Ext_A^{i,j}(k,k)=0$ if $i\neq j$.
\end{enumerate}

Now assume $A=T(V)/(R)$ is a more general algebra which is not necessarily quadratic,
i.e. $R\subset \sum_{i\geq 1} V^{\ot i}$. We are interested in the case when $H^*(A)$ is
generated as an algebra by $H^1(A)$ \emph{and} $H^2(A)$. Following Priddy, we shall call
such algebras \emph{semi-Koszul}. The quintessential example is the truncated polynomial
algebra $\f[x]/(x^{p})$, whose cohomology is generated by a one-dimensional exterior
class $z$ and a two dimensional polynomial class $u$. In this example, $u=-\bp (z)$. In
general, it will often be the case that the generators from $H^2$ are related to those of
$H^1$ by some algebraic Bockstein operation or Massey product. Since we allow $A$ to not
be quadratic, none of the results cited above for Koszul algebras necessarily hold for
semi-Koszul algebras.
%In fact, they are defined somewhat more generally, but it is shown in lofwall and bgs
%that any koszul algebra is quadratic. so that knocks from consideration many of our algebras.
It is unknown at this time whether there are any interesting relationships between $A$
and $H^*(H^*A)$, or anything like a semi-Koszul resolution.

We will focus on locally finite, connected, cocommutative Hopf algebras over $k=\f$.
Consequently, we will be able to use algebraic Steenrod operations. In Section \ref{main
theorem chapter}, we analyze extensions of Hopf algebras $F\ra A\ra B$, where
$F=\f[x]/(x^p)$ is central in $A$. There is a first quadrant spectral sequence converging
to $H^*(A)$ with $E_2^{**}\simeq H^*(B)\ot H^*(F)$.  The behavior of this spectral
sequence is determined by the differential $d_2(1\ot z)=\mu\in H^2(B)$, where $z\in
H^1(F)$. We prove in Theorem \ref{main theorem} that $A$ is semi-Koszul provided that $B$
is semi-Koszul, $\bp(\mu)=0$ and the annihilator ideal of $\mu$ is either zero or
generated in dimension one.

Section \ref{filtration} delves into Hopf algebras arising from $p$-groups. For a finite
$p$-group $G$, the mod-$p$ lower central series filtration $\{\Gamma^i G\}$ has as
associated graded object a restricted Lie algebra $\L G$. The universal restricted
enveloping algebra $\VL G$ is isomorphic to the associated graded algebra
%$E^{\scriptstyle 0\, } \f G$ \ $E^{\scriptscriptstyle 0\, } \f G$ \ $E^{\scriptstyle 0}(\f G)$
$\eo (\f G)$  of the group ring $\f G$ with respect to the augmentation ideal filtration
\cite{Quillen}. Working with Lie algebras is helpful because many of the algebraic
Steenrod operations vanish there, in particular $\pti{0}$ \cite{Priddy-primary}. In
Section \ref{discussion}, we explore central extensions of $p$-groups $H\ra G\ra Q$,
where $H\simeq\z$.  Using Theorem \ref{main theorem}, we show that $\VL G$ is semi-Koszul
if $Q$ is abelian (Theorem \ref{base is abelian}). In Section \ref{vistas}, we prove a
corresponding statement for Hopf algebras which do not necessarily arise from $p$-groups.
Given a central extension of Hopf algebras $F\ra A\ra B$ with $F= \f[x]/(x^p)$, we show
that $\eo A$ is semi-Koszul provided that $B$ is commutative.

A rank two $p$-group is a finite $p$-group whose largest elementary abelian subgroup has
rank two. In Section \ref{rank 2} we show that $\VL G$ is semi-Koszul for all rank two
$p$-groups $G$ with $p\geq 5$.  One would hope to show that this holds for any finite
$p$-group, but such is not the case.  A counterexample is given in \ref{counterexample}.
%These are tractable because they have two "basic generators".


Section \ref{algebraic preliminaries} introduces notation and covers the necessary
algebraic preliminaries such as algebraic Steenrod operations and the Cartan-Eilenberg
spectral sequence associated with a central extension of algebras. We relegate to the
Appendix definitions of the cobar complex $C^*(A)$, Massey products in $H^*(A)$ and
compuations in the cohomology of $\f[x]\big/(x^{p^n})$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Algebraic Preliminaries}\label{algebraic preliminaries}

%\subsection{Homological Algebra}\label{homological algebra}

In this section we introduce notation and conventions.  For more details on the
cohomology of algebras, the reader is referred to Adams \cite[Chapter 2]{Adams}.  We also
summarize algebraic Steenrod operations, the Cartan-Eilenberg spectral sequence, and Lie
algebras.

Let $k=\f$ be the field of $p$ elements. By an \emph{algebra} $A$ we mean a
non-negatively graded, associative $k$-algebra with product $\mu:A\ot A\ra A$, unit $\eta
: k\ra A$ and augmentation $\epsilon : A\ra k$. We assume $A$ is locally finite, that is
each $A_n$ is finite dimensional over $k$. We also assume $A$ is \emph{connected}, i.e.
$A_0=k$. Let $I=I(A)=\ker\epsilon$ denote the augmentation ideal. For $A$ bigraded
algebra

Tensor products are taken over $k$, unless otherwise noted. The tensor product of two
algebras $A$ and $B$ is defined by $(A\ot B)_n=$ $\sum_{i+j=n}A_i\ot B_j$, with
multiplication $(a\ot b)(x\ot y)=\sign{\deg b\, \deg x}ax\ot by$.  If the algebras in
question are bigraded, we define $(A\ot B)_{n,m}= \sum_{\genfrac{}{}{0pt}{}{i+s=n}{j+t=m}}   A_{i,j}\ot
B_{s,t}$, and the total degree is used in the multiplication formula. For $x\in A_{s,t}$,
$x$ has \emph{homological} degree (or dimension) $s$, \emph{internal} degree $t$ and
\emph{total} degree $s+t$. By \emph{commutativity} of $A$ we shall mean
\emph{graded-commutativity}, also called \emph{anticommutativity}: $ab=(-1)^{\deg a\,\deg
b}ba$.

A \emph{Hopf algebra} is an algebra with \emph{coproduct}, or \emph{diagonal,} $\psi:A\ra
A\ot A$ which is an algebra map.  We shall assume that $\psi$ is (graded) coassociative
and cocommutative. Since the algebra $A$ is assumed to be connected, it has an antipode
$\sigma:A\lra A$, but we shall not make use of it. See \cite{Milnor-Moore} for more
details on Hopf algebras and their structure.

%In general, whenever two objects $a$ and $b$ are permuted, the sign $(-1)^{\deg
%a\,\deg b}$ is introduced.

Let $\Ext_A^i(-,k)$ denote the $i$-th right derived functor of $\Hom_A (-,k)$ in the
category of graded left $A$-modules.  The \emph{cohomology} $H^*(A)$ of $A$ is defined to
be $\Ext_A^*(k,k)$, where $k$ is given an $A$-module structure by the augmentation map
$\epsilon$. The Yoneda product makes this a graded algebra \cite{Adams}. %agrees with the Yoneda product
We recall some basic facts about the cohomology of algebras.

\begin{enumerate}
\item Let $A,B$ be locally finite algebras. Then $H^*(A\ot B)\simeq H^*(A)\ot
H^*(B)$.
\item If $A$ is a connected cocommutative
Hopf algebra with unit, then $H^*(A)$ is anticommutative.
\end{enumerate}

We introduce the following definitions:
\begin{definition}\mbox{}

\begin{enumerate}
\item A graded algebra $C$ is called \emph{bigenerated} if it is generated as an
algebra by elements of homological degree 1 and 2.
\item An algebra $A$ is \emph{semi-Koszul} if $C=H^*(A)$ is bigenerated.
\end{enumerate}
\end{definition}

We shall make use of a reindexed form of \emph{algebraic Steenrod operations}. In
\cite[Theorem 11.8]{May}, May defines operations $\p^i :H^{s,t}(A)\ra
H^{s+(2i-t)(p-1),\,pt}(A)$ for $p\geq 3$.  These operations are zero if $2i<t$ or
$2i>s+t$. We would like to reindex these so that $\p^i$ raises homological degree by
$2i(p-1)$ and is trivial for $i<0$, $2i>s$. (May denotes these reindexed operations by
$\pti{i}$.) This regrading has the effect of making $\pti{0}$ the first possibly
non-trivial operation on $H^*(A)$. It also eliminates all operations on $H^{s,t}(A)$ for
$p>2$ when $t$ is odd. This won't be of concern to us as most of our algebras will be
concentrated in even degrees.

\begin{theorem}\label{steenrod}
 Let $A$ be a cocommutative Hopf algebra over $\f$.  There exist natural
homomorphisms
\begin{enumerate}
\item $\sqit{i}:H^{s,t}(A)\lra H^{s+i,2t}(A)$ for $p=2$,
\item  $\pti{i}:H^{s,2t}(A)\lra H^{s+2i(p-1),2pt}(A)$ and\\
       $\bpit{i}:H^{s,2t}(A)\lra H^{s+2i(p-1)+1,2pt}(A)$ for $p\geq 3$
\end{enumerate}
 with the following properties:
\begin{enumerate}
\item $\sqit{i}=0$ if $i<0$ or $i>s$,\\
      $\pti{i}=0$ if $i<0$ or $2i>s$,\\
      $\bpti{i}=0$ if $i<0$ or $2i\geq s$;
\item $\sqit{i}(x)=x^2$ if $i=s$,\\
      $\pti{i}(x)=x^p$ if $2i=s$.
\item \emph{Cartan formulae}\\
$\sqit{i}(xy)=\sum_{j=0}^i \sqit{j}(x)\sqit{i-j}(y)$,\\
$\pti{i}(xy)=\sum_{j=0}^i \pti{j}(x) \pti{i-j}(y)$,\\
  $\bpit{i}(xy)=\sum_{j=0}^i \bl\bpit{j}(x)\pti{i-j}(y)+\pti{j}(x)\bpit{i-j}(y)\br$;
\item \emph{Adem relations}
\begin{enumerate}
\item If $a<2b$,\\
 $\sqit{a}\sqit{b}=\sum_i \binom{b-i-1}{a-2i}\sqit{a+b-i}\, \sqit{i},$
\item If $a<pb$,\\
 $\widetilde{\beta^\epsilon \mathscr{P}}^{ \scriptstyle{a}} \pti{b}
=\sum_i(-1)^{a+i}\binom{(b-i)(p-1)-1}{a-pi} \widetilde{\beta^{\scriptstyle{\epsilon}}
\mathscr{P}}^{ {\scriptstyle{a+b-i}}}\, \pti{i}$,
\item If $a\leq pb$,\\
$\widetilde{\beta^\epsilon \mathscr{P}}^{ \scriptstyle{a}} \widetilde{\mathscr{P}}^{
{}^{\scriptstyle{b}} }=(1-\epsilon)\sum_i (-1)^{a+i}
\binom{(b-i)(p-1)-1}{a-pi} \widetilde{\beta\p}^{\scriptstyle{a+b-i}} \,\pti{i}$ \\
\phantom{${\beta^\epsilon \mathscr{P}}^{ \scriptstyle{a}} \widetilde{\mathscr{P}}^{
{}^{\scriptstyle{b}} }=$} $- \sum_i (-1)^{a+i}\binom{(b-i)(p-1)-1}{a-pi-1}
\widetilde{\beta^\epsilon\p }^{ {\scriptstyle{a+b-i}}}\, \bpit{i},$\\
\end{enumerate}
where $\epsilon=0$ or $1$, the binomial coefficients are taken modulo $p$ and, by abuse
of notation, $\widetilde{\beta^{\scriptstyle{0}}\p}^{ {\scriptstyle{s}} }=\pti{s}$ and
$\widetilde{\beta^{\scriptstyle{1}}\p}^{  {\scriptstyle{s}} }=\bpit{s}$.
\end{enumerate}
\end{theorem}

\begin{remark}
One can actually define operations $\pti{i/2}$, $\bpti{i/2}$ on elements of odd internal
degree such that the Cartan formulae and Adem relations still make sense.  See
\cite[Appendix 1.5]{Ravenel} for details.
\end{remark}


%%%%%%%%%%%%%%% CARTAN-EILENBERG  everything connected here.%%%%%%%%%%%%%%%%%%%%%%%%5

A \emph{decreasing filtration} of $A$ is a sequence of $k$-modules $A=F^0 A\supset F^1
A\supset F^2 A\supset\dotsb$ such that $F^s A\cdot F^t A\subset F^{s+t}A$. Let $\rho_s$
denote the natural map $F^s A\twoheadrightarrow F^{s/s+1}A=F^sA\big/F^{s+1}A$. We will
sometimes suppress the $A$ when it is clear from context. We will say that elements in
$F^{s+i}$ are of \emph{lower} filtration level than elements in $F^s$, even though the
indices are greater. Let the \emph{filtration degree} of an element $x\in A$ be the
largest $i$ such that $x\in F^i A$ (if such an $i$ exists).
% Bigraded filtrations?

Let $F\subset A$ be a subalgebra. We say $F$ is \emph{normal} in $A$ if $I(F)\cdot
A=A\cdot I(F)$. In this case, there is a well defined  quotient $B= A\big/ A\cdot I(F)$
denoted by $A//F$.  In the case that $F$ is a sub-\emph{Hopf} algebra, $B$ is also a Hopf
algebra. We say $F$ is \emph{central} in $A$ if each element of $F$ anticommutes with
each element of $A$. By an \emph{extension} or \emph{exact sequence} $F\ra A\ra B$ of
algebras we shall mean that $F$ is normal in $A$ and $B\simeq A//F$.
%trigraded?
\begin{theorem}\label{Cartan-Eilenberg} Let $F \overset{i}{\ra} A \overset{\pi}{\ra} B$ be a
central extension of graded, connected algebras.  There is a first quadrant spectral sequence converging to $H^*(A)$ with
\be
\item $E_2^{s,t}\simeq H^s(B)\otimes H^t(F)$.
\item $E_\infty^{s,t} \simeq F^sH^{s+t}(A)\big/ F^{s+1}H^{s+t}(A)$ where $\{F^s\}$ is a
decreasing filtration of $H^*(A)$.
%
\item The ring structure of $H^s(B)\ot H^t(F)$ is defined by \\
$(x\ot y)(z\ot w)=\sign{(t+t')(s+s')} xz\ot yw$ where $y\in H^{t,t'}(F)$, $z\in
H^{s,s'}(B)$.
%
\item $d_r:E_r^{s,t}\ra E_r^{s+r,t-r+1}$ is a graded derivation, meaning \\ %respects internal grading?
$d_r(ab)=d_r(a)\cdot b +\sign{\deg a}a \cdot d_r(b)$.
%
\item The map $H^s(B)\simeq E_2^{s,0}\twoheadrightarrow E_\infty^{s,0}\simeq F^sH^s(A)
\rightarrowtail H^s(A)$ corresponds to the induced map $\pi^*:H^*(B)\ra H^*(A)$. \\
      The map $H^t(A)=F^0H^t(A)\simeq E_\infty^{0,t}\rightarrowtail E_2^{0,t}\simeq H^t(F)$
corresponds to the induced map $i^*:H^*(A)\ra H^*(F)$.
%
\item \emph{Kudo transgression theorem.} Assume, in addition, that
$F \overset{i}{\ra} A \overset{\pi}{\ra} B$
is an extension of Hopf algebras over $\f$.
 The \emph{transgression} $d_r:E_r^{0,r-1}\ra E_r^{r,0}$ anticommutes with Steenrod operations:
if $x\in E_r^{0,r-1}$ and $\theta$ is a Steenrod operation of homological
 degree $i$, then $\theta(x)$ survives until $E_{r+i}^{0,r+i-1}$ and
 $d_{r+i}\bl\theta (x)\br=\sign{i}\theta\bl d_r(x)\br$.
\ee
\end{theorem}

Since the algebras $F$, $A$ and $B$ are graded, the spectral sequence is actually
trigraded, with the third grading being the internal degree.  We have suppressed this
third grading, as it is preserved by the differentials $d_r$. This spectral sequence was
used by Adams \cite[2.3.1]{Adams}. Ravenel \cite[A1.3.14]{Ravenel} calls it the
\emph{Cartan-Eilenberg} spectral sequence.  For (6) see \cite[Proposition
2.3]{Wilkerson}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Lie Algebras}

The Lie algebras we shall work with are all graded over either $\Z$ or $\f$.  A
\emph{restricted} Lie algebra is a Lie algebra over $\f$ with restriction $\xi$.

For $L$ a restricted Lie algebra, let $\Tens(L)$ be the tensor algebra on $L$, and let
$J\subset \Tens(L)$ be the two sided ideal generated by elements of the form $x\ot
y-(-1)^{\deg x\,\deg y}y\ot x-[x,y]$ and $x^{\ot p}-\xi(x)$ for $x,y\in L$.
\begin{definition}\label{VL}
The \emph{universal restricted enveloping algebra} of $L$ is defined to be $\V
L=\Tens(L)/J$.
\end{definition}
We can put a Hopf algebra structure on $\V L$ by setting $\psi(x)=x\ot 1+1\ot x$,
$\epsilon(x)=0$ for $x\in L$. The cohomology of a Lie algebra $L$ is defined to be
$H^*(\V L)$.
\begin{theorem}[{ \cite[5.3]{Priddy-primary}, \cite[Theorem 8.5]{May} }]
Let $L$ be a Lie algebra over $\f$. Then the Steenrod operations $\pti{0}$ and $\sqit{0}$
are identically zero on $H^*(\V L)$.
\end{theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{An Extension Theorem}\label{main theorem chapter}% should come after group chapter?

Let $A$ be a cocommutative  Hopf algebra over $\f$. In this section, we will find
sufficient conditions for $H^*(A)$ to be bigenerated. Let $z\in A$ be a central primitive
element of height $p$. This means $z$ anticommutes with every element of $A$, $z^p=0$ and
$z^i\neq 0$ if $i<p$. Let $F=\f[z]\big/(z^p)$ be the central sub-Hopf algebra of $A$
generated by $z$. Let $B=A//F$.  We have the central extension of  Hopf algebras
\beq\label{main theorem Hopf algebra extension}
F\lra A\lra B
\enq
with corresponding spectral sequence
\begin{equation}\label{main theorem Hopf algebra spectral sequence}
E_2^{s,t}\simeq H^s(B)\ot H^t(F)\Rightarrow H^{s+t}(A).
\end{equation}
%
Let $n$ be the internal degree of $z\in F$.  Note that when $p$ is odd, $n$ must be even.
We recall that the cohomology of the fiber $F$ is
\begin{align*}
H^*(F) &\simeq \Lambda[\zeta]\ot \f[\epsilon],& \zeta&\in H^{1,n},
\,\epsilon\in H^{2,np},& \text{with }\bp(\zeta)&= -\epsilon & p \text{ odd,}\\
H^*(F) &\simeq \ff[\zeta],& \zeta&\in H^{1,n},& \text{with }\sqit{1}(\zeta)&=\zeta^2 &
p=2.
\end{align*}
%
See Theorem \ref{non-zero Steenrod operations} for more details on the Steenrod
operations.

\begin{theorem}[Main Theorem] \label{main theorem} Let $F,A,B$ be as in
\eqref{main theorem Hopf algebra extension}. In the spectral sequence \eqref{main theorem
Hopf algebra spectral sequence}, let $d_2(\zeta)=\mu\in H^{2,n}(B)$. Assume
\be
\item $\bp(\mu)=0$ or $\sq{\mu}=0$.
\item The annihilator ideal of $\mu$ in $H^*(B)$ is either zero or is generated in dimension
one.
\item $B$ is semi-Koszul.
\ee
Then $A$ is semi-Koszul.
\end{theorem}

\noindent\textit{Proof.} By the transgression theorem \ref{Cartan-Eilenberg}(6),
\begin{align*}
d_3(\epsilon)&=-\bp(\mu), \quad p \text{ odd},\\
d_3(\zeta^2) &=\sq (\mu), \quad\, p=2.
\end{align*}
If $\bp(\mu)=0$ or $\sq(\mu)=0$ then
\begin{align*}
&d_{2 p+1}(\epsilon^p) =\pti{1}\bl d_3(\epsilon)\br=0 \quad\text{ or}\\
&d_5(\zeta^4) =\sqit{2}\bl d_3(\zeta^2)\br=0.
\end{align*}
Likewise for all other powers of $\epsilon$ or $\zeta$.  Thus in this case, the spectral
sequence \eqref{main theorem Hopf algebra spectral sequence} collapses at $E_3$.

In order for $H^*(A)$ to be bigenerated, we need $E_\infty$ to be bigenerated. Assume
$\bp(\mu)=0$ or $\sq(\mu)=0$. If $\mu$ is a zero divisor in $H^*(B)$, we may run into the
following problem. Let $\eta\in H^i(B)$ be in the annihilator ideal of $\mu$. Then
%
$$d_2(\eta\ot\zeta)=d_2(\eta)\ot\zeta +\sign{i}\eta \ot d_2(\zeta)=\pm\eta\ot\mu=0.$$
%
Nothing can hit $\eta\ot\zeta\in E_r^{i,1}$ because $d_r\equiv 0$ for $r\geq 3$, and
$d_2\equiv 0$ from the $E_2^{*,2}$ line. Thus $\eta\ot\zeta$ is a non-bounding permanent
cycle of dimension $i+1$. If $i=1$, then $\eta\ot\zeta\in E_\infty^{1,1}$ is a
two-dimensional generator of $E_\infty$.  If $i>1$ on the other hand, $\eta\ot\zeta$ is a
possibly indecomposable element in $E_\infty^{i,1}$ of homological dimension larger than
$2$. (It will be indecomposable if, for example, $\eta$ is indecomposable in $H^*(B)$.)
Consequently $E_\infty$---and thus $H^*(A)$---is not bigenerated.

Assume then that the annihilator ideal of $\mu$ in $H^*(B)$ is either zero or is
generated multiplicatively by elements of dimension one.
%$\Ann_{H^*(B)}(\mu)$
Assume also that $H^*(B)$ is bigenerated. Then $E_\infty$ has the following possible set
$S$ of multiplicative generators:
%beq\label{S}
\be
\item $1\ot \epsilon$ for $p$ odd, $1\ot\zeta^2$ for $p=2$;
\item elements of the form $\eta\ot\zeta$, where $\eta\in H^1(B)$ and $\eta\mu=0$;
\item elements $\sigma\ot 1$, $\nu\ot 1$, where $\sigma\in H^1(B)$ and $\nu\in H^2(B)$;
\item $1\ot \zeta$ (if $d_2\equiv 0$).
\ee

All these elements have homological degree less than or equal to two, and thus $E_\infty$
 is  bigenerated.  Recall that
$H^*(A)$ has a decreasing filtration $F^*H^*(A)$, and that we denote the natural
projection $F^sH^n(A)\ra F^{s/s+1}H^n(A)\simeq E_\infty^{s,n-s}$ by $\rho_s$. Choose a
set of representatives $T\subset H^*(A)$ for each of the types of generators in $S$:
%\beq\label{T}
\be
\item $e\in F^0H^2(A)$;
\item $x\in F^1 H^2(A)$;
\item $\alpha\in F^1H^1(A)$, $v\in F^2 H^2(A)$;
\item $z\in F^0H^1(A)$.
\ee
%\enq

We claim that $H^*(A)$ is generated as an algebra by $T$, from which it follows that $A$
is semi-Koszul. Let $S'$=$S\cup \{1\ot 1\}$, $T'=T\cup\{1\}$. The proof will proceed by
reverse induction on filtration degree $s$ in $H^n(A)$. Let $y\in F^nH^n(A)$. Then $y\in
H^n(B)$, which is bigenerated. Now let $y\in F^sH^n(A)$, where $s<n$, and assume
$F^iH^n(A)$ is bigenerated for $i>s$. Projecting $y$ to $F^{s/s+1}H^n(A)\simeq
E_\infty^{s,n-s}$, $\rho_s(y)$ can be written as a sum of products
$f_i\lambda_{i_1}\lambda_{i_2}\dotsm\lambda_{i_n}$ where $f_i\in k$ and $\lambda_{i_j}\in
S'$. So $y$ is a sum of products $f_i\tilde{\lambda}_{i_1}\dotsm \tilde{\lambda}_{i_n}$
and a term $\gamma$, where $\tilde{\lambda}_{i_j}\in T'$ are lifts of $\lambda_{i_j}$
under $\rho$, and $\gamma\in F^{s+1}H^n(A)$.  By induction, $\gamma$ is also a linear
combination of products of elements from $T'$. Thus $y$ is of the desired form. \qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Hopf algebras arising from $p$-groups}\label{filtration}

The mod-$p$ lower central series of a group $G$ gives rise to a restricted Lie algebra
$L$, whose universal restricted enveloping algebra $VL$ is a Hopf algebra.  Starting with
a central extension of $p$-groups $H\ra G\ra Q$, we take enveloping algebras of the
associated Lie algebras and get a central extension of Hopf algebras $F\ra A\ra B$. In
Section \ref{discussion} we show that $A$ is semi-Koszul if $Q$ is abelian and
$H\simeq\z$. In Section \ref{vistas}, we generalize to extensions of Hopf algebras $F\ra
A\ra B$ which do not necessarily arise from extensions of $p$-groups.  We show $\eo A$ is
semi-Koszul if $B$ is commutative.

\subsection{Definitions} Let $G$ be a group, and let $G=F^1G\supseteq
F^2G\supseteq \dotsb$ be a descending sequence of subgroups (also called a filtration)
with $F^iG\unrhd F^{i+1}G$, \,$F^iG\big/F^{i+1}G$ abelian. Let
$\mathrm{gr}.^FG=\bigoplus_i F^iG\big/F^{i+1}G$ be the \emph{associated graded} object,
and let ${\mathrm{gr}_i}^FG=F^iG\big/F^{i+1}G$.  When the filtration is understood, we
shall simply write $\gr G$ and $\mathrm{gr}_i G$. Recall the natural surjection $\rho_i
:F^iG\twoheadrightarrow \mathrm{gr}_i G$.

For $H,\, K$ subgroups of $G$, let $(H,\, K)$ be the subgroup generated by all
commutators of the form $(h,\, k)=\inv{h}\inv{k}hk$ for $h\in H,\, k\in K$. Let $H^n$ be
the subgroup generated by $h^n$ for $h\in H$. An \emph{$N$-sequence} is a normal series
$G=G_1\unrhd G_2\unrhd G_3\unrhd \dotsb$ such that $(G_i,\, G_j)\subset G_{i+j}$
\cite{Lazard}. The successive quotients $\mathrm{gr}_i G=G_i\big/ G_{i+1}$ of an
$N$-sequence are abelian, since $(G_i,G_i)\subset G_{2i}\subset G_{i+1}$. The associated
graded abelian group $\gr G=\bigoplus_n G_n\big/ G_{n+1}$ has the structure of a Lie
algebra over $\Z$, with bracket $\lb\rho_r x,\rho_s y\rb =\rho_{r+s}\bl(x,y)\br$. A
\emph{$p$-restricted} $N$-sequence is an $N$-sequence $\{G_n\}$ such that ${G_n}^p
\subset G_{pn}$.  The associated graded group $\gr G$ is then a vector space over $\f$.
We can define a restriction $\xi\bl\rho_r(x)\br=\rho_{pr}\bl x^p\br$ on $\gr G$ which
makes it into a restricted Lie algebra over $\f$.

The fastest descending $p$-restricted $N$-sequence of $G$ is the \emph{mod-$p$ lower
central series} $\bigl\{ \Gamma^iG \bigr\}$: (see \cite{Lazard}, \cite{Blackburn2})
%
\begin{equation}\label{filtration def}
 \Gamma^{2n-1}G=\Gamma^{2n}G =
 \langle (\gamma_1 , \dots , \gamma_r )^{p^u} \text{ such that } r\cdot p^u
 \geq n \rangle,
\end{equation}
%
where $(\gamma_1 , \dots , \gamma_r)=
 \bl\gamma_1,\bl \dotso \bl\gamma_{r-2},(\gamma_{r-1},\gamma_r)\br\dotso\br$.
The associated object $\mathrm{gr}.^\Gamma G$ is thus a restricted Lie algebra over $\f$.
Let $\L(G)=\L G=\mathrm{gr}.^{\Gamma}G$. It can be easily checked that $\mathscr{L}$ is a
functor from the category of groups to the category of restricted Lie algebras over $\f$.

The definition we have given here is slightly different from that of \cite{Quillen} or
\cite{Blackburn2}.  Our $\Gamma^{2n}$ equals their $\Gamma^n$. We have doubled the
filtration degree in order not to have to worry about signs for $p$ odd.

%In what follows, $k=\f$. unnecesary
Let $kG$ be the group ring of $G$ over $k$, with augmentation $\epsilon(g)=1$ for $g\in
G$. A diagonal $\psi:kG\ra kG\ot kG$ is induced by the map $g\ra g\ot g$, giving $kG$ the
structure of a cocommutative Hopf algebra over $k$. Let $I=\ker\epsilon$ be the
augmentation ideal of $kG$.  Filter $kG$ as follows: $F^{2n-1}=F^{2n}=I^n$ for $n>0$. The
associated graded algebra $\eo (kG)=\mathrm{gr}.^FkG$ has the structure of a restricted
Lie algebra over $k$ inherited from the multiplication in $kG$. Specifically,
$[\rho_{r}x,\rho_{s}y]= \rho_{r+s}(xy-yx),\, \xi(\rho_{r}x)=\rho_{rp}(x^p)$. Since $\psi$
respects the filtration on $kG$, it induces a cocommutative diagonal on $\eo (kG)$.  It
is clear that $\eo (kG)$ is generated as an algebra by $\eo_2 (kG)=I/I^2$. These elements
are all primitive by reason of dimension.  Thus $\eo (kG)$ is primitively generated, and
by a theorem of Milnor-Moore \cite[Theorem 6.11]{Milnor-Moore}, $\eo (kG)$ is isomorphic
to the restricted universal enveloping algebra of its restricted Lie subalgebra
$L=\eo_2 kG$.%\mathrm{gr}_2 kG

The map $\psi:\L G\ra\eo (kG)$ given by $\rho_r(x) \mapsto\rho_r(x-1)$ is a restricted
Lie algebra homomorphism. In fact, the image of $\psi$ lies in the primitive elements
$\mathcal{P}\bigl(\eo (kG)\bigr)$ of $\eo (kG)$.
%
\begin{theorem}[{\cite[Theorem 1.4]{Quillen}}]\label{Quillen's theorem} The induced map
 $$\L G\longrightarrow \mathcal{P}\bl\eo (kG)\br$$
is an isomorphism of restricted Lie algebras.
\end{theorem}
%
  Therefore,
\begin{equation}\label{previous equation}
\V(\L G) \simeq \V\bigl(\mathcal{P}(\eo (kG))\bigr) \simeq \eo (kG)
\end{equation}
as Hopf algebras over $k$.

Let $\VL G$ denote $\V(\mathrm{gr}.^{\Gamma}G)$. Studying the cohomology $H^*\bl\eo
(kG)\br$ over $k$ is thus equivalent to studying $H^*(\VL G)$. The advantage of working
with cohomology of Lie algebras is that the Steenrod operations $\sqit{0}\equiv 0$,
$\pti{0}\equiv 0$  there (\cite{Priddy-primary}, \cite[Theorem 8.5]{May}), which greatly
simplifies the situation.

We shall need the following result.
\begin{lemma}\label{VL for G abelian}
Let $G$ be a finite abelian $p$-group, then $\VL G$ is a semi-Koszul algebra.
\end{lemma}
\begin{proof}
 Let $G=\zn{n}$. It is clear that
\begin{equation*}
\VL G\simeq \eo (\f G)\simeq \f[x] \big/ \bl x^{p^n}\br
\end{equation*}
For a general finite $p$-group $G=\zn{n_1}\times\dots\times\zn{n_k}$, then
\begin{equation*}
\VL G\simeq \f[x_1,\dots, x_k]\big/\bl x_i^{p^{n_i}} \br
\end{equation*}
Thus
\begin{equation*}
H^*(\VL G)\simeq \Lambda[z_1,\dots ,z_k]\ot\f[e_1,\dots,e_k],
\end{equation*}
where $z_i\in H^{1,2}$,\, $e_i\in H^{2,2p^{n_i}}$.  If $p=2$ and some $n_i=1$, replace
$\Lambda[z_i]\ot\f[e_i]$ by $\ff[z_i]$.
\end{proof}


\subsection{An Extension Theorem}\label{discussion}

Let $G$ be a non-trivial finite $p$-group, and let $k=\f$.  There is an element $x$ of
order $p$ in the center of $G$. Let $H\simeq\z$ be the subgroup generated by $x$. We have
the central extension
\begin{equation}\label{group extension}
H\overset{f}{\lra} G\overset{g} {\lra}Q=G/H
\end{equation}
Our plan is to apply the $\mathscr{L}$ functor, and then the $\mathscr{V}$ functor, to
(\ref{group extension}) in the hope of obtaining an exact sequence of Hopf algebras of
the form \eqref{main theorem Hopf algebra extension}.

First, note that $g(\Gamma^n G)=\Gamma^n Q$.  Refilter $H$ by
$F^nH=\inv{f}\bl\Gamma^nG\br$.  According to Theorem 2.4 of \cite{Lazard},
the following is an exact sequence of graded Lie algebras over $\Z$.
\begin{equation}\label{seq}
\gr^FH\overset{\ol{f}}{\lra} \gr G \overset{\ol{g}}{\lra} \gr Q
\end{equation}
Now $\mathrm{gr}.^FH$ is generated by $\ol{x}\in F^{2n/2n+1}H$, where $2n$ is the
filtration degree of $f(x)$ in $G$. Since $\L H$ is generated by $\ol{x}\in
\Gamma^{2/3}H$, $\mathrm{gr}.^FH\simeq \L H$ as ungraded Lie algebras.  The sequence
\eqref{seq} is actually a sequence of graded restricted Lie algebras and---if we regrade
$\L H$ appropriately---is isomorphic to
\begin{equation}\label{Lie algebra extension}
\L H\lra \L G \lra \L Q.
\end{equation}

Now let
\begin{equation}
L_1\overset{i}{\lra} L_2 \overset{\pi}{\lra} L_3
\end{equation}
be an exact sequence of graded restricted Lie algebras over $\f$. Recall that $\V L_i$ is
defined to be the quotient of $\Tens(L_i)$ by some specific ideal $J_i$. It is clear that
$\pi(J_2)=J_3$, and that $\inv{i}(J_2)=J_1$. Therefore,
\begin{equation}
\Tens(L_1)\big/J_1 \lra \Tens(L_2)\big/J_2 \lra \Tens(L_3)\big/J_3
\end{equation}
is an exact sequence of Hopf algebras over $k$.

We have just proven
\begin{proposition} Let $H, G,$ and $Q$ be as in (\ref{group extension}).
Regrade $\VL H$ as in \eqref{Lie algebra extension}.
Then the following sequence of Hopf algebras is exact:
\beq\label{VL extension}
\VL H\lra \VL G\lra \VL Q
\enq
\end{proposition}

Since $\vl H\simeq \f[x]\big/(x^p)$, the central extension \eqref{VL extension} is of the
form $F\ra A\ra B$, as in \eqref{main theorem Hopf algebra extension}. Associated to this
extension is the Cartan-Eilenberg spectral sequence
 \eqref{main theorem Hopf algebra spectral sequence}:
%
\begin{equation}\label{algebra spectral sequence}
\begin{split}
E_2&\simeq H^*(\VL Q)\ot \Lambda[z]\ot\f[e]\Rightarrow H^*(\VL G), \quad p \text{
odd},\\
E_2&\simeq H^*(\VL Q)\ot \ff[z]\Rightarrow H^*(\VL G), \qquad\qquad p=2.
\end{split}
\end{equation}
%
Let us now examine conditions on the group $G$ which will assure that the
assumptions of Theorem \ref{main theorem} are satisfied.
%
\begin{theorem} In the spectral sequence \eqref{algebra spectral sequence}, assume
$\mu=d_2(z)$ is a sum of products of 1 dimensional elements.  That is, assume $\mu$ is
\emph{decomposable}. Assume also that Conditions 2 and 3 of Theorem \ref{main theorem}
hold. Then $\VL G$ is semi-Koszul.
\end{theorem}
\begin{proof} This result follows from the fact that
\begin{equation}\label{proof that bp}
\begin{aligned}\
\bp(\sigma\tau)&=\bp(\sigma)\pti{0}(\tau)+\pti{0}(\sigma)\bp(\tau)=0 \text{ and}\\
\sqit{1}(\sigma\tau)&=\sqit{1}(\sigma)\sqit{0}(\tau)+\sqit{0}(\sigma)\sqit{1}(\tau)=0,
\end{aligned}
\end{equation}
since $\pti{0}\equiv 0$ and $\sqit{0}\equiv 0$ on the cohomology of a Lie algebra.
Condition $1$ of Theorem \ref{main theorem} is thus satisfied.
\end{proof}
%
\begin{theorem}\label{base is abelian}
If $Q$ is abelian, then $\VL G$ is semi-Koszul.
\end{theorem}
\begin{proof}
By Corollary \ref{VL for G abelian}, $H^*(\VL Q)$ is isomorphic to a tensor product of
algebras of the form $\Lambda[z_i]\ot\f[e_i]$ (and $\ff[z]$ for $p=2$), so $H^2(\VL Q)$
is spanned by $\{z_iz_j, e_i\}$. Now $\bp(z_iz_j)=0=\sqit{1}(z_iz_j)$ by \eqref{proof
that bp} and $\bp(e_i)=0$ by reason of internal degree.  Thus Condition $1$ of Theorem
\ref{main theorem} is satisfied. Condition 2 is satisfied because the annihilator ideal
of $H^2(\VL Q)$ is generated by one-dimensional elements $\{z_i\}$. Finally, Condition 3
is satisfied since $H^*(\VL Q)$ is bigenerated.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Generalization to Hopf Algebras}\label{vistas}

The results in \S \ref{discussion} can be generalized somewhat to Hopf algebras.  Let $A$
be a Hopf algebra over $\f$.  Assume $A$ contains a central, primitive element $x$ of
height $p$. This means $x$ anticommutes with every element of $A$, $x^p=0$, and $x^i\neq
0$ if $i<p$.  If $p$ is odd, $x$ is forced to have even internal degree. Let
$M=\f[x]/(x^p)$ be the central sub-Hopf algebra of $A$ generated by $x$.

We have the central extension of graded Hopf algebras
\begin{equation}\label{general extension}
M \overset{f}{\lra} A \overset{g}{\lra} B.
\end{equation}
%
For an augmented algebra $C$, let $\{F^{2n-1}C=F^{2n}C=(IC)^n\}$ be the modified
augmentation ideal filtration, and let $\eo C=\mathrm{gr.}^FC$. Since
$\epsilon^B(g(y))=\epsilon^A(y)$, we see that $g(IA)\subset IB$, and more generally,
$g(IA^{\,n})\subset IB^{\,n}$ .  Thus $g$ respects the augmentation ideal filtration, and
we can form the maps
\begin{gather*}
\bar{g}_{2n} :F^{2n/2n+1}A= IA^{\,n}\big/IA^{\,n+1}\lra
IB^{\,n}\big/IB^{\,n+1} = F^{2n/2n+1}B,\\
\eo g:\eo A\lra \eo B.
\end{gather*}
Each $\bar{g}_{2n}$ is a map of graded modules, while $\eo A$ and $\eo B$ are
\emph{bigraded} algebras (by internal degree and filtration degree). It is easy to check
that $\eo g$ is surjective.  If $\bar{g}_{2n}(\bar{y})=0$, this means $g(y)\in
F^{2n+1}B$. Since $g$ is surjective, $y$ can be written as $y=a+b$, where $a\in
F^{2n+1}A$ and $b\in \ker g\cap F^{2n}A$. Since $\ker g=\image f$, $b=f(b')$ for some
$b'\in M$. Unfortunately, we cannot tell if $b'\in F^{2n}M$. If we refilter $M$ by
$\hat{F}^{2n}M=\inv{f}\bl F^{2n}A\br$, we obtain the exact sequence of bigraded algebras
$$\mathrm{gr}.^{\hat{F}}M\lra \eo A\lra \eo B.$$
In our case, $\gr^{\hat{F}} M\simeq\eo M \simeq M$, except that the generator of
$\gr^{\hat{F}} M$ has filtration degree equal to the filtration degree of $f(x)$ in $A$.
With the understanding that $M$ is regraded as such,
%regrade $M,\eo A,\eo B$ by total degree, i.e. $A_n
%:=\sum_{i+j=n} A_{i,j}$. Since the filtration degree of each element in $A$ is even, this will not change much.
let us investigate the extension
\beqs
M \lra \eo A\lra \eo B
\enqs
of primitively generated Hopf algebras with associated Cartan-Eilenberg spectral sequence
\beq\label{yet another spectral sequence}
E_2^{s,t}\simeq H^s(\eo B)\ot H^t(M)\Lra H^{s+t}(\eo A).
\enq

We recall that a Hopf algebra $C$ is \emph{monogenic} if it
is generated by one element $x$.

\begin{proposition}\label{monogenic algebras}\cite{Milnor-Moore} A monogenic Hopf algebra
over $\f$ has the form
\be
\item\label{exterior monogenic} An exterior algebra $\Lambda[x]=\f[x]/(x^2)$, with
$\deg x$ odd for $p$ odd.
\item A truncated polynomial algebra $A=\f[x]\big/\bigl(x^{p^n}\bigr)$ as in \eqref{truncated
polynomials}.
\item A polynomial algebra $\f[x]$ with $\deg x$ even for $p$ odd.
\ee
\end{proposition}
%
In all cases, $x$ is primitive by reason of dimension. We have excluded the case $p=2$,
$n=1$ from $(2)$, as $\ff[x]/(x^2)\simeq \Lambda[x]$.
%
\begin{lemma} {\cite[Proposition 2.2]{Wilkerson}} For each monogenic Hopf algebra $C$ above,
\be
\item $H^*(C)\simeq\f[z]$, with $z\in H^{1,\deg x}$.
\item $H^*(C)\simeq\Lambda[z]\ot\f[e]$, with $z\in H^{1,\,\deg x}$, $e\in H^{2,\,p^n \! \deg x}$.
\item $H^*(C)\simeq\Lambda[z]$, with $z\in H^{1,\deg x}$.
\ee
\end{lemma}


\begin{theorem}[Borel, \cite{Milnor-Moore}] A locally finite, connected, primitively
generated, commutative Hopf algebra over $\f$ is
isomorphic to a tensor product of monogenic Hopf algebras.
\end{theorem}

\begin{theorem}\label{Hopf algebra main theorem} Let $M,A,B$ be as in \eqref{general extension}. If $B$ is
commutative, then $\eo A$ is semi-Koszul.
\end{theorem}
\begin{proof} The fiber $M$ has cohomology generated by $z\in H^1(M)$ and possibly
$e\in H^2(M)$. The element $e$ is related to $z$ by the Steenrod operation $\theta$,
where $\theta$ is one of $\bp$ or $\sqit{1}$. The spectral sequence \eqref{yet another
spectral sequence} will collapse if $d_3(e)=\theta\bl d_2(z)\br=0$. Since $B$ is
commutative, so is $\eo B$. By Borel's theorem, $\eo B\simeq \otimes_i B_i$, where the
$B_i$ are monogenic Hopf algebras.
%of the form $\Lambda[x]$, $\f[x]\big/\bigl(x^{p^n}\bigr)$ or $\f[x]$.
Then $H^*(\eo B)\simeq H^*(B_1)\ot\dotsb\ot H^*(B_n)\simeq \Lambda[z_i]\ot\f[z_j']\ot
\f[e_k]$ with $z_i,\, z_j'\in H^1$, $e_k\in H^2$. We can show $\theta \equiv 0$ on
$H^2(\eo B)$ by using the same argument as in Theorem \ref{base is abelian}.
\end{proof}

\begin{remark}
A theorem of Wilkerson \cite{Wilkerson} states that a \emph{finite dimensional}, graded,
connected, cocommutative Hopf algebra $A\neq k$ always contains a nontrivial central
monogenic sub-Hopf algebra $C$. In the case that $C$ is a truncated polynomial algebra of
height $p^n$, we can always find a central primitive element $x\in A$ of height $p$. The
conditions of Theorem \ref{Hopf algebra main theorem} then hold.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%Rank 2 p-groups %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rank Two $p$-Groups}\label{rank 2}

In this section we show that the cohomology of the Lie algebra associated with a rank $2$
$p$-group for $p\geq 5$ is bigenerated.  A rank 2 $p$-group is a finite $p$-group whose
largest elementary abelian subgroup has dimension 2. In the case $p\geq 5$, Blackburn
\cite{Blackburn} has classified these as belonging to one of the following four families:
abelian, metacyclic, $C(r)$ and $G(r,e)$ of order $p^r$. A metacyclic $p$-group $P$ is an
extension of a cyclic group by a cyclic group. A presentation for $C(r)$ is given by
%
$$C(r)=\langle a,b,c \; \vert \; a^p=b^p
=c^{p^{r-2}}=1,\, (a,b)=c^{p^{r-3}}, \, c\in \mathcal{Z}(C)\rangle$$
%
where $r\geq 3$, $(a,b)=\inv{a}\inv{b}ab$, and $\mathcal{Z}(C)$ is the center of $C$. A
presentation for $G(r,e)$ is given by
%
$$G(r,e)=\langle f,g,h \; \vert \; f^p=g^{p^{r-2}}=h^p= \lp g,h\rp=1,\, \lp
f,h^{\sss{-1}}\rp=g^{ep^{r-3}},\, \lp f,g\rp=h\rangle$$ where $r\geq 4$,\, $e$ is 1 or a
quadratic non residue mod $p$.

These also define rank 2 $p$-groups for $p=3$. We include the families for $p=3$ in the
calculations to follow because they are the source of most of the interesting behavior.
%We also include the metacyclic 2-groups in $\S \ref{metacyclic 2-groups}$. These form a
%large subclass of the rank two 2-groups.

We will be performing explicit calculations in the cobar complex and with Massey
products. See the Appendix for a review of these concepts.
%
\subsection{$G$ of type $C(r)$}\label{section 1}\mbox{}

Since $c^{p^{r-3}}$ is a central element of order $p$, we have the central
extension

$$\z\lra G\lra \z\times\z\times\zn{r-3}.$$
As the base is abelian, Theorem \ref{base is abelian} tells us that $\VL G$
is semi-Koszul.  We shall now explicitly calculate $H^*(\VL G)$.

\subsubsection{Case 1: $r\geq4$}

\begin{alignat*}{2}
\notag\Gamma^2 &= \bigl\langle a, b, c \bigr\rangle = G &\qquad \Gamma^{2/3}
&= \f\langle \overline{a},\overline{b},\overline{c} \rangle\\
%
\notag\Gamma^{2p} &= \langle c^p \rangle \simeq \mathbb{Z}/(p^{r-3})
&\qquad
\Gamma^{2p/2p+1} &= \f\langle \overline{c^p}\rangle\\
%
& \, \vdots & & \, \vdots \\
%
\notag\Gamma^{2p^{r-3}} &= \langle c^{p^{r-3}}\rangle \simeq
\z &\qquad \Gamma^{2p^{r-3}/2p^{r-3}+1} &=
\f\langle\overline{c^{p^{r-3}}}\rangle
 \\
%
\notag & & \qquad \Gamma^{i/i+1} &=0 \text{ for } i\neq 2 p^j,\, j\leq r-3.
\end{alignat*}

Since $a^{-1}b^{-1}ab= c^{p^{r-3}} \in \Gamma^{2p^{r-3}}$ falls below $\Gamma^4$, $\L G$
is an abelian restricted Lie algebra over $\f$.  Moreover, $\xi\bigl(
\overline{a}\bigr)=\xi\bigl(\ol{b}\bigr)=0,\, \xi\bigl(\overline{c^{p^i}}\bigr)=
\overline{c^{p^{i+1}}}$. Let $x=\ol{a},\, y=\ol{b},\, z_i=\overline{c^{p^{i-1}}}$ for
$1\leq i\leq r-2$. Then
%
\begin{align*}
\VL G&=\Tens\{x,y,z_i\} \big/
 \left(\begin{smallmatrix}
   x^p=y^p=0, \, z_i^p=z_{i+1}\\
   z_{r-2}^p=0, \,  x,y,z_i \text{ commute }
 \end{smallmatrix}\right) \\
&\simeq \f[x,y,z]\big/\bl x^p,y^p,z^{p^{r-2}}\br \quad \text{as Hopf algebras.}
\end{align*}
%
So $H^* ( \VL C(r) )\simeq \Lambda[x,y,z]\otimes\f[u,v,w] \up{where} x,y,z\in
\Ext^{1,2}$, and $u,v\in Ext^{2,2p}$ and $w\in Ext^{2,2p^{r-2}}$.  As usual, the
one-dimensional classes $x,y$ are related to the two dimensional classes $u,v$
 by the algebraic Bockstein $-\bp$, and $w=\langle z\rangle^{p^{r-2}}$.  See Appendix
\ref{calculations} for details.
\smallskip

\subsubsection{Case 2: $r=3$}\mbox{}

Let $G=C(3)=\bigr\langle a,b,c \, \vert
\, a^p=b^p =c^p=1,\, (a,b)=c\in \mathscr{Z}(G)\bigr\rangle$.
This case is more interesting, as the filtration is different.
The filtration degree of $c$ is 4, which makes $\L G$ non-abelian: $\bigl[ \ol{a},\ol{b}
\bigr]=\ol{c}\in\Gamma^{4/5}$. We still have  $\xi(\overline{a})=\xi\bigl(\ol{b}\bigr)=0$.
Thus $\VL G \simeq \Tens\{a,b,c\}/J$, where $J$ is the ideal generated by the relations
$\bigl\{ab-ba=c, \,ac=ca, \,bc=cb, \,a^p=b^p=c^p=0\bigr\}$. The central extension of groups:
%
\beq\label{group extension for C}
\begin{CD}
\Z/(p) @>>z\mapsto c> G @>>a,b\mapsto x,y> \Z/(p) \times \Z/(p)
\end{CD}
\enq
%
induces the central extension of algebras
%
\beq\label{algebra extension for C}
\begin{CD}
\f[z]\big/(z^p) @>>z\mapsto c> VLG @>>a,b\mapsto x,y> \f[x,y]\big/(x^p,y^p)
\end{CD}
\enq
%
which we abbreviate as $F \lra A \lra B$. This sequence will be exact as \emph{graded} algebras
if we set $\deg z=4$.  The induced Cartan-Eilenberg spectral sequence has
$E_2$ term
%
\begin{equation}\label{ss for C}
E_2^{*,*}\bl C(3)\br=H^*(B)\otimes H^*(F) \simeq
\Lambda[\sigma,\tau]\!\otimes\!\f[\mu,\nu]\, \otimes\, \Lambda[
\zeta]\!\otimes\!\f[\epsilon]
\end{equation}
%
$$\begin{aligned}[t]
\sigma, \tau &\in E_2^{1,0,2} \ &\mu, \nu&\in E^{2,0,2p}_2 \\
\zeta &\in E_2^{0,1,4}   &\epsilon &\in E_2^{0,2,4p}
\end{aligned}
\qquad
\begin{aligned}[t]
\mu =& -\bp(\sigma)=\langle\sigma\rangle^p, \quad \nu = -\bp(\tau)=\langle\tau\rangle^p\\
\epsilon =& -\bp(\zeta)=\langle\zeta\rangle^p
\end{aligned}$$
\smallskip

The third superscript is the internal degree. The elements $\sigma\otimes 1$ and
$\tau\otimes 1$ in $E^{1,0}_2$ are permanent cycles converging to nonzero classes
$\alpha$ and $\beta$ in $H^{1,2}(A)$. Put a reverse lexicographic order on $A:\, c>b>a$.
Choose the basis for $A$ induced by this ordering: $\mathscr{B}=\bigl\{c^i b^j a^k \
\vert \, 0\leq i,j,k\leq p-1\bigr\}$.  The nontrivial relations are generated by the
following:
\beqs
ab=ba+c, \quad ac=ca, \quad bc=cb, \quad a^p=b^p=c^p=0.
\enqs
Let $\mathscr{B}^*=\{x^*\vert x\in \mathscr{B}\}$ denote the dual basis, i.e.,
\beqs
\langle x^*, y\rangle =
\begin{cases}
1 &\quad \text{if } y=x,\\ 0&\quad \text{otherwise.}
\end{cases}
\enqs
Identify $\alpha$, $\beta$ with the classes $\bigl[a^*\bigr]$, $\bigl[b^*\bigr]$ in the
cobar complex $C^*(A)$. Now, $\delta\bigl[c^*\bigr]=\bigl[a^*\vert b^*\bigr]\,
\Rightarrow\, \alpha\beta=0 \up{in} E_\infty^{2,0}$.  This implies
\beq\label{d2 for C}
d_2(1 \otimes\zeta)=\lambda \cdot \sigma\otimes\tau
\enq
for some $\lambda\in\f^\times$,
because that is the last differential which can hit $E_2^{2,0}$.  We can ignore
the constant $\lambda$, all that matters is that it is nonzero.  Since $\sigma\ot\tau=
1\ot\sigma\cdot 1\ot\tau$ is a product, \eqref{proof that bp} implies that
the spectral sequence \eqref{ss for C} collapses at $E_3$. Moreover, it
can be easily seen that
\be
\item $d_2:\Tens (\mu,\nu)\otimes\zeta\epsilon^k\lra
\Ideal(\sigma\tau)\otimes\epsilon^k$ is never zero,
\item $d_2 \equiv 0$ on $\Ideal(\sigma,\tau)\ot\zeta\epsilon^k \subset E_2^{*,2k+1},$
\item $d_2\equiv 0$ on the $\epsilon^k$ lines.
\ee

The permanent classes
\begin{gather}
S=\{\sigma\ot 1, \, \tau\ot 1, \, \mu\ot 1, \, \nu \ot 1, \, \sigma \ot \zeta, \,
\tau \ot \zeta, \, 1 \ot \epsilon\}
\label{generators 1}
\intertext{in $E_\infty$ converge to elements}
T=\{\alpha,\beta\, u, v, x_1, x_2,e \} \label{generators 2}
\end{gather}
in $H^*(A)$, with $\alpha,\,\beta\in H^{1,2}(A),\: u,\,v\in H^{2,2p}(A),\:
x_1,\,x_2\in H^{2,6}(A)$ and $e\in H^{2,4p}(A)$.
Every element in $E_\infty$ can be expressed as a sum of products from
$S$, for example, $\sigma\mu^i\nu^j\ot\zeta\epsilon^k=
(\sigma \ot \zeta)(\mu\ot 1)^i (\nu\ot 1)^j (1\ot\epsilon)^k$. Therefore,

\begin{theorem}
$T$ generates $H^*(A)=H^*\bl\VL C(3)\br$ multiplicatively.
\end{theorem}

We shall look at the relations in $H^*(A)$ in the following section.

\subsection{Relations in $H^*\bl\VL C(3)\br$}\mbox{}

We will use a theorem of Ravenel \cite[Appendix 1.4]{Ravenel} on Massey products in
spectral sequences.  Refer to \ref{massey products} for the definition of Massey
products.  The formulation is somewhat complicated, but the main idea, due to Kahn
\cite{Kahn}, is that we do not want any higher ``cross-differentials'' to interfere with
$d_2$. Since $d_r=0$ for $r\geq 3$, this is not an issue.  Thus we can safely state that
if $a_i\in E_2^{*,*}$ converge to $u_i$ in $H^*(A)$, then the Massey product $\langle
a_1,\dots,a_n \rangle\in E_3$ (when defined) will converge to $\langle u_1,
\dots,u_n\rangle$, up to indeterminacy of course. We shall also make use of the following
``juggling theorem":
\begin{lemma} {\cite[A1.4.6]{Ravenel}}
If  $\langle u_1,u_2, u_{3}\rangle$ and  $\langle u_2,u_3, u_4\rangle$
are defined, then
\beqs
u_1 \langle u_2,u_3, u_4\rangle=\langle \ol{u}_1,\bar{u}_2, \ol{u}_{3}\rangle u_4,
\enqs
where $\ol{u}=(-1)^{\deg u+1}u$ and $\deg u$ means total degree of $u$.
\end{lemma}

\begin{proposition}\label{relations} In $H^*(\VL C(3))$, the
multiplicative relations are generated by the following:

\begin{enumerate}
\item $\alpha^2=\alpha\beta=\beta^2=0$.\label{eq0}
\item $\alpha x_2=-\beta x_1.$\label{eq1}
%
\item $\alpha x_1=
\begin{cases}
\beta u,\quad &p=3,\\
0,\quad &p>3.
\end{cases}\qquad
\beta x_2=
\begin{cases}
-\alpha v,\quad &p=3,\\
0,\quad &p>3.
\end{cases}$\label{eq2}
%
\item $x_1^2,x_1 x_2, x_2^2=\begin{cases}-ux_2, -uv, vx_1, \quad &p=3,\\
0,0,0,\quad &p>3.\end{cases}$\label{eq3}
\end{enumerate}

\noindent In addition, we have the following Massey products

\begin{enumerate}\addtocounter{enumi}{4}
\item $x_1 = \langle\alpha, \alpha, \beta\rangle=-1/2\langle \alpha,
\beta, \alpha\rangle=\langle\beta, \alpha, \alpha\rangle.\label{eq4}$
%
\item $x_2=-\langle\alpha,\beta,\beta\rangle=1/2\langle\beta,\alpha,\beta\rangle
=-\langle\beta,\beta,\alpha\rangle.\label{eq5}$
%
\item $\langle\alpha\rangle^p=-\bp(\alpha)=u, \quad \langle\beta\rangle^p=-\bp(\beta)=v.
\label{eq6}$
%
\end{enumerate}
\end{proposition}

We will prove the last three first.

\noindent\textit{Proof of} \ref{relations}.\ref{eq4},  \ref{relations}.\ref{eq5}.

Using the notation of \S  \ref{massey products}, let $(U,\delta)=\bl E_2^{**}, d_2\br$.
The elements $a_1=a_2=\sigma\ot 1, \, a_3=\tau\ot 1$ are in $E_2^{1,0}$, and have total
degree 1. Therefore $\ol{a_i}=a_i$. Now $a_1 a_2=\sigma^2\ot 1=0,$ and $a_2 a_3
=\sigma\ot\tau = d_2(1\ot\zeta)$, so we can take $u_1=0$,\, $u_2=1\ot\zeta$. Then
$\langle a_1,a_2,a_3 \rangle=\sigma\ot 1\cdot 1\ot\zeta + 0\cdot\tau\ot 1 =
\sigma\ot\tau$, which converges to $x_1$.  The indeterminacy is $\alpha H^1 + H^1
\beta$, which is 0 since $H^1(A)=\f\langle\alpha,\beta\rangle$ and $\alpha^2=\alpha\beta=\beta^2=0$.

$\langle\beta, \alpha, \alpha\rangle=\tau\ot 1\cdot 0 +(-1\ot\zeta)\cdot\sigma\ot 1
=\sigma\ot\zeta$.
The minus sign comes from the fact that $\beta\alpha=-\alpha\beta$.  It disappears
because $1\ot\zeta$ and $\sigma\ot 1$ anticommute.

$\langle \alpha,\beta, \alpha\rangle=\sigma\ot 1\cdot(-1\ot\zeta)+1\ot\zeta\cdot
\sigma\ot 1=-2\,\sigma\ot\zeta$.

Now let $U=C^{**}(A)$, the cobar complex of $A$.  $\alpha$ is represented by $a_1=a_2=[a^*]$,
$\beta$ by $a_3=[b^*]$.  Let us calculate $\langle\alpha, \alpha, \beta\rangle$ in $U$.
$u_1=[{a^2}^*],\, u_2 =[c^*]$, and $\langle a_1,a_2,a_3 \rangle=[a^*\vert c^*]
+[{a^2}^*\vert b^*]$.  We now have a representative for $x_1$ in $C^*(A)$.

Other representatives are
\begin{subequations}\label{Massey for x1}
\begin{align}
x_1&=-\recip{2}\langle \alpha,\beta,\alpha\rangle=\frac{1}{2}\bl [a^*\vert c^*]-
[c^*\vert a^*]-[a^*\vert {ba}^*]\br\\
x_1&=\langle\beta,\alpha,\alpha\rangle=[b^*\vert{a^2}^* ] -[c^* \vert a^* ]+
[{ba}^* \vert a^* ]
\end{align}
\end{subequations}
The procedure for $x_2$ is similar.  Some representatives are
\begin{subequations}\label{Massey for x2}
\begin{align}
x_2&=-\langle\alpha,\beta,\beta\rangle=-[a^*\vert{b^2}^*]-[c^*\vert b^*]\\
x_2&=-\langle\beta,\beta,\alpha\rangle=-[b^*\vert{ba}^*]+[b^*\vert c^*]-[{b^2}^*\vert a^*]\\
x_2&=\recip{2}\langle\beta,\alpha,\beta\rangle=\recip{2}\bl[b^*\vert c^*]+[{ba}^*\vert b^*]
-[c^*\vert b^*]\br.
\end{align}
\end{subequations}\qed

\noindent\textit{Proof of} \ref{relations}.\ref{eq6}.

Follows from naturality of the algebraic Steenrod operations.

\noindent\textit{Proof of} \ref{relations}.\ref{eq0}.

Follows from relations in $E_\infty$.

\noindent\textit{Proof of} \ref{relations}.\ref{eq1}.

$\alpha x_2=-\alpha\langle\alpha,\beta,\beta\rangle=-\langle\ol{\alpha},\ol{\alpha},\ol{\beta}\rangle
\beta=-x_1 \beta =-\beta x_1$

\noindent\textit{Proof of} \ref{relations}.\ref{eq2}.

$\alpha x_1=\alpha\langle\alpha,\alpha,\beta\rangle=\langle\ol{\alpha},\ol{\alpha},
\ol{\alpha}\rangle\beta=\langle\alpha,\alpha,\alpha\rangle\beta=u\beta$
for $p=3$. For larger primes, $\langle\alpha\rangle^3=0$, since for
$\langle\alpha\rangle^p$ to be defined, we need all lower symmetric Massey
products to vanish. Similarly, $\beta
x_2=-\beta\langle\beta,\beta,\alpha\rangle=-\langle\beta,\beta,\beta\rangle
\alpha=-v\alpha$ for $p=3$, while it vanishes for larger primes. In either
situation, the products fall to the lower filtration level
$F^3H^3(A)=E_\infty^{3,0}$.\qed

\noindent\textit{Proof of} \ref{relations}.\ref{eq3}.

Since ${(\sigma\ot\zeta)}^2 =-\sigma^2\ot\zeta^2=0$ in $E_\infty^{2,2}\simeq
F^{2/3}H^4(A)=F^2 H^4(A)\big/F^3 H^4(A)$, we deduce that $x_1^2\in F^3 H^4(A)$.  Since
$F^3 H^4(A)$ is spanned by $\bigl\{u x_1, ux_2, vx_1, vx_2, u^2,$ $uv, v^2\bigr\}$, we
can express $x_1^2$ as a linear combination
\begin{equation}\label{expr}
x_1^2=c_1 u x_1+c_2 ux_2+c_3 vx_1+c_4 vx_2+c_5 u^2+c_6 uv +c_7 v^2, \quad
c_i\in \f.
\end{equation}
\noindent The element $x_1^2$ has internal degree $12$, while the elements on the right
hand side of (\ref{expr}) have internal degree larger than or equal to $2p+6$.  For
$p>3$, we are forced to conclude that $x_1^2=0$.  As usual, the case $p=3$ requires
special attention.

Let $p=3$. Multiplying both sides of (\ref{expr}) by $\alpha$ gives us $\alpha
x_1^2=\beta ux_1$ on the left, and $c_1\beta u^2-c_2\beta u x_1 +c_3\beta uv-c_4\beta
vx_1+c_5\alpha u^2+ c_6\alpha uv+c_7\alpha v^2$ on the right. The summands are linearly
independent in $H^5(A)$ because their representatives in $E_\infty$ are linearly
independent. Therefore, $c_2=-1$, and all the other $c_i$ are zero. Starting with linear
combinations in $H^4(A)$ for $x_1 x_2$ and $x_2^2$, then multiplying by $\alpha$, will
yield the other two identities.\qed

These relations all come about from the fact that $\sigma^2=\tau^2=\zeta^2=0,\,
\sigma\tau=-\tau\sigma$ in $E_\infty$.  Since there are no other identities of this type
in $E_\infty$, there are no new relations in $H^*(A)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  G(4,e)  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$G$ of type $G(r,e)$}\mbox{}

We next turn our attention to groups $G$ of type
$$G(r,e)=\langle f,g,h \; \vert \; f^p=g^{p^{r-2}}=h^p= \lp g,h\rp=1,\, \lp
f,h^{\sss{-1}}\rp=g^{ep^{r-3}},\, \lp f,g\rp=h\rangle.$$
 We will soon see that for our purposes, we can take $e=1$.
\begin{lemma}\label{center}
 $g^p\in\mathcal{Z}(G).$
\end{lemma}
\begin{proof} Since $(f,g)=h$, we have $\inv{f}\inv{g}f=h\inv{g}$, and
\beq
(f,g^{p})=f^{-1}g^{-p}f g^{p}={\lp f^{-1}g^{-1}f\rp}^{p} g^p =\bl hg^{-1}\br^p
g^p=h^p g^{-p} g^p=1
\enq
The second to last equality follows because $(g,h)=1$.
\end{proof}

We will investigate the exact sequence
%
\begin{equation}\label{yae}
\begin{CD}
\Z/(p) \langle d\rangle @>>d\mapsto g^{p^{r-3}}> G(r) @>>>
Q(r-1)
\end{CD}
\end{equation}
This is a central extension by Lemma \ref{center}. As the base is not abelian, we cannot
use Theorem \ref{base is abelian}.
%
The cokernel $Q$ has generators $\tilde{f},\tilde{g},\tilde{h}$ and relations:
\begin{equation*}
\tilde{f}^{\,p}=\tilde{h}^{\,p}=\bigm( \tilde{g},\tilde{h} \bigm)=1,\,
 \bigm( \tilde{f},\tilde{h}^{\,\scriptstyle{-1}} \bigm)=\tilde{g}^{\,ep^{r-3}}=1,\;
 \bigm( \tilde{f},\tilde{g} \bigm)=\tilde{h}.
\end{equation*}
%
Therefore $Q(r-1)\simeq\langle a,b,c \; \vert \; a^p=b^{p^{r-3}}=c^p=1,\,
(a,b)=c\in \mathscr{Z}(Q)\rangle,$ where $a=\tilde{f}$, $b=\tilde{g}$,
$c=\tilde{h}$. For $r=4$,  $Q(r-1)=Q(3)\simeq C(3)$. We will focus on this case
presently. The case $r>4$ is taken up in section \ref{case 2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\subsubsection{Case 1: $r=4$}\label{counterexample}\mbox{}

For the extension \eqref{yae}, we have
\begin{equation}\label{G4 extension}
\begin{CD}
  \z @>>d\mapsto g^p>  G(4) @>>f,g,h\mapsto a,b,c>   Q=C(3)
\end{CD}
\end{equation}
%
with $f,g,h,d$ in filtration degree $2,2,4,2p$ respectively. The Lie algebra $\L G$  is
generated by $\left\{ x=\ol{f}, y_1=\ol{g}, y_2=\overline{g^p}, z=\ol{h}\right\}$. Since
$2+4=2p\Leftrightarrow p=3$, $\lb\,\ol{f},\ol{h}\,\rb=0$ for $p>3$. Since
$\overline{\inv{h}}=-\ol{h}$,
$\lb\,\ol{f},\ol{h}\,\rb=-\lb\,\ol{f},\overline{\inv{h}}\,\rb=-\overline{g^{ep}}
=-e\overline{g^p}$ for $p=3$. Summarizing, the non-trivial structure of $\L G$ is as
follows
\begin{equation*}
\xi(y_1)=y_2,\quad [x,y_1]=z,\quad [x,z]=
\begin{cases}
-ey_2 \quad & p=3\\
 0         \quad & p>3.
\end{cases}
\end{equation*}
For $p=3$, $e$ can only equal 1, so we see that $e$ doesn't really matter.

Taking the usual quotient of $\Tens\{x,y_1,y_2,z\}$ by the ideal $J$ of relations as in
Definition \ref{VL},
\begin{equation}\label{VLG for G4}
\VL G \simeq
\begin{cases}
\Tens\{f,g,h\}\big/
   \left(\begin{smallmatrix}
f^p=g^{p^2}=h^p=0, \, fg=h+gf \\
    fg^p=g^p f,\, fh=hf,\, gh=hg\end{smallmatrix}\right) &\quad p>3,\\
\vspace{-3mm}\\
\Tens\{f,g,h\}\big/
   \left(\begin{smallmatrix} f^p=g^{p^2}=h^p=0,\, fg=h+gf\\
    fg^p=g^p f,\, fh=-g^p+hf,\, gh=hg\end{smallmatrix}\right)   &\quad p=3.
\end{cases}
\end{equation}
%
under the mapping $x\mapsto f,\, y_1\mapsto g,\, y_2\mapsto g^p,\, z\mapsto h$.

The central extension of Hopf algebras:
\begin{equation}\label{algebra extension for G4}
\begin{CD}
\f[d]\big/(d^p) @>i>d\mapsto g^p> \VL G @>\pi>f,g,h\mapsto a,b,c> \VL Q=\VL C(3).
\end{CD}
\end{equation}
denoted by $F\stackrel{i}{\lra} A\stackrel{\pi}{\lra} B$ gives us a spectral sequence
\begin{equation}\label{ss for G4}
E_2^{*,*}\bl G(4)\br\simeq H^*(B) \ot \left(\Lambda[z]\ot\f[w]\right) \Rightarrow H^*(A).
\end{equation}
where $z\in E_2^{0,1,2p}$ and $w\in E_2^{0,2,2p^2}$. We shall suppress tensor notation in
spectral sequences (e.g $1\ot z=z$) from now on.
%
\begin{proposition}\label{d2 of z} In the spectral sequence \eqref{ss for G4}
 $$d_2(z)=
\begin{cases}
v-x_1, \quad &p=3\\
v,     \quad &p>3.
\end{cases}$$
\end{proposition}
%
\begin{proof}
Put an ordering $h>g>f$ on $A=\VL G$. A basis for $A$ consists of $\{h^i g^j f^k\vert
0\leq i,k\leq p-1 ,\, 0\leq j\leq p^2-1\}$. Using the relations \eqref{VLG for G4} for
$A$, we see that in the cobar complex $C^*(A)$
\begin{equation}
\label{diff}
\delta [{g^p}^*]=\begin{cases}
\sum_{i=1}^{p-1}[{g^i}^*\vert {g^{p-i}}^*] -[f^*\vert h^*]-[{f^2}^*\vert g^*]&\quad p=3,\\
\sum_{i=1}^{p-1}[{g^i}^*\vert {g^{p-i}}^*]&\quad p>3.
\end{cases}
\end{equation}
Under the edge map $\pi^*: H^*(B)\lra H^*(A)$, $v\mapsto
\cl\bl\sum_{i=1}^{p-1}[{g^i}^*\vert {g^{p-i}}^*]\br$ and $x_1\mapsto
\cl\bl[f^*\vert h^*]+[{f^2}^*\vert g^*]\br$. The only differential that
can cause the right hand side in \eqref{diff} to be zero is $d_2(z)$.
\end{proof}
%
\begin{proposition}  \label{bp}
$\bp(v)=\bp(x_1)=0$ in $H^3(B)$.
\end{proposition}
\begin{proof}
$v, x_1 \in H^{2,2p}(B)$. The image of $\bp : H^{2,2p}(B)\lra H^{3,2p^2}$ is zero because the internal degrees
of the elements of $H^3(B)$ are all less than or equal to $2+4p$, which is less
than $2p^2$.
\end{proof}
%
\begin{proposition}\label{zero divisor}
$d_2(z)$ is not a zero divisor in $H^*(B)$.
\end{proposition}
\begin{proof} Assume $d_2(z)y=0$ for some nonzero element $y\in H^{s+t}(B)$ of filtration
degree $s$.  Project $y$ to $\bar{y}\in F^{s/s+1}H^{s+t}(B)\simeq E_\infty^{s,t}\bl
C(3)\br$.

For $p>3$, the equation $vy=0$ projects to $ \nu\bar{y}=0$. Since $\nu$ is not a zero
divisor in $E_\infty\left( C(3)\right)$, $\bar{y}=0$.  This means $y\in F^{s+1}H^*(B)$,
which is a contradiction.

For $p=3$, we have $(v-x_1)y=0$, or $vy=x_1y$. This projects to $\nu\bar{y}
=\sigma\zeta\bar{y}$. Note that $\nu\bar{y}\in E_\infty^{s+2,t}$, while $\sigma
\zeta\bar{y}\in E_\infty^{s+1,t+1}$.  The two cannot possibly be equal, unless of course
they are both zero.  But $\nu$ is not a zero divisor, and the same contradiction ensues.
\end{proof}

We conclude
\begin{proposition}\label{result}
The spectral sequence \eqref{ss for G4} collapses at $E_3$, and
$$H^*(A)\simeq
\begin{cases}
 H^*(B)\big/(v-x_1)\ot\f[w],\quad &p=3,\\
 H^*(B)\big/(v)\ot\f[w],\quad &p>3.
\end{cases}$$
\end{proposition}

\begin{theorem}\label{cohomology of G4}
$H^*(\VL G(4))$ is generated by $\{\alpha, \beta, u, x_1, x_2, e, w\}$, with
$\alpha,\beta\in H^{1,2}$,
$x_1, x_2\in H^{2,6}$, $u\in H^{2,2p}$,  $e\in H^{2,4p}$, and $w\in
H^{2,2p^2}$.
%The bidegrees of the generators are as in
%\eqref{generators 2}, with $w\in H^{2,2p^2}$.
The relations are generated by
%
\begin{enumerate}
\item $\alpha^2=\alpha\beta=\beta^2=0$. %\label{eq0}
\item $\alpha x_2=-\beta x_1.$ %\label{eq1}
%
\item $\alpha x_1=
\begin{cases}
\beta u,\quad &p=3,\\
0,\quad &p>3.
\end{cases}\qquad
\beta x_2=
\begin{cases}
-\beta u,\quad &p=3,\\
0,\quad &p>3.
\end{cases}$  %\label{eq2}
%
\item $x_1^2,x_1 x_2, x_2^2=\begin{cases}-ux_2, -ux_1, -ux_2, \quad &p=3,\\
0,0,0,\quad &p>3.\end{cases}$%\label{eq3}
\end{enumerate}
%
In particular, $\VL G(4)$ is semi-Koszul. The Massey products of Proposition \ref{relations}
still hold, except now
$\langle\beta\rangle^p=x_1$ for $p=3$, and
$\langle\beta\rangle^p=0,\ \langle\beta\rangle^{p^2}=w$ for $p>3$.
\end{theorem}
%
\begin{proof} This is simply a matter of setting $v=0$ or $v=x_1$ in the relations
for $H^*\bl \VL C(3)\br$ (Proposition \ref{relations}). The class $w$ is
formally the coboundary of $\bigl[{g^{p^2}}^*\bigr]$. For $p>3$, this is
equal to the representative of $\langle\beta\rangle^{p^2}$ in the cobar
complex.
\end{proof}

\begin{remark}
Notice that $x_2^2=- u x_2$ implies $x_2+u$ is a zero divisor, with $x_2(x_2+u)=0$.
If we had a central extension of 3-groups
$\Z/(3) \lra K \lra G(4)$
and the induced spectral sequence \eqref{algebra spectral sequence} had
$d_2(z)=x_2+u$
then $x_2\ot z\in E_\infty^{2,1}$ would be an indecomposable element of dimension 3.  In this
case, $\VL K$ would not be semi-Koszul.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  R>4         %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection{Case 2: $r>4$}\label{case 2}\mbox

Now let $G=G(r,e),\, r>4,\, p\geq 3$. The situation is
\begin{gather*}
\begin{CD}
\Z/(p)  \langle d\rangle @>>d\mapsto g^{p^{r-3}}> G(r) @>>f,g,h\mapsto a,b,c>
Q(r-1)
\end{CD}\\
\text{where }\, Q(r-1)\simeq\langle a,b,c \; \vert \; a^p=b^{p^{r-3}}=c^p=1,\,\lb a,b\rb=c\in \mathscr{Z}(Q)\rangle
\end{gather*}
The filtration degrees of $f,g,h,d$ are $2,2,4,2p^{r-3}$ respectively.

Let us first deal with finding $H^*(\VL Q(r-1))$. There is a central extension for $Q$
\beqs
\begin{CD}
\z\langle z\rangle @>>z\mapsto c> Q\langle a,b,c\rangle @>>a,b\mapsto x,y> \z
\times \zn{r-3}\langle x, y\rangle
\end{CD}
\enqs
which is very similar to \eqref{group extension for C}. The induced algebra
extension
\beqs
\begin{CD}
\f[z]\big/(z^p) @>>> \VL Q(r-1) @>>> \f[x,y]\big/\bl x^p,y^{p^{r-3}}\br, \quad \deg z=4
\end{CD}
\enqs
gives us the spectral sequence
$$E_2^{**}\bl Q(r-1)\br
\simeq \Lambda[\sigma,\tau]\!\otimes\!\f[\mu,\nu]\, \otimes\, \Lambda[
\zeta]\!\otimes\!\f[\epsilon]$$ with $\sigma,\tau,\mu,\nu,\zeta,\epsilon$ as in \eqref{ss
for C} except $\nu$ has internal degree $2p^{r-3}$ and $\nu=\langle
\tau\rangle^{p^{r-3}}$. Just as in \eqref{d2 for C},\, $d_2(\zeta)=\sigma\tau$ and the
spectral sequence collapses at $E_3$.
%
\begin{theorem}
$H^*\bl\VL Q(r-1)\br$ is generated by $\{\alpha,\beta,x_1,x_2,u,v,e\}$, with
$\alpha,\beta\in H^{1,2}$, $x_1, x_2\in H^{2,6}$, $u\in H^{2,2p}$, $v\in H^{2,2p^{r-3}}$
and $e\in H^{2,4p}$. Relations are generated by
%
\begin{enumerate}
\item $\alpha^2=\alpha\beta=\beta^2=0$.
\item $\alpha x_2=-\beta x_1.$
%
\item $\alpha x_1=
\begin{cases}
\beta u,\quad &p=3,\\
0,\quad &p>3.
\end{cases}\qquad
\beta x_2=0.$
%
\item $x_1^2,x_1 x_2, x_2^2=\begin{cases}-ux_2,0, 0, \quad &p=3,\\
0,0,0,\quad &p>3.\end{cases}$
%
\end{enumerate}
The Massey products of Proposition \ref{relations}
still hold, except $\langle\beta\rangle^{p^{r-3}}=v$ for all $p$.
%$\bp(\beta)=\langle\beta\rangle^p=0$
%
\end{theorem}
%
\begin{proof}
The proofs of  \eqref{eq0} and \eqref{eq1} are the same as in \ref{relations}. For the
second half of \eqref{eq2}, $\beta x_2=-\beta\langle
\beta,\beta,\alpha\rangle=-\langle\beta,\beta,\beta\rangle\alpha=0$ even for
$p=3$.
% because for $\langle \beta\rangle^9$ to be defined, we need
%$\langle\beta\rangle^i=0$ for $i<9$.
In the proof of \eqref{eq3} for $p=3$, we are reduced to solving the equation
\beq\label{expr2}
x_ix_j=c_1 ux_1+c_2ux_2+c_5u^2, \quad  c_k\in \f,
\enq
by reason of internal dimension.  Multiplying both sides by $\alpha$ and equating coefficients yield
the desired identities.  For example, $\alpha x_2^2=-\beta x_1 x_2=0$, while the right side of
\eqref{expr2} becomes $c_1\beta u^2-c_2\beta u x_1+c_5 \alpha u^2$.
\end{proof}

We now turn to finding $H^*(\VL G(r))$. Notice that since
$(f,h^{-1})=g^{p^{r-3}}$ falls below $\Gamma^6 G$, $\lb\ol{f},\ol{h}\rb=0$ in
$\L G$ even for $p=3$.
Thus \eqref{VLG for G4} becomes
\beqs
\VL G(r)\simeq \Tens\{f,g,h\}\Big/\binom{f^p=g^{p^{r-2}}=h^p=0,\,fg=h+gf}
{ fg^p=g^pf, \, fh=hf,\, gh=hg }.
\enqs
%
As in \eqref{algebra extension for G4}, setting $\deg d=2p^{r-3}$ gives us the graded
 algebra extension
%
\beqs
\begin{CD}
\f[d]/(d^p) @>i>> \VL G(r) @>\pi>> \VL Q(r-1)
\end{CD}\enqs
%
which induces the analogue of \eqref{ss for G4}
%
\beq\label{ss for G(r)}
E_2^{*,*}\bl G(r)\br\simeq H^*\bl\VL Q(r-1)\br \ot \Lambda[z]\ot\f[w].
\enq
%
\begin{proposition} In the spectral sequence \eqref{ss for G(r)},
$d_2(z)=v$.
\end{proposition}
\begin{proof}
In the cobar complex $C^*\bl\VL G(r)\br$, $\delta \bigl[{g^{p^{r-3}}}^*\bigr]=\sum_{i=1}^{p^{r-3}-1}
\bigl[{g^i}^*\vert {g^{p^{r-3}-i}}^*\bigr]$, which is the representative of
$\pi^*(v)$.
Thus $d_2(z)=v$ for all $p$.
\end{proof}
%
Proceeding in the same way as in Propositions \ref{bp}, \ref{zero divisor} and
\ref{result}, we obtain
\begin{proposition}
$\bp(v)=0$, and $v$ is not a zero divisor in $H^*\bl\VL Q(r-1)\br$. Thus the spectral sequence
\eqref{ss for G(r)} collapses at $E_3$ and
\beqs
H^*\bl \VL G(r)\br\simeq E_\infty \simeq H^*\bl\VL Q(r-1)\br\big/(v)\otimes
\f[w]
\enqs
\end{proposition}
%
Finally,
\begin{theorem}
$H^*\bl\VL G(r)\br$ is generated by $\alpha, \beta, u, x_1, x_2, e, w$. The bidegrees
of the generators are as in Theorem \eqref{cohomology of G4}, except $w\in
H^{2,2p^{r-2}}$.
Relations are as in Theorem \ref{cohomology of G4} except now $\beta x_2=x_1
x_2=x_2^2=0$, $\langle\beta\rangle^{p^{r-2}}=w$ for all $p$.
In particular, $H^*\bl \VL G(r)\br$ is semi-Koszul.
\end{theorem}


%%%%%%%%%%*********************************************************************
%%%%%%%%%%*********************************************************************

\subsection{$G$ Metacyclic}

Let $G$ be a metacyclic $p$-group. That is, there is an extension
$$
\begin{CD}
\zn{m}@>>> G @>>> \zn{n}
\end{CD}
$$
For $p$ odd,  any  metacyclic $p$-group $G$ has presentation
\beqs
G =P(m,n,q,l)=\langle x,y \; \vert \; x^{p^m}=1,\, y^{p^n}=x^{p^q},\,
(x,y)=x^{p^l}\rangle
\enqs
for integers $m,n,l,q\geq 1$ satisfying $(p^l+1)^{p^n}\equiv 1 \mod p^m$ and
$(p^l+1)p^q\equiv p^q\mod p^m$\, \cite{Dietz-odd}.  We can get all groups up to
isomorphism by requiring $l\leq m$, $q\leq m$, $n+l\geq m$ and $q+l\geq m$. Note that
$y^{p^{m+n-q}}=1$.

It is clear that $\L G$ is generated by $x_i=\overline{x^{p^{i-1}}}\in
\Gamma^{2p^{i-1}/2p^{i-1}+1},\, 1\leq i\leq m$, and $y_j=\overline{y^{p^{j-1}}}\in
\Gamma^{2p^{j-1}/2p^{j-1}+1}$,
$1\leq j\leq n+m-q$, some of which may be zero.
\begin{lemma}\label{metacyclic lemma 1} For all $i,j$, \,$[x_i,y_j]=0$.
\end{lemma}
\begin{proof}
$\inv{y}xy=x^{p^l+1}\, \Ra\, y^{-p^j}x^{p^i}y^{p^j}=x^{p^i(p^l+1)^{p^j}}\, \Ra\, \bl
x^{p^i},y^{p^j}\br=x^{p^i\left( (p^l+1)^{p^j}-1\right)}.$ The exponent is $
p^i\bl(p^l+1)^{p^j}-1\br=p^i\bl p^{lp^j}+\dots +p^j p^l+1-1\br=p^{i+j+l}a,\quad a\in \Z.
$ Therefore, $\bl x^{p^i},y^{p^j}\br=(x^a)^{p^{i+j+l}}\in \Gamma^{2p^{i+j+l}}\subsetneq
\Gamma^{2(p^i+p^j)}$, even for $i=j=0$.
\end{proof}

 We can break the analysis of $\L G$ down into three cases:

\noindent\textbf{Case 1:}\, $n<q$.

$\xi(y_j)=y_{j+1}$ for $j< n$. Since $y^{p^n}=x^{p^q}$ falls down to a lower
filtration level, $\xi(y_{n})=\overline{y^{p^n}}=0$, and there are no $y_j$ for $j> n$. The other
generators
$\{x_i\}_{1\leq i\leq m}$ have restriction $\xi(x_i)=x_{i+1}$ for $i<m$ and
$\xi(x_m)=0$. Then
\beqs
\VL G\simeq \f[x,y]\big/\bigl(x^{p^m},y^{p^n}\bigr) \quad \deg x=\deg y=2.
\enqs

\noindent\textbf{Case 2:}\, $n>q$.

By the same reasoning as above, t here are no $x_i$ for $i>q$. $\xi(x_i)=x_{i+1}$ for $i<
q$, $\xi(x_{q})=0$. $\xi(y_j)=y_{j+1}$ for $j< n+m-q$, and $\xi(y_{n+m-q})=0$. So
\beqs
\VL G\simeq \f[x,y]\big/\bl x^{p^q},y^{p^{n+m-q}} \br \quad \deg x=\deg y=2.
\enqs

\noindent\textbf{Case 3:}\, $n=q$.

In this case, $x^{p^q}=y^{p^n}$ in $\Gamma^{2p^q}=\Gamma^{2p^n}$. Let
$z=y x^{p^{m-n}-1}$.
Then
$(x,z)=\inv{x}x^{1-p^{m-n}}\inv{y}x y x^{p^{m-n}-1}=x^{p^l}=(x,y)$.
\begin{claim} $z^{p^n}$=1.
\end{claim}
\begin{proof}
It can be shown by induction that $(yx^a)^{b}=y^b x^{a[(p^l+1)^b-1]/p^l}$. For
$a=p^{m-n}-1,\, b=p^n$, the exponent of $x$ is
$$\bl p^{m-n}-1 \br\bl (p^l+1)^{p^n}-1\br/p^l=\bl p^{m-n}-1 \br
p^{n+l}c/p^l =( p^{m}-p^{n})c,$$ where
$c=1+\recip{p^n}\dbinom{p^n}{2}p^l+\recip{p^n}\dbinom{p^n}{3}p^{2l}+\dotsb =1+p^l d$, for
some $d\in\Z$. Then $z^{p^n}=y^{p^n} x^{( p^{m}-p^{n})c}=x^{p^n (-c+1)}=x^{p^{n+l}d}=1$
since $n+l\geq m$.
\end{proof}
%$c=1+\dfrac{p^{n}-1}{2}p^l+\dfrac{p^n-1\cdot p^n-2}{2\cdot 3}p^{2l}$

So $G\simeq \langle x,z\vert x^{p^m}=1,\, z^{p^n}=1,\, (x,z)=x^{p^l}\rangle$.
Generators for
$\L G$ are $\{x_i\}_{1\leq i\leq m}$ and $\{z_j\}_{1\leq j\leq n}$, and
\beqs
\VL G\simeq \f[x,z]\big/\bl x^{p^m},z^{p^{n}}\br \quad \deg x=\deg z=2.
\enqs
Note that the transformation $y\mapsto z$ in $G$ corresponds to the map $y_1\mapsto
z_1=y_1-x_1$ in $\L G$.

\begin{theorem} Let $G=P(m,n,q,l)$, then
\begin{gather*}
H^*(\VL G)\simeq \Lambda[x,y]\ot\f[u,v],\ x,y\in H^{(1,2)},\,  u=\langle x\rangle^a\in
H^{(2, 2a)},\,  v=\langle y\rangle^b\in H^{(2,2b)}
\end{gather*}
where $a=p^m,\, b=p^n$ if $q\geq n$ and $a=p^q,\, b=p^{m+n-q}$ if $q<n$.
\end{theorem}

We have finally proven
\begin{theorem} Let $p\geq 5$, and let $G$ be a finite $p$-group of rank 2. Then
$\VL G$ is semi-Koszul.
\end{theorem}

\appendix
\section{}
\subsection{The Cobar Complex}\label{cobar complex}

We summarize the construction of the \emph{cobar complex} of $A$, which can be used to
calculate the cohomology of $A$.  Refer to Priddy \cite[Section 1]{Priddy} for more
details.

As in Section \ref{algebraic preliminaries}, let $A$ be an algebra with augmentation
ideal $I$. The cobar complex of $A$, denoted by $C^*(A)$, is given by $C^s(A)=(I^*)^{\ot
s}$. A typical element is written $\bigl[\alpha_1\vert\dotsb\vert\alpha_s\bigr]$ and has
bidegree $(s,t)$, where $t=\sum_{i=1}^s \deg\alpha_i$ is the internal degree inherited
from $A$.  Let $C^{s,t}$ be spanned by elements of bidegree $(s,t)$. There is a
differential $\delta : C^{s,t}(A)\longrightarrow C^{s+1,t}(A)$, which may be computed as
follows. Let $\mu^*$ be the composite $A^* \stackrel{\mu^*_A}{\lra} \bigl(A\otimes
A\bigr)^* \stackrel{\theta^{-1}}{\lra} A^*\otimes A^*$, where $\mu^*_A$ is dual to the
multiplication map $\mu_A \text{ of } A$, and $\theta$ is the isomorphism defined more
generally by
\begin{equation*}
\theta :M^*\ot N^*\lra \bl M\ot N\br^*, \quad \theta(f\ot g)(m\ot n)=(-1)^{\deg g\,\deg
m} f(m)g(n)
\end{equation*}
for $A$-modules $M$ and $N$. If $\mu^*(\alpha)=\sum_r \alpha^{'}_r \otimes
\alpha^{''}_r$, then $\delta$ is given by
\begin{equation}\label{delta}
\delta \bigl( \bigl[ \alpha_1 \vert \dotsb \vert \alpha_s \bigr] \bigr) = -\sum_{1\leq i
\leq s; r} (-1)^{e_{i,r}} \bigl[ \alpha_1 \vert \cdots \vert \alpha^{'}_{i,r} \vert
\alpha^{''}_{i,r} \vert \cdots \vert\alpha_s \bigr],
\end{equation}
where $e_{i,r}$ is the total degree of $\bigl[\alpha_1 \vert \cdots \vert
\alpha^{'}_{i,r} \bigr]$. The cohomology groups of $A$ are then $H^{s,t}(A)=H^s\bl
C^{*,t}(A),\delta\br$.


$C^*(A)$ has the structure of a differential graded algebra with product
\begin{equation*}
[\alpha_1\vert \dotsb\vert \alpha_s]\cup [\beta_1\vert \dotsb\vert \beta_{s'}]=
[\alpha_1\vert \dotsb\vert \alpha_s \vert \beta_1 \vert \dotsb\vert \beta_{s'}].
\end{equation*}
The induced product in cohomology coincides with the Yoneda Ext product as given in
\cite[2.2]{Adams}.

\subsection{Massey Products}\label{massey products}
The following material on Massey products can be found in \cite[A1.4]{Ravenel} and
\cite{Kraines}. We recall the definition of the 3-fold Massey product . Let $(U,\delta)$
be a differential bigraded algebra. Let $\alpha_i \in H^{s_i,t_i}(U)$ be represented by
$a_i$, and assume $\alpha_1\alpha_2= \alpha_2\alpha_3=0$. Let $s=s_1+s_2 +s_3, \, t=t_1+
t_2 + t_3$. For any element $a\in U$, let $\bar{a}=(-1)^{\deg(a)+1}a$ where, as usual,
degree means total degree.  Let $\delta\bl u_i\br=\bar{a}_i a_{i+1}$. Then the Massey
product $\langle\alpha_1,\alpha_2,\alpha_3\rangle\in H^{s-1,t}(U)$ is the class
represented by the cocycle $\bar{a}_1 u_2+\ol{u}_1a_3$. This 3-fold product is not well
defined because the choices made in its construction are not unique. The choices of $a_i$
don't matter, but the $u_i$ could each be altered by adding a cocycle $x_i\in
U^{s_i+s_{i+1}-1, t_i+t_{i+1}}$. This would alter
$\langle\alpha_1,\alpha_2,\alpha_3\rangle$ by an element of the form $\alpha_1 \llbracket
x_2\rrbracket+\llbracket x_1\rrbracket\alpha_3\in \alpha_1 H^{s_2+s_3-1,t_2+t_3}(U) +
H^{s_1+s_2-1,t_1+t_2}(U)\,\alpha_3=\Ind\langle\alpha_1,\alpha_2,\alpha_3\rangle$. By
$\llbracket x\rrbracket$ we mean the class of $x$ in $H^*(U)$. The group
$\Ind\langle\alpha_1,\alpha_2,\alpha_3\rangle$ is called the \emph{indeterminacy} of
$\langle\alpha_1,\alpha_2,\alpha_3\rangle$. If the indeterminacy is trivial, then
$\langle\alpha_1,\alpha_2,\alpha_3\rangle$ is a single class in $H^**(U)$.

\begin{definition} With notation as above, $\langle\alpha_1,\alpha_2,\alpha_3\rangle\subset
H^{s-1,t}(U)$ is the coset of $\Ind\langle\alpha_1,\alpha_2,\alpha_3\rangle$ represented
by $\bar{a}_1 u_2+\ol{u}_1a_3$.
\end{definition}

Similarly, one can define $k$-fold Massey products $\langle \alpha_1, \dots, \alpha_k
\rangle$ for $k>3$. Let $\alpha_i\in H^{s_i,\,t_i}$ be represented by $a_i$, and let
$s_{i,j}=\sum_{r=i+1}^{j}(s_r -1)$, \,$t_{i,j}=\sum_{r=i+1}^{j} t_r$,\, $s=s_{1,k}+2$,
and $t= t_{1,k}$.
%Finally let $a_1, \hdots, a_k$ be representative cocycles of $\alpha_1, \hdots, \alpha_k$ respectively.

\begin{definition}
A collection of cochains $\mathcal{A}=\bigl\{a_{i,j}\in
U^{s_{i,j}+1,\,t_{i,j}}\bigr\}_{\scriptscriptstyle{0\leq i < j \leq k} }$ is said to be a
\emph{defining system} for the $k$-fold Massey product $\langle \alpha_1, \dots, \alpha_k
\rangle$ if
\begin{align*}
a_{i-1,i} & =a_i \text{ for } i=1, \dots, k,\\
%a_{i,j}& \in U^{s_{i,j}+1,\,t_{i,j}}, \text{ and}\\
\delta a_{i,j}& =\sum_{i<r<j}\overline{a}_{i,r} a_{r,j}.\\
\intertext{The cocycle $c\bigl(\mathcal{A}\bigr) \in U^{s,t}$ defined by} c(\mathcal{A})&
= \sum_{r=1}^{k-1}\overline{a}_{1,r} a_{r,k},
\end{align*}
is called the \emph{related cocycle} of $\mathcal{A}$.  The $k$-fold Massey product
$\langle \alpha_1, \dots, \alpha_k \rangle\subset H^{s,t}(U)$ consists of all classes
$\left\llbracket c\bigl(\mathcal{A}\bigr)\right\rrbracket$, where $\mathcal{A}$ ranges
over all defining systems of $\langle \alpha_1, \dots, \alpha_k \rangle$.
\end{definition}

%The cochain product $\langle a_1,\hdots,a_k \rangle$ then consists of all cochains $c(A)$ such that A is...
%$u\in H^{s,t}(U)$ for which there exists a defining system $\mathcal{A}$ such that
%c\left(\mathcal{A}\right)$ represents $u$.

It can be shown that the set $\langle \alpha_1, \dots, \alpha_k \rangle$ does not depend
on the choice of representatives of the $\alpha_i$. The $k$-fold product is then a
cohomology operation of $k$ variables that is defined if all the lower products $\langle
\alpha_i,\dots, \alpha_j\rangle$ for $0<j-i<k-1$ are defined and contain zero.  Here the
double product $\langle \alpha_i,\alpha_{i+1}\rangle$ is understood to be the regular
product $\alpha_i\alpha_{i+1}$. The indeterminacy  can get quite complicated, however
\cite[A1.4.4]{Ravenel}. Kraines has constructed a `restricted' $k$-fold Massey product on
the cohomology of topological spaces which has no indeterminacy \cite[Section
3]{Kraines}. This construction can be carried over to the Hopf algebra setting
\cite[Section 3]{Kochman}.
%mimicked for any differential bigraded algebra $U$.
If $\alpha_i=\alpha$ for all $i$ and $\alpha^2=0$, one can define the \emph{symmetric}
Massey product $\langle\alpha\rangle^k$.

\begin{definition} Let $\alpha\in H^{s,t} (U)$ be a class such that $\alpha^2=0$.
A collection of cocycles
 $\mathcal{A}=\bigl\{a_i\in U^{i(s-1)+1,it}\bigr\}_{i=1}^{k-1}$  is called a \emph{defining system} for the
symmetric  Massey product $\langle\alpha\rangle^k$ if
\begin{align*}
a_1 & \text{ represents } \alpha,\\
%a_i& \in U^{i(s-1)+1,it}, \text{ and}\\
\delta a_i& = \sum_{r=1}^{i-1}\overline{a}_r a_{i-r} \text{ for } 2\leq i\leq r-1.\\
\intertext{The cocycle $c\bigl(\mathcal{A}\bigr)\in U^{k(s-1)+2, kt}$ defined by}
c\bigl(\mathcal{A}\bigr)& =\sum_{r=1}^{k-1}\overline{a}_r a_{k-r} \in U^{k(s-1)+2, kt}
\end{align*}
is called the \emph{related cocycle} of $\mathcal{A}$.  The symmetric product
$\langle\alpha\rangle^k$ is then the set of all classes $u\in H^{k(s-1)+2,kt}(U)$ which
are represented by some $ c\bigl(\mathcal{A}\bigr)$, where $\mathcal{A}$ is a defining
system for $\langle\alpha\rangle^k$. If $\langle\alpha\rangle^k$ is defined and equal to
a single homology class then $\langle\alpha\rangle^k$ is said to be defined with
\emph{zero indeterminacy}.
\end{definition}

\begin{lemma} Let $p$ be odd, and let $A$ be a Hopf algebra over $\f$.
\be
\item If $\alpha\in H^{2s+1,t}(A)$, then $\langle\alpha\rangle^p$ is defined with
zero indeterminacy \cite[Lemma 14]{Kochman}.
\item If $\alpha\in H^{1,t}(A)$, then $\langle\alpha\rangle^p=-\bp(x)$ \cite[Remark 11.11]{May}.
\ee
\end{lemma}


\subsection{Calculations}\label{calculations}

Let
\begin{equation}\label{truncated polynomials}
\begin{aligned}
A &=\f[x]\big/\bigl(x^{p^n}\bigr), \quad &\deg x &=m \text{ even, } n\geq 1,\quad p
\text{ odd,\
}\\
A &=\mathbb{F}_2[x]\big/(x^{2^n}),\quad &\deg x &= m,\, n>1,\quad p=2.
\end{aligned}
\end{equation}
The augmentation in both cases is defined by $\epsilon (x)=0$. It is known that the
cohomology of such a truncated polynomial algebra is the tensor product of an exterior
algebra on a one-dimensional class and a polynomial algebra on a two dimensional class.
We would like to identify specific generators in the cobar complex $C^*(A)=\Tens\bl
I(A)^*\br$.

Choose for $I(A)$ the natural basis $\{x^i, \ 0 < i < p^n \}$. Let $\{y_i \}$ be the dual
basis for $I^*$. Then
\begin{equation}\label{a}
\delta \bigl[y_k\bigr]= \sum_{\substack{i+j=k\\ 0<i<k}} \bigl[y_i \vert y_{j} \bigr].
\end{equation}

Clearly, in $C^{1,2m}(A)$,\, $z = [y_1]$ is a cocycle. Let $e= \sum_{i=1}^{p^n -1} [y_i
\vert y_{p^n-i}]$. A routine calculation shows that $e$ is a nonbounding cocycle. By
abuse of notation, we can thus take $z$ and $e$ to be the generators of $H^*(A)$.

\begin{lemma}\label{Massey product for u} Let $A$ be a truncated polynomial
algebra as in \eqref{truncated polynomials}. Then $H^{*}(A) \simeq
\Lambda(z)\otimes\f[e]$, with $z\in H^{1,m}$, $e\in H^{2,mp^n}$ as defined above.
Moreover, the symmetric Massey product ${\langle z\rangle}^{p^n}=e$ with zero
indeterminacy.
\end{lemma}

\begin{proof}
Choosing $a_i=[y_i]$ gives a defining system for $\langle z\rangle^{p^n}$. Its related
cocycle is $\sum_{r=1}^{p^n-1}\bigl[y_r\vert y_{p^n-r}\bigr]$, which represents $e$.
There is no indeterminacy because for each $i$, there is no other choice of $a_i$.
\end{proof}
%
\begin{remark}\label{remark} If $p=2$, $n=1$, then $A=\mathbb{F}_2[x]/(x^2)\simeq \Lambda[x]$, and
$H^*(A)\simeq \mathbb{F}_2[z]$ with $z=[x^*]\in H^{1,\,\deg x}$. %Also, $\sqit{1}(z)=z^2$.
\end{remark}
%

Summarizing,

\begin{theorem}\label{non-zero Steenrod operations} Let $A=\f[x]/(x^{p^n})$,
then the following Steenrod operations and symmetric Massey products are present in
$H^*(A)$:
\begin{enumerate}
\item $\langle z\rangle^{p^n}=e$,\, $\pti{1} e=e^p$ for $p$ odd,
\item $\bp z=-e$ for $p$ odd, $n=1$,
\item $\langle z\rangle^{2^n}=e$,\, $\sqit{2}(e)=e^2$ for $p=2$, $n\geq 2$,
\item $\sqit{1}(z)=z^2$ for $p=2$, $n=1$.
\end{enumerate}
\end{theorem}
%
\begin{proof} These are consequences of the axioms in Theorem \ref{steenrod} and of the
results just stated.
\end{proof}

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\bibitem{BGS}
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\bibitem{Kraines}
David Kraines, \emph{Massey higher products}, Trans. Amer. Math. Soc.
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\bibitem{Lofwall}
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\bibitem{Milnor-Moore}
John~W. Milnor and John~C. Moore, \emph{On the structure of {H}opf algebras},
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\bibitem{Priddy}
Stewart~B. Priddy, \emph{Koszul resolutions}, Trans. Amer. Math. Soc.
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\bibitem{Priddy-primary}
\bysame, \emph{Primary cohomology operations for simplicial {L}ie algebras},
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\bibitem{Quillen}
Daniel~G. Quillen, \emph{On the associated graded ring of a group ring}, J.
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\bibitem{Ravenel}
Douglas~C. Ravenel, \emph{Complex cobordism and stable homotopy groups of
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Clarence Wilkerson, \emph{The cohomology algebras of finite-dimensional {H}opf
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\end{thebibliography}

\end{document}

math0303327.extracted_bib
\begin{thebibliography}{99}

\bibitem{Adams}
J.~F. Adams, \emph{On the non-existence of elements of {H}opf invariant one},
  Ann. of Math. (2) \textbf{72} (1960), 20--104.

\bibitem{BGS}
Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, \emph{Koszul
  duality patterns in representation theory}, J. Amer. Math. Soc. \textbf{9}
  (1996), no.~2, 473--527. \MR{96k:17010}

\bibitem{Blackburn}
Norman Blackburn, \emph{Generalizations of certain elementary theorems on
  $p$-groups}, Proc. London Math. Soc. (3) \textbf{11} (1961), 1--22.

\bibitem{Dietz-odd}
Jill Dietz, \emph{Stable splittings of classifying spaces of metacyclic
  $p$-groups, $p$ odd}, J. Pure Appl. Algebra \textbf{90} (1993), no.~2,
  115--136.

\bibitem{Blackburn2}
Bertram Huppert and Norman Blackburn, \emph{Finite groups. {I}{I}},
  Springer-Verlag, Berlin, 1982.

\bibitem{Kahn}
Daniel~S. Kahn, \emph{${\rm Cup}-i$ products and the {A}dams spectral
  sequence}, Topology \textbf{9} (1970), 1--9.

\bibitem{Kochman}
Stanley~O. Kochman, \emph{Symmetric {M}assey products and a {H}irsch formula in
  homology}, Trans. Amer. Math. Soc. \textbf{163} (1972), 245--260. \MR{48
  \#9721}

\bibitem{Kraines}
David Kraines, \emph{Massey higher products}, Trans. Amer. Math. Soc.
  \textbf{124} (1966), 431--449.

\bibitem{Lazard}
Michel Lazard, \emph{Sur les groupes nilpotents et les anneaux de {L}ie}, Ann.
  Sci. Ecole Norm. Sup. (3) \textbf{71} (1954), 101--190.

\bibitem{Lofwall}
Clas L{\"o}fwall, \emph{On the subalgebra generated by the one-dimensional
  elements in the {Y}oneda {E}xt-algebra}, Algebra, algebraic topology and
  their interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183,
  Springer, Berlin, 1986, pp.~291--338. \MR{88f:16030}

\bibitem{May}
J.~Peter May, \emph{A general algebraic approach to {S}teenrod operations}, The
  Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E.
  Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970),
  Springer, Berlin, 1970, pp.~153--231.

\bibitem{Milnor-Moore}
John~W. Milnor and John~C. Moore, \emph{On the structure of {H}opf algebras},
  Ann. of Math. (2) \textbf{81} (1965), 211--264.

\bibitem{Priddy}
Stewart~B. Priddy, \emph{Koszul resolutions}, Trans. Amer. Math. Soc.
  \textbf{152} (1970), 39--60.

\bibitem{Priddy-primary}
\bysame, \emph{Primary cohomology operations for simplicial {L}ie algebras},
  Illinois J. Math. \textbf{14} (1970), 585--612.

\bibitem{Quillen}
Daniel~G. Quillen, \emph{On the associated graded ring of a group ring}, J.
  Algebra \textbf{10} (1968), 411--418.

\bibitem{Ravenel}
Douglas~C. Ravenel, \emph{Complex cobordism and stable homotopy groups of
  spheres}, Academic Press Inc., Orlando, FL, 1986.

\bibitem{Wilkerson}
Clarence Wilkerson, \emph{The cohomology algebras of finite-dimensional {H}opf
  algebras}, Trans. Amer. Math. Soc. \textbf{264} (1981), no.~1, 137--150.

\end{thebibliography}