Algebras 6.163 -- 6.167 give a classification of algebras of order $p^{6}$
with presentations \[ \langle a,b,c\,|\,ca-baa,\,cb,\,pa-\lambda baa-\mu bab,\,pb+\nu baa+\xi
bab,\,pc,\,\text{class }3\rangle \]%
with $\lambda ,\mu ,\nu ,\xi\neq 0$. Most of these algebras are terminal,
and we need a slightly different classification of these algebras from that
given in the classification of nilpotent Lie rings of order $p^{6}$, so as
to classify the capable ones. It turns out that $\frac{5}{2}p-\frac{9}{2}+% \frac{1}{2}\gcd (p-1,4)$ of these algebras are capable, and that they have a
total of $\frac{1}{2}p^{3}+2p^{2}-5p+\frac{1}{2}+\frac{p}{2}\gcd (p-1,4)$
descendants of order $p^{7}$ and $p$-class 4.
Let $L$ have the presentation above, and suppose that $a^{\prime },b^{\prime
},c^{\prime }$ generate $L$ and satisfy similar relations, but with
(possibly) different $\lambda ,\mu ,\nu ,\xi $. Then \begin{eqnarray*}
a^{\prime } &=&\alpha a+\gamma c, \\
b^{\prime } &=&\delta b+\varepsilon c, \\
c^{\prime } &=&\alpha\delta c \end{eqnarray*}%
modulo $L_{2}$ and \begin{eqnarray*}
pa^{\prime } &=&\frac{\lambda }{\alpha\delta }b^{\prime }a^{\prime
}a^{\prime }+\frac{\mu }{\delta ^{2}}b^{\prime }a^{\prime }b^{\prime }, \\
pb^{\prime } &=&\frac{\nu }{\alpha ^{2}}b^{\prime }a^{\prime }a^{\prime }+% \frac{\xi }{\alpha\delta }b^{\prime }a^{\prime }b^{\prime } \end{eqnarray*}%
or \begin{eqnarray*}
a^{\prime } &=&\alpha b+\gamma c, \\
b^{\prime } &=&\delta a+\varepsilon c, \\
c^{\prime } &=&\alpha\delta c \end{eqnarray*}%
modulo $L_{2}$ and \begin{eqnarray*}
pa^{\prime } &=&\frac{\xi }{\alpha\delta }b^{\prime }a^{\prime }a^{\prime }+% \frac{\nu }{\delta ^{2}}b^{\prime }a^{\prime }b^{\prime }, \\
pb^{\prime } &=&\frac{\mu }{\alpha ^{2}}b^{\prime }a^{\prime }a^{\prime }+% \frac{\lambda }{\alpha\delta }b^{\prime }a^{\prime }b^{\prime }. \end{eqnarray*}
So we can take $\lambda =1$ and $\mu =1$ or $\omega $ (or any other fixed
integer which is not a square $\func{mod}p)$. Given these values of $\lambda
,\mu $ it turns out that the algebra is terminal unless $\xi =1$ or $\xi
=\mu\nu $.
So we have two families of capable algebras of order $p^{6}$:
\[ \langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\mu bab,\,pb+\nu baa+bab,\,pc,\,\text{%
class }3\rangle , \]
For the first family of descendants we consider transformations of the form% \begin{eqnarray*}
a^{\prime } &=&\pm a+\gamma c, \\
b^{\prime } &=&\pm b+\varepsilon c, \\
c^{\prime } &=&c, \end{eqnarray*}%
where if $\mu\nu =1$ we need $\gamma =\mu\varepsilon $, and
transformations of the form \begin{eqnarray*}
a^{\prime } &=&\alpha b+\gamma c, \\
b^{\prime } &=&\alpha ^{-1}a+\varepsilon c, \\
c^{\prime } &=&c, \end{eqnarray*}%
where $\alpha ^{2}\nu =\mu $, and where if $\mu\nu =1$ we need $\gamma =\mu \varepsilon $. For these transformations we have% \begin{eqnarray*}
y &\rightarrow &\pm y+\gamma t+\gamma\mu ^{-1}+\varepsilon\\
z &\rightarrow &\pm z+\varepsilon t-\nu\gamma\mu ^{-1}+\nu\varepsilon
-2\varepsilon\mu ^{-1}, \\
t &\rightarrow &t, \end{eqnarray*}%
and% \begin{eqnarray*}
y &\rightarrow &-\alpha z-\gamma t+\varepsilon -\nu\gamma +2\gamma\mu
^{-1}, \\
z &\rightarrow &-\alpha ^{-1}y-\varepsilon t-\gamma\mu ^{-1}\nu
-\varepsilon\mu ^{-1}, \\
t &\rightarrow &-t-\nu +\mu ^{-1}. \end{eqnarray*}
For the second family of descendants we can assume that $\mu\nu\neq 1$. We
consider transformations of the form% \begin{eqnarray*}
a^{\prime } &=&\pm a+\mu\varepsilon c, \\
b^{\prime } &=&\pm b+\varepsilon c, \\
c^{\prime } &=&c, \end{eqnarray*}%
and, when $\mu\nu =-1$ and $p=1\func{mod}4$, transformations of the form% \begin{eqnarray*}
a^{\prime } &=&\alpha b+\mu\varepsilon c, \\
b^{\prime } &=&-\alpha ^{-1}a+\varepsilon c, \\
c^{\prime } &=&-c, \end{eqnarray*}%
where $\alpha ^{2}=-\mu ^{2}$. For these transformations we have% \begin{eqnarray*}
y &\rightarrow &\pm y+\mu\varepsilon t+2\varepsilon , \\
z &\rightarrow &\pm z+\varepsilon t-2\nu\varepsilon , \\
t &\rightarrow &t, \end{eqnarray*}%
and% \begin{eqnarray*}
y &\rightarrow &-\alpha z-\mu\varepsilon t-2\varepsilon , \\
z &\rightarrow &\alpha ^{-1}y-\varepsilon t+2\nu\varepsilon , \\
t &\rightarrow &t. \end{eqnarray*}
There is a \textsc{Magma} program to compute representative sets of
parameters in notes6.163.m.
\end{document}
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