GAP (Sophus) - Chapter 1: The theory

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### 1. The theory

The Sophus package was originally designed to aid the author to classify some small-dimensional nilpotent Lie algebras over small fields. The classification follows the ideas that were used to classify small $p$-groups by O'Brien [O90]. The theory developed by O'Brien could easily be adopted to Lie algebras, and the details of this new theory can be found in [S]. Here we only summarise the main ideas, so that the user can understand the procedures implemented in this package. In this section $L$ denotes a finitely generated, and hence finite-dimensional, nilpotent Lie algebra. Suppose that $L$ has nilpotency class $c$, and hence the lower central series is as follows:

\[ L= \gamma_1(L)>\gamma_2(L)=L'>\gamma_3(L)>\cdots >\gamma_c(L)>\gamma_{c+1}(L)=0. \]

We say that a basis $\mathcal B=\{b_1,\ldots,b_n\}$ for $L$ is compatible with the lower central series if there are indices $1=i_1<i_2<\cdots<i_{c}<n$ such that $\{b_{i_k},\ldots,b_n\}$ is a basis of $\gamma_k(L)$ for $k\in\{1,\ldots,c\}$. We compute the structure constants table with respect to this basis, that is, we compute coefficients $\alpha_{i,j}^k$ for $1\leq i<j<k\leq n$ such that

\[ [b_i,b_j]=\sum_{k=j+1}^n\alpha_{i,j}^k b_k. \]

Suppose that $b_i\in\gamma_j(L)\setminus\gamma_{j+1}(L)$. Then we say that the number $j$ is the weight of the basis element $b_i$.

Note that in the nilpotent Lie algebra $L$ minimal generating sets have the same size, namely the dimension of $L/L'$. If $\dim L/L'=d$ then we call $L$ a $d$-generator algebra. We call a basis $\mathcal B$ a nilpotent basis if the following hold.

- The basis $\mathcal B$ is compatible with the lower central series.

- For each $b_i\in \mathcal B$ with weight $w\geq 2$ there are $b_{j_1},\ b_{j_2}\in\mathcal B$ with weight 1 and $w-1$, respectively such that $b_i=[b_{j_1},b_{j_2}]$. The product $[b_{j_1},b_{j_2}]$ is called the definition of $b_i$.

A Lie algebra $K$ is said to be a central extension of $L$ if $L\cong K/I$ for some ideal $I$ such that $I\leq Z(K)\cap K'$. Suppose that $c$ denotes the nilpotency class of $L$. Then a Lie algebra $K$ is an immediate descendant of $L$ if $K$ has class $c+1$ and $K/\gamma_{c+1}(K)\cong L$. As in this case $\gamma_{c+1}(K)\leq Z(K)\cap K'$, it follows that an immediate descendant $K$ is a central extension of $L$. If $s=\dim \gamma_{c+1}(K)$ then $K$ is said to be a step-$s$ immediate descendant of $L$.

Let $L$ be a $d$-generator nilpotent Lie algebra with class $c$, and let $F$ be a free Lie algebra of rank $d$. Choose an ideal $I$ of $F$ such that $L\cong F/I$. Then the Lie algebra $L^*=F/[I,F]$ is called the Lie cover of $L$. The Lie multiplicator in $L^*$ is the subspace $I/[I,F]$ and the Lie nucleus is $\gamma_c(L^*)$. It clear from the definition that $L^*/M\cong L$. It is verified in [S] that, up to isomorphism, the Lie cover, the Lie multiplicator and the Lie nucleus are determined by the isomorphism type of $L$. Further, each central extension of the nilpotent Lie algebra $L$ is a quotient of the Lie cover $L^*$. Thus it is possible to obtain all such descendants by first computing the Lie cover; this procedure is explained in [S]. Similar ideas can be used to compute the automorphism group of a nilpotent Lie algebra, and to verify isomorphism between two nilpotent Lie algebras; see [S] for details.

The main functions in Sophus are thus able to compute

- a nilpotent basis for a nilpotent Lie algebra;

- the cover of a nilpotent Lie algebra;

- the immediate descendants of a nilpotent Lie algebra;

- the full automorphism group of a nilpotent Lie algebra.

There is also a function in the package to check if two nilpotent Lie algebras are isomorphic. After repeated applications of the immediate descendants algorithm, it is, in theory, possible to list all nilpotent Lie algebras of a given dimension over a prime field $\mathbb F_p$. Of course, this computation requires relatively large computational resources, and quickly becomes unfeasible as the dimension or the characteristic $p$ grows.

The Sophus package was written for the GAP~4 computer algebra system. In many procedures it is very important that we can compute the stabiliser of a subspace under some matrix group action. This is carried out using the procedures implemented in the autpgrp package [EO]. Hence this package is required to run Sophus.

The current version of Sophus deals with general nilpotent Lie algebras over finite prime fields. If you are to compute with Lie algebras obtained from group algebras via the bracket operation, then another GAP package LAGUNA [VS] may also offer some very efficient methods.

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