20.32 LIE: Functions for the Classification of Real n-Dimensional Lie Algebras

LIE is a package of functions for the classification of real $n$-dimensional Lie algebras. It consists of two modules: liendmc1 and lie1234. With the help of the functions in the liendmcl module, real $n$-dimensional Lie algebras $L$ with a derived algebra $L^{(1)}$ of dimension 1 can be classified.

Authors: Carsten and Franziska Schöbel

20.32.1 liendmc1

With the help of the functions in this module real $n$-dimensional Lie algebras $L$ with a derived algebra $L^{(1)}$ of dimension $1$ can be classified. $L$ has to be defined by its structure constants $c_{ij}^k$ in the basis $\{X_1,\dots ,X_n\}$ with $[X_i,X_j]=c_{ij}^k{X}_k$. The user must define an array lienstrucin(n,n,n) with n being the dimension of the Lie algebra $L$. The structure constants lienstrucin($i,j,k$):=$c_{ij}^k$ for $i<j$ should be given. Then the procedure liendimcom1 can be called. Its syntax is:

$⟨$number$⟩$ corresponds to the dimension $n$. The procedure simplifies the structure of $L$ performing real linear transformations. The returned value is a list of the form

(i)

{LIE_ALGEBRA(2),COMMUTATIVE(n-2)} or

(ii)

{HEISENBERG(k),COMMUTATIVE(n-k)}

with $3\le k\le n$, $k$ odd.

The concepts correspond to the following theorem (LIE_ALGEBRA(2)$\to L_2$, HEISENBERG(k)$\to H_k$ and COMMUTATIVE(n-k)$\to C_{n-k}$):
Theorem. Every real $n$-dimensional Lie algebra $L$ with a 1-dimensional derived algebra can be decomposed into one of the following forms:

(1)

$C(L)\cap L^{(1)}=\{0\}:L_2\oplus C_{n-2}$ or

(2)

$C(L)\cap L^{(1)}=L^{(1)}:H_k\oplus C_{n-k}(k=2r-1,r\ge 2)$,

with

1.

$C(L)=C_j\oplus (L^{(1)}\cap C(L))$ and dim$C_j=j$ ,

2.

$L_2$ is generated by $Y_1,Y_2$ with $[Y_1,Y_2]=Y_1$ ,

3.

$H_k$ is generated by $\{Y_1,\dots ,Y_k\}$ with
$[Y_2,Y_3]=\cdots =[Y_{k-1},Y_k]=Y_1$.

(cf. [Sch93])

The returned list is also stored as lie_list. The matrix lientrans gives the transformation from the given basis $\{X_1,\dots ,X_n\}$ into the standard basis $\{Y_1,\dots ,Y_n\}$: $Y_j=($LIENTRANS${)}_j^k{X}_k$.

A more detailed output can be obtained by turning on the switch tr_lie: before the procedure liendimcom1 is called.

The returned list could be an input for a data bank in which mathematical relevant properties of the obtained Lie algebras are stored.

20.32.2 lie1234

This part of the package classifies real low-dimensional Lie algebras $L$ of the dimension $n:=$dim$L=1,2,3,4$. $L$ is also given by its structure constants $c_{ij}^k$ in the basis $\{X_1,\dots ,X_n\}$ with $[X_i,X_j]=c_{ij}^k{X}_k$. An ARRAY LIESTRIN($n,n,n$) has to be defined and LIESTRIN($i,j,k$):=$c_{ij}^k$ for $i<j$ should be given. Then the procedure lieclass can be called whose syntax is:

$⟨$number$⟩$ should be the dimension of the Lie algebra $L$. The procedure stepwise simplifies the commutator relations of $L$ using properties of invariance like the dimension of the centre, of the derived algebra, unimodularity etc. The returned value has the form:

   {LIEALG(n),COMTAB(m)},

where $m$ corresponds to the number of the standard form (basis: $\{Y_1,\dots ,Y_n\}$) in an enumeration scheme. The corresponding enumeration schemes are listed below (cf. [Sch92],[Mac99]). In case that the standard form in the enumeration scheme depends on one (or two) parameter(s) $p_1$ (and $p_2$) the list is expanded to:

   {LIEALG(n),COMTAB(m),p1,p2}.

This returned value is also stored as lie_class. The linear transformation from the basis $\{X_1,\dots ,X_n\}$ into the basis of the standard form $\{Y_1,\dots ,Y_n\}$ is given by the matrix liemat: $Y_j=($LIEMAT${)}_j^k{X}_k$.

By turning on the switch tr_lie before the procedure lieclass is called the output contains not only the list lie_class but also the non-vanishing commutator relations in the standard form.

By the value $m$ and the parameters further examinations of the Lie algebra are possible, especially if in a data bank mathematical relevant properties of the enumerated standard forms are stored.

20.32.3 Enumeration schemes for lie1234

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