20.31 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.

20.31.1 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 \(\{X_1,\ldots ,X_n\}\) 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:

\(\langle \)number\(\rangle \) 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 \(3\leq k\leq n\), \(k\) odd.

The concepts correspond to the following theorem (LIE_ALGEBRA(2)\(\rightarrow L_2\), HEISENBERG(k)\(\rightarrow H_k\) and COMMUTATIVE(n-k)\(\rightarrow C_{n-k}\)):
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)

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

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 \(\{Y_1,\ldots ,Y_k\}\) 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 \(\{X_1,\ldots ,X_n\}\) into the standard basis \(\{Y_1,\ldots ,Y_n\}\): \(Y_j=(\)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.

20.31.2 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 \(\{X_1,\ldots ,X_n\}\) 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:

\(\langle \)number\(\rangle \) 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: \(\{Y_1,\ldots ,Y_n\}\)) 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 \(\{X_1,\ldots ,X_n\}\) into the basis of the standard form \(\{Y_1,\ldots ,Y_n\}\) 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.

20.31.3 Enumeration schemes for lie1234

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