LIE: Functions for the classiﬁcation of real n-dimensional Lie algebras
Date: 2020-10-07 18:10:00

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### 16.36 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.

LIE is a package of functions for the classiﬁcation of real n-dimensional Lie algebras. It consists of two modules: liendmc1 and lie1234.
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 cijk in the basis {X1,…,Xn} with [Xi,Xj] = cijkXk. 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):=cijk for i < j should be given. Then the procedure LIENDIMCOM1 can be called. Its syntax is:

LIENDIMCOM1(<number>).

<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 ≤ k ≤ n, k odd.
The concepts correspond to the following theorem (LIE_ALGEBRA(2)→ L2, HEISENBERG(k)→ Hk and COMMUTATIVE(n-k)→ Cn−k):
Theorem. Every real n-dimensional Lie algebra L with a 1-dimensional derived algebra can be decomposed into one of the following forms:
(i) C(L) ∩ L(1) = {0} : L2 ⊕ Cn−2 or
(ii) C(L) ∩ L(1) = L(1) : Hk ⊕ Cn−k (k = 2r − 1, r ≥ 2), with

1. C(L) = Cj ⊕ (L(1) ∩ C(L)) and dim Cj = j ,
2. L2 is generated by Y 1,Y 2 with [Y 1,Y 2] = Y 1 ,
3. Hk is generated by {Y 1,…,Y k} with
[Y 2,Y 3] = ⋯ = [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 {X1,…,Xn} into the standard basis {Y 1,…,Y n}: Y j = (LIENTRANS)jkXk.
A more detailed output can be obtained by turning on the switch TR_LIE:

ON 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.
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 cijk in the basis {X1,…,Xn} with [Xi,Xj] = cijkXk. An ARRAY LIESTRIN(n,n,n) has to be deﬁned and LIESTRIN(i,j,k):=cijk for i < j should be given. Then the procedure LIECLASS can be performed whose syntax is:

LIECLASS(<number>).

<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,…,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) p1 (and p2) 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 {X1,…,Xn} into the basis of the standard form {Y 1,…,Y n} is given by the matrix LIEMAT: Y j = (LIEMAT)jkXk.

By turning on the switch TR_LIE:

ON 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.
Enumeration schemes for lie1234

returned list LIE_CLASS | the corresponding commutator relations
LIEALG(1),COMTAB(0) | commutative case
LIEALG(2),COMTAB(0) | commutative case
LIEALG(2),COMTAB(1) | [Y 1,Y 2] = Y 2
LIEALG(3),COMTAB(0) | commutative case
LIEALG(3),COMTAB(1) | [Y 1,Y 2] = Y 3
LIEALG(3),COMTAB(2) | [Y 1,Y 3] = Y 3
LIEALG(3),COMTAB(3) | [Y 1,Y 3] = Y 1,[Y 2,Y 3] = Y 2
LIEALG(3),COMTAB(4) | [Y 1,Y 3] = Y 2,[Y 2,Y 3] = Y 1
LIEALG(3),COMTAB(5) | [Y 1,Y 3] = −Y 2,[Y 2,Y 3] = Y 1
LIEALG(3),COMTAB(6) | [Y 1,Y 3] = −Y 1 + p1Y 2,[Y 2,Y 3] = Y 1,p1≠0
LIEALG(3),COMTAB(7) | [Y 1,Y 2] = Y 3,[Y 1,Y 3] = −Y 2,[Y 2,Y 3] = Y 1
LIEALG(3),COMTAB(8) | [Y 1,Y 2] = Y 3,[Y 1,Y 3] = Y 2,[Y 2,Y 3] = Y 1
LIEALG(4),COMTAB(0) | commutative case
LIEALG(4),COMTAB(1) | [Y 1,Y 4] = Y 1
LIEALG(4),COMTAB(2) | [Y 2,Y 4] = Y 1
LIEALG(4),COMTAB(3) | [Y 1,Y 3] = Y 1,[Y 2,Y 4] = Y 2
LIEALG(4),COMTAB(4) | [Y 1,Y 3] = −Y 2,[Y 2,Y 4] = Y 2,
| [Y 1,Y 4] = [Y 2,Y 3] = Y 1
LIEALG(4),COMTAB(5) | [Y 2,Y 4] = Y 2,[Y 1,Y 4] = [Y 2,Y 3] = Y 1
LIEALG(4),COMTAB(6) | [Y 2,Y 4] = Y 1,[Y 3,Y 4] = Y 2
LIEALG(4),COMTAB(7) | [Y 2,Y 4] = Y 2,[Y 3,Y 4] = Y 1
LIEALG(4),COMTAB(8) | [Y 1,Y 4] = −Y 2,[Y 2,Y 4] = Y 1
LIEALG(4),COMTAB(9) | [Y 1,Y 4] = −Y 1 + p1Y 2,[Y 2,Y 4] = Y 1,p1≠0
LIEALG(4),COMTAB(10) | [Y 1,Y 4] = Y 1,[Y 2,Y 4] = Y 2
LIEALG(4),COMTAB(11) | [Y 1,Y 4] = Y 2,[Y 2,Y 4] = Y 1

returned list LIE_CLASS | the corresponding commutator relations
LIEALG(4),COMTAB(12) | [Y 1,Y 4] = Y 1 + Y 2,[Y 2,Y 4] = Y 2 + Y 3,
| [Y 3,Y 4] = Y 3
LIEALG(4),COMTAB(13) | [Y 1,Y 4] = Y 1,[Y 2,Y 4] = p1Y 2,[Y 3,Y 4] = p2Y 3,
| p1,p2≠0
LIEALG(4),COMTAB(14) | [Y 1,Y 4] = p1Y 1 + Y 2,[Y 2,Y 4] = −Y 1 + p1Y 2,
| [Y 3,Y 4] = p2Y 3,p2≠0
LIEALG(4),COMTAB(15) | [Y 1,Y 4] = p1Y 1 + Y 2,[Y 2,Y 4] = p1Y 2,
| [Y 3,Y 4] = Y 3,p1≠0
LIEALG(4),COMTAB(16) | [Y 1,Y 4] = 2Y 1,[Y 2,Y 3] = Y 1,
| [Y 2,Y 4] = (1 + p1)Y 2,[Y 3,Y 4] = (1 − p1)Y 3,
| p1 ≥ 0
LIEALG(4),COMTAB(17) | [Y 1,Y 4] = 2Y 1,[Y 2,Y 3] = Y 1,
| [Y 2,Y 4] = Y 2 − p1Y 3,[Y 3,Y 4] = p1Y 2 + Y 3,
| p1≠0
LIEALG(4),COMTAB(18) | [Y 1,Y 4] = 2Y 1,[Y 2,Y 3] = Y 1,
| [Y 2,Y 4] = Y 2 + Y 3,[Y 3,Y 4] = Y 3
LIEALG(4),COMTAB(19) | [Y 2,Y 3] = Y 1,[Y 2,Y 4] = Y 3,[Y 3,Y 4] = Y 2
LIEALG(4),COMTAB(20) | [Y 2,Y 3] = Y 1,[Y 2,Y 4] = −Y 3,[Y 3,Y 4] = Y 2
LIEALG(4),COMTAB(21) | [Y 1,Y 2] = Y 3,[Y 1,Y 3] = −Y 2,[Y 2,Y 3] = Y 1
LIEALG(4),COMTAB(22) | [Y 1,Y 2] = Y 3,[Y 1,Y 3] = Y 2,[Y 2,Y 3] = Y 1

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