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16.35 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 Sch÷bel.

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 dimCj = 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. [?])
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 :=dimL = 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. [?],[?]). 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,p10
  
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,p10
  
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,p20
  
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,p20
  
LIEALG(4),COMTAB(15)[Y 1,Y 4] = p1Y 1 + Y 2,[Y 2,Y 4] = p1Y 2,
[Y 3,Y 4] = Y 3,p10
  
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,
p10
  
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

Bibliography

[1]   M.A.H. MacCallum. On the classification of the real four-dimensional lie algebras. 1979.

[2]   C. Schoebel. Classification of real n-dimensional lie algebras with a low-dimensional derived algebra. In Proc. Symposium on Mathematical Physics ’92, 1993.

[3]   F. Schoebel. The symbolic classification of real four-dimensional lie algebras. 1992.