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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 c_{ij}^{k} in the basis {X_{1},…,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
with 3 ≤ k ≤ n, k odd.
The concepts correspond to the following theorem (LIE_ALGEBRA(2)→ L_{2},
HEISENBERG(k)→ H_{k} and COMMUTATIVE(n-k)→ 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:
(i) C(L) ∩ L^{(1)} = {0} : L_{2} ⊕ C_{n-2} or
(ii) C(L) ∩ L^{(1)} = L^{(1)} : H_{k} ⊕ C_{n-k} (k = 2r - 1,r ≥ 2), with
1. C(L) = C_{j} ⊕ (L^{(1)} ∩ C(L)) and dimC_{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},…,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 {X_{1},…,X_{n}} into the standard basis {Y _{1},…,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.
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 c_{ij}^{k} in the basis
{X_{1},…,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 performed 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:
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) p_{1} (and p_{2}) the list is expanded to:
This returned value is also stored as LIE_CLASS. The linear transformation from the basis {X_{1},…,X_{n}} into the basis of the standard form {Y _{1},…,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.
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} + p_{1}Y _{2},[Y _{2},Y _{3}] = Y _{1},p_{1}≠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} + p_{1}Y _{2},[Y _{2},Y _{4}] = Y _{1},p_{1}≠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}] = p_{1}Y _{2},[Y _{3},Y _{4}] = p_{2}Y _{3}, |
p_{1},p_{2}≠0 | |
LIEALG(4),COMTAB(14) | [Y _{1},Y _{4}] = p_{1}Y _{1} + Y _{2},[Y _{2},Y _{4}] = -Y _{1} + p_{1}Y _{2}, |
[Y _{3},Y _{4}] = p_{2}Y _{3},p_{2}≠0 | |
LIEALG(4),COMTAB(15) | [Y _{1},Y _{4}] = p_{1}Y _{1} + Y _{2},[Y _{2},Y _{4}] = p_{1}Y _{2}, |
[Y _{3},Y _{4}] = Y _{3},p_{1}≠0 | |
LIEALG(4),COMTAB(16) | [Y _{1},Y _{4}] = 2Y _{1},[Y _{2},Y _{3}] = Y _{1}, |
[Y _{2},Y _{4}] = (1 + p_{1})Y _{2},[Y _{3},Y _{4}] = (1 - p_{1})Y _{3}, | |
p_{1} ≥ 0 | |
LIEALG(4),COMTAB(17) | [Y _{1},Y _{4}] = 2Y _{1},[Y _{2},Y _{3}] = Y _{1}, |
[Y _{2},Y _{4}] = Y _{2} - p_{1}Y _{3},[Y _{3},Y _{4}] = p_{1}Y _{2} + Y _{3}, | |
p_{1}≠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} |
[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.
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