![]() LD1 := SimpleLieAlgebraData("sl(3)", alg1, labelformat = "gl", labels = ) Initialize the Lie algebra sl 3, the Lie algebra of trace-free 3×3 matrices. With(DifferentialGeometry): with(LieAlgebras): The keyword arguments labelformat, labels allow for the labeling of the basis of the abstract Lie algebra which characterizes the basis elements in terms of their standard matrix elements. This choice is preferred for roots space computations. Q 1 = 0 I q 0 I q 0 0 0 0 I p − q or Q 2 = I p 0 0 − I qįor the quadratic form preserved by these algebras. Two versions of the Lie algebras su p, q and so p, q are available, corresponding the choices ![]() Subalgebras of any of these Lie algebras can be calculate using the command MatrixSubalgebras. The command StandardRepresentation generates the standard matrix representations of these algebras.Ĭartan matrices, Dynkin diagram, Satake diagrams, positive roots can easily be found for each of the simple Lie algebra. The precise definitions and examples of each of these Lie algebras are found in SimpleLieAlgebraDataDetails. The Other are classical matrix algebras which are often used in Lie theory and differential geometry. The Lie algebras A, B, C, D, F, G are all simple Lie algebras. Gl n, &reals, gl n, &complexes, sl n, ℂ, u p, q, so n, ℂ, sp n, &complexes, sol n, nil n G 2, 14, 0 or g 2, compact, g 2, 6, 8 or g 2, split (two versions) So p, q, p + q = 2 m + 1 (two versions), p ≥ q ![]() Sl n, su p, q (two versions), su * n, p ≥ q This command returns the structure equations (see LieAlgebraData ) for any one of the following Lie algebras: Different standard basis for some of the Lie algebras can be specified with the keyword version. Options - ( optional) keyword arguments labelformat, labels which specify the labelling of the basis for the Lie algebra. SimpleLieAlgebraData ( algtype, algname, option )Īlgtype - a string, describing the type and dimension of a classical matrix algebraĪlgname - an unassigned name or a string, the name of the classical matrix algebra to be constructed ![]() LieAlgebras - obtain the structure equations for a classical matrix Lie algebra ![]()
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