Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them

Details Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them

2006-01-09

arxiv.org/abs/math/0601184v1

Theoret.Math.Phys 149(1): 1299-1311 (2006)

Mathematics

Representation Theory

14 pages, 6 figures

Scientific paper

Among the simple finite dimensional Lie algebras, only sl(n) possesses two automorphisms of finite order which have no common nonzero eigenvector with eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow one to generate all of sl(n) under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all n times n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't Hooft's work on confinement operators in QCD, and various other instances. Here I give an algorithm which both generates sl(n) and explicitly describes a set of defining relations. For simple (up to center) Lie superalgebras, analogs of Sylvester generators exist only for sl(n|n). The relations for this case are also computed.

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