Sylvester-’t Hooft generators of sl(n) and gl(n|n), and relations between them
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Submission date: 24. Aug. 2006
published in: Theoretical and mathematical physics, 149 (2006) 1, p. 1299-1311
DOI number (of the published article): 10.1007/s11232-006-0119-0
with the following different title: Sylvester-'t Hooft generators and relations between them for sl(n) and gl(n|n)
MSC-Numbers: 17A70, 17B01, 17B70
Keywords and phrases: Defining relations, Lie algebras, Lie superalgebras
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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 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 close to simple (up to nontrivial center and outer derivations) Lie superalgebras, analogs of Sylvester generators exist only for gl(n|n). The relations for this case are also computed.