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