Sylvester-’t Hooft generators and relations between them for $\mathfrak{s}\mathfrak{l}(n)$ and $\mathfrak{g}\mathfrak{l}(n|n)$

Abstract

Among the simple finite-dimensional Lie algebras, only $\mathfrak{s}\mathfrak{l}(n)$ has two finite-order automorphisms that have no common nonzero eigenvector with the eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow generating all of $\mathfrak{s}\mathfrak{l}(n)$ 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×n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, the orthogonal decompositions of Lie algebras, ’t Hooft’s work on confinement operators in QCD, and various other instances. Here, we give an algorithm that both generates $\mathfrak{s}\mathfrak{l}(n)$ and explicitly describes a set of defining relations. For simple (up to the center) Lie superalgebras, analogues of Sylvester generators exist only for $\mathfrak{g}\mathfrak{l}(n|n)$. We also compute the relations for this case.