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

Abstract

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.

14 pages, 6 figures

References in corpus (2)

• Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices
• Vertex Ring-Indexed Lie Algebras

Cited by in corpus (6)

• Quantum Rabi model for N-state atoms
• General phase spaces: from discrete variables to rotor and continuum limits
• Superintegrability of Geodesic Motion on the Sausage Model
• Cartan matrices and presentations of Cunha and Elduque superalgebras
• Quantum models with spectrum generated by the flows of polynomial zeros
• Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix