Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
MiS Preprint
89/2006
Sylvester-'t Hooft generators of $sl(n)$ and $gl(n|n)$, and relations between them
Christoph Sachse
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 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.
