Representations of two-dimensional compatible Lie algebras

Abstract: A compatible Lie algebra is a vector space equipped with two Lie products such that any linear combination of them is also a Lie product. These algebras arose from the related class of compatible Poisson algebras in the context of mathematical physics and Hamiltonian mechanics. In this talk, we start by stating some basic definitions and results about compatible Lie algebras. We then present counterexamples to analogues of some of the most important theorems in Lie algebra theory, namely the theorems of Weyl and Levi, highlighting how compatible Lie algebras can behave very differently from Lie algebras. We then move on to studying the representation theory of a family of simple two-dimensional compatible Lie algebras. We construct a family of irreducible representations for each algebra of this family, and thereafter, we focus on one specific simple two-dimensional compatible Lie algebra in order to make the computations simpler and results easier to state and prove. In this setting, we prove a Clebsch-Gordan formula for the irreducible representations previously described, and we also exhibit a second family of representations, this time "breaking" Weyl's theorem (i.e., reducible and indecomposable representations over the field of complex numbers). Time permitting, we finish by discussing the failure of further central results from Lie algebra theory in this broader context, including the characterization of semisimple algebras and the Whitehead Lemmas.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar