UAlberta-PIMS Mathematics and Statistics Colloquium: Ivan Angiono

A good way to understand the structure of a finite group is through its representations. The family of representations of a group on a fixed field has several properties: it is closed by finite direct sums, by tensor products, it contains the dual space and the tensor product of two representations is (naturally) isomorphic to the tensor product of the same representations in the opposite order. These properties are abstracted to give the notion of a symmetric tensor category. When the characteristic of the field does not divide the order of the group, the category of representations is semisimple: every representation is written as a direct sum of simple representations. The smallest example where this does not happen is of a cyclic group of prime order over a field whose characteristic is exactly that prime. Through a process of semisimplification we obtain a quotient category which is semisimple, and still tensor symmetric, which is called the Verlinde category Ver_p. Being symmetric, objects from Lie theory can be considered, which give rise to new tensor categories via their representations. We are particularly interested in Lie algebras in. Due to a celebrated result by Deligne, symmetric tensor categories of moderate growth over (algebraically closed) fields of characteristic zero correspond to categories of representations of affine algebraic supergroups. Once we move to positive characteristic, we need to take into account the Verlinde category Ver_p: Coulembier-Etingof-Ostrik proved recently that every such symmetric tensor category is the one of representations of an algebraic group in Ver_p, under some restrictions. Thus, we wonder how to describe algebraic groups in Ver_p, which in turn leads to the question of how to obtain Lie algebras in Ver_p. 

This talk is based on joint works with J. Plavnik and G. Sanmarco where we look for examples of these Lie algebras. We prove the existence of contragredient Lie algebras in symmetric tensor algebras generalizing Kac-Moody construction of Lie (super) algebras, which at the same time give a description of some examples obtained previously by semisimplification of usual Lie algebras and provide new Lie algebras in Ver_p.