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 correspond to pairs of usual algebraic groups and Lie algebras in Ver_p, as described by Venkatesh. This 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 'semi simplifying' usual Lie algebras and provide new Lie algebras in Ver_p.
Classical random matrix theory focuses on the study of highly structured models (e.g. Wigner and Wishart matrices) which are presented as a sequence of random matrices defined for every dimension, whose asymptotic (i.e. as the dimension goes to infinity) spectral properties must be understood in detail. However, modern problems in data and computer science require only a coarser understanding of the random matrices in question, but necessitate nonasymptotic results in settings where the models are less structured and do not necessarily belong to a prescribed sequence of matrices. In this work we provide tools for analyzing the spectral distribution of any self-adjoint noncommutative polynomial evaluated in arbitrary independent unitarily invariant Hermitian random matrices of a fixed dimension. Our results can be interpreted as a quantitative version of Voiculescu's celebrated asymptotic freeness result. This is joint work with Chi-Fang Chen and Joel Tropp.
Non-asymptotic Random Matrix Theory for Unitarily Invariant Matrices