I intend to overview classifications of simple Lie (super)algebras of finite dimension and of polynomial growth. Various properties of complex Lie superalgebras resemble same of modular Lie algebras. I will encourage to consider these classifications without fanaticism: certain non-simple Lie (super)algebras, ‘close’ to simple ones, are often ‘better’ for us than simple ones.
Interesting features of deformations: semi-trivial deformations and (in super setting) odd parameters.
I’ll formulate classification of finite-dimensional simple complex Lie superalgebras, odd parameters including.
I’ll formulate a definition of Lie superalgebra suitable for any characteristic and classification of simple (finite-dimensional) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gradings modulo 2. I’ll mention conjectures on classification of modular Lie algebras and superalgebras.
Is it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms of the Poisson Lie (super)algebras, Lie (super)algebras of ‘matrices of complex size’, etc.
Examples. Double extensions of simple Lie (super)algebras are definitely ‘more interesting’ than the simple objects they extend.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
