In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis. Fifteen years ago, Boe, Kujawa and the speaker introduced the concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the concept of a BBW parabolic subalgebra.
Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫– ⨁ 𝔣 ⨁ 𝔫+ with 𝔟 = 𝔣 ⨁ 𝔫– where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical ‘simple’ Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R ∙ indBG (-). The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how this treatment can further unify the study of the representation theory of classical Lie superalgebras.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
