Centres of centralizers of nilpotent elements in simple lie superalgebras

Han, Leyu (2020). Centres of centralizers of nilpotent elements in simple lie superalgebras. University of Birmingham. Ph.D.

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Abstract

Let g = g¯ 0 ⊕ g¯ 1 be a nite-dimensional simple basic classical Lie superalgebra over C. Let G be the reductive algebraic group over C such that Lie(G) = g¯ 0. Suppose e ∈ g¯ 0 is nilpotent. In this thesis, we calculate the centralizer ge of e in g and its centre z(ge) especially. We begin by recalling basic notions of Lie algebras and Lie superalgebras, such as root system and Dynkin diagrams. Once this is achieved, we look into further detail about the structure of basic classical Lie superalgebras of type A(m,n), B(m,n), C(n), D(m,n), D(2,1;α), G(3) and F(4) to calculate bases for ge and z(ge). Note that for Lie superalgebras of type A(n,n), we consider sl(n|n) instead of psl(n|n). For the above types of Lie superalgebras, we also determine the labelled Dynkin diagram with respect to e. After considering the structure of z(ge) under the adjoint action of Ge, we prove theorems relating the dimension of (z(ge))Ge and the labelled Dynkin diagram.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):