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

Abstract Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a finite-dimensional simple basic classical Lie superalgebra over \mathbb{C}. Let G be the reductive algebraic group over \mathbb{C} such that \mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. In this thesis, we calculate the centralizer \mathfrak{g}^{e} of e in \mathfrak{g} and its centre \mathfrak{z}(\mathfrak{g}^{e}) 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;\alpha), G(3) and F(4) to calculate bases for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}). Note that for Lie superalgebras of type A(n,n), we consider \mathfrak{sl}(n|n) instead of \mathfrak{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 \mathfrak{z}(\mathfrak{g}^{e}) under the adjoint action of G^{e}, we prove theorems relating the dimension of \left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}} and the labelled Dynkin diagram.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):