PBW deformations arising from algebraic groups

Mackscheidt, Verity; Fourier, Ghislain Paul Thomas (Thesis advisor); Levandovskyy, Viktor (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

The PBW theorem, named after the authors Poincaré, Birkhoff and Witt, states the following: For a Lie algebra $\mathbb{g}$, the associated graded algebra of its universal enveloping algebra, $\text{gr}~ U(\mathbb{g})$, is isomorphic to its symmetric algebra $S(\mathbb{g})$. The main idea of this correspondence is to deform a graded algebra - here, $S(\mathbb{g})$ - in such a way that it loses the grading, yet the associated graded algebra of this deformation is still unchanged. This PBW theorem motivates the definition of PBW deformations, which should be understood as exactly those deformations which preserve the associated graded algebra. PBW deformations have been defined in different contexts, and in this work, we will focus on those PBW deformations that stem from algebraic groups. In the course of this work, we will define them and summarise the main known results so far - that is, deformations are PBW if and only if the deformation map fulfils an equivariance condition and a generalised Jacobi identity. This is an important result that has been discussed by various authors, most importantly for our context by Etingof-Gan-Ginzburg. However, these notions so far have been translated into explicit conditions on the deformation map only in special cases. There are still numerous interesting open questions in the area of PBW deformations. This work focuses on answering the two following main questions: 1) How do we translate the known conditions on PBW deformation maps into explicit parametrisations of these maps? 2) What does the center of PBW deformations look like? The first question will be approached in the setting of orthosymplectic groups. We will follow a combinatorial approach in Deligne's interpolation category. In that manner, we firstly describe a combinatorial basis of Hom-spaces in which the deformation maps live and impose the Jacobi identity there. This will be done combinatorially in terms of certain arc diagrams. From this, we obtain certain conditions that we can translate onto the deformation maps, in order to find explicit results on what a PBW deformation look like. This result then holds true for all orthosymplectic groups, and recovers special cases found by Etingof-Gan-Ginzburg (2005) and Tsymbaliuk (2015). We will approach the second question in different steps, which will be done for different levels of generality. This leads to an explicit formula for central elements for infinitesimal Hecke algebras of $\mathbb{so}_2$, and a conjecture for a whole family of central elements for infinitesimal Hecke algebras of $\mathbb{so}_n$.