Symplectic reflection algebras and Poisson geometry

Martino, Maurizio (2006) Symplectic reflection algebras and Poisson geometry. PhD thesis, University of Glasgow.

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Abstract

The subject of this thesis is the Poisson geometry of varieties associated to the centres of symplectic reflection algebras in the PI case. In particular it focuses on describing the symplectic leaves of these varieties. Chapter 1 introduces the theory of symplectic reflection algebras. We introduce the classical objects of rings of invariants and skew group rings and describe deformations of these and present some of their basic properties following [26]. We highlight a dichotomy in the theory and focus our attention to the PI case. The framework for studying the representation theory is developed and the close connection with Poisson geometry is explained using results of [11]. Poisson algebras are introduced and the fundamental notion of stratifying Poisson varieties by symplectic leaves is explained. Although symplectic leaves are a well-known concept in the field of Poisson manifolds (see [55] and [79]) we examine them in the context of Poisson algebraic geometry as described in [11]. At the heart of Chapter 2 is the example of representations of deformed preprojective algebras and we view different aspects of the geometry of these varieties. The necessary invariant theory is introduced, which involves discussing the categorical quotient and its properties as in [68] or [48]. Crucial to us will be the stratification by orbit type. Moment maps and Marsden-Weinstein reductions for symplectic varieties are introduced following the approach of [15]. Combinatorial aspects of quivers are examined, in particular this includes the description by root vectors of orbit type strata for representations of preprojective algebras, as given in [16] and [17]. Finally hyper- Kahler manifolds make an appearance and provide a means for establishing the existence of the local normal form of the moment map, as was proved in [63]. The local normal form plays a crucial role in our main theorem concerning symplectic leaves, Theorem 4.2. Chapter 3 combines filtered and gi'aded techniques with the properties of symplectic leaves to establish that the associated variety of a Poisson prime ideal of the centre of a syraplectic reflection algebra is irreducible. The arguments used are ba.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Additional Information: Adviser: Ken Brown
Keywords: Mathematics
Date of Award: 2006
Depositing User: Enlighten Team
Unique ID: glathesis:2006-74084
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 23 Sep 2019 15:33
Last Modified: 23 Sep 2019 15:33
URI: http://theses.gla.ac.uk/id/eprint/74084

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