Symplectic Reflection Algebras

By Kenneth A. Brown. Irish Mathematical Society Bulletin, 2003

56 citations recorded by the source corpus.

1.1. In this paper I shall describe a beautiful class of associative algebras, the symplectic reflection algebras of the title, introduced in a recent paper of Etingof and Ginzburg [11]. These algebras are deformations of, and so retain some of the properties of, skew group algebras of finite groups, so it’s with the latter class of algebras with which we begin in Section 2, in particular with the skew group algebra S(V )∗Γ of a finite group Γ acting linearly on a finite dimensional complex vector space V , and hence acting on the symmetric algebra of V , S(V ). One can deform such an algebra by destroying the commutativity of S(V ) using a skew-symmetric bilinear form κ : V × V −→ CΓ, as we explain in (4.1); but in general when this is done the algebra which results is too “small”, in the sense that there is no natural vector space bijection with S(V ) ∗ Γ. Remarkably, however, if Γ consists of symplectic automorphisms of the symplectic space (V, ω) then there is a class of forms κ derived from ω for which the resulting deformations do have a natural linear bijection with S(V ) ∗ Γ—that is, there is a “PBW theorem” in this setting, which we state and explain in (4.1). The definition of this class of forms requires the concept of a symplectic reflection, the analogue in the symplectic world of the pseudo-reflections on an ordinary vector space. So we define and discuss these symplectic reflections in Section 3, before going on to state the PBW theorem of Etingof and Ginzburg and

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5281a96df8373b84d0e34254ae025ab796b9b78b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

  1. I. Gordon
  2. K. Goodearl

In Geometry and Topology

  1. I. Gordon
  2. Stephen Griffeth
  3. V. Ginzburg
  4. Pavel Etingof
  5. J. T. Stafford
  6. E. Gorsky
  7. I. Losev
  8. Y. Berest

In Statistical and Nonlinear Physics

  1. I. Gordon
  2. V. Ginzburg
  3. J. T. Stafford

Citations in the corpus