Symplectic Reflection Algebras
Symplectic Reflection Algebras
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辛反射代数
DOI：
10.33232/bims.0050.27.50
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
K. Brown
中科院分区：
文献类型：
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作者：
K. Brown
1.1.在本文中，我将描述一类美丽的结合代数，即标题中的辛反射代数，这是在 Etingof 和 Ginzburg 最近的一篇论文中介绍的[11]。这些代数是有限群的斜群代数的变形，因此保留了有限群的某些属性，因此我们在第 2 节中开始讨论后一类代数，特别是有限群 Γ 的斜群代数 S(V )*Γ 线性作用于有限维复向量空间 V ，因此作用于 V 的对称代数， S(V)。我们可以通过使用斜对称双线性形式 κ : V × V −→ CΓ 破坏 S(V ) 的交换性来使这样的代数变形，如我们在 (4.1) 中所解释的；但一般来说，当这样做时，结果的代数太“小”，因为不存在与 S(V ) ∗ Γ 的自然向量空间双射。然而，值得注意的是，如果 Γ 由辛空间 (V, ω) 的辛自同构组成，则存在一类从 ω 导出的形式 κ，其所产生的变形确实与 S(V ) ∗ Γ 具有自然的线性双射——也就是说，在这种情况下存在一个“PBW 定理”，我们在 (4.1) 中陈述和解释。此类形式的定义需要辛反射的概念，即辛世界中普通向量空间上的伪反射的类似物。因此，我们在第 3 节中定义并讨论这些辛反射，然后继续陈述 Etingof 和 Ginzburg 的 PBW 定理以及
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