Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads
Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads
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DOI：
10.4310/mrl.2001.v8.n3.a12
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发表时间：
2000-05
影响因子：
1
通讯作者：
V. Ginzburg
中科院分区：
文献类型：
--
作者：
V. Ginzburg
Quiver 簇最近出现在数学的各个不同领域，例如 Kac-Moody 代数和量子群的表示论、4 流形上的瞬时子以及克莱因奇点的解析。在本文中，我们证明了许多重要的仿射箭袋变体，例如Calogero-Moser空间，可以作为共伴轨道嵌入到适当的无限维李代数的对偶中。特别是，所讨论的李代数对箭袋簇有无穷小的传递作用。我们的构造基于 Kontsevich 的“非交换辛几何”形式主义的扩展。我们表明，这种形式主义在更一般的 P 几何框架中获得了最充分和最自然的表述，P 几何是任意循环 Koszul 运算上的代数的“非交换几何”。
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.