Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads

By V. Ginzburg. 2000

132 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4766e2173c3fbfefe7e815a94465eafb194afbf4

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In Algebra and Number Theory

1. B. Keller
2. V. Ginzburg
3. Osamu Iyama
4. Harm Derksen
5. Michel van den Bergh
6. Pavel Etingof
7. Jerzy Weyman
8. I. Losev

In Geometry and Topology

1. B. Keller
2. V. Ginzburg
3. Osamu Iyama
4. Michel van den Bergh
5. Harm Derksen
6. Jerzy Weyman
7. Pavel Etingof
8. I. Losev

In Mathematical Physics

1. V. Ginzburg
2. Michel van den Bergh
3. Pavel Etingof
4. I. Losev
5. Harm Derksen
6. Jerzy Weyman
7. W. Crawley‐Boevey
8. I. Gordon

Citations in the corpus

• Noncommutative geometry and quiver algebras (2005, background): “ X a∈Q [pa,a] = 0 . Proof of the Lemma is straightforward and is left to the reader. 8.4 Necklace bracket. The symplectic form ωgives a Lie bracket on P♮, known as the Necklace Lie bracket, see [BLB],[Gi]. To write explicit formulas for this bracket it is convenient to introduce the following notati”
• Calabi-Yau algebras and superpotentials (2010, methodology): “ are certainly heavily inspired by [13, 21, 23]. More precise references will be given below. 10.1. A reminder on non-commutative symplectic geometry. We introduce some notions from [5, 22]. See also [4, 14, 32, 33]. Let Abe a graded algebra over PC(le). Put DR l(A) = TA(Ω1A)/[TA(Ω1A),TA(Ω1A)] Here ”
• Quivers with potentials and their representations I: Mutations (2007, background): “r the convenience of the reader, we present complete independent proofs. Victor Ginzburg informed us that in the context of path algebras of quivers, cyclic derivatives were introduced and studied in [5, 24], and that Proposition 3.7 is a consequence of the geometric interpretation of J(S) given in ”
• Calabi-Yau algebras (2006, unknown): “gebra of Qand view R= CI⊂CQas the subalgebra formed by trivial paths. The commutator quotient, F cyc, of the algebra F= CQmay be identified with the vector space spanned by cyclic paths in Q, cf. e.g. [Gi1]. Generalizing formulas (1.5.7) and (1.3.1), one defines, for any edge x∈Q, linear maps F cyc →F”