Noncommutative geometry and quiver algebras

By W. Crawley‐Boevey, Pavel Etingof, V. Ginzburg. 2005

155 citations recorded by the source corpus.

Abstract We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B , which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, Π = Π ( B ) , that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space Π / [ Π , Π ] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where Π is the preprojective algebra associated with a quiver of non-Dynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of Π in terms of the Lie algebra Π / [ Π , Π ] .

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

View Paper: https://www.semanticscholar.org/paper/d75f8aa702141df2df3fc48d348b6fadcada438f

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

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Representation theory in complex rank, II (2014, unknown): “g of invariants of msquare matrices A 1,...,Am is generated by traces of cyclic words of these matrices, and these traces are asymptotically independent when the matrix size goes to inﬁnity (see e.g. [CEG], Section 11). For Ot and Sp 2t, the proof is similar; namely, one needs to use the well known ”
The Coulomb Branch of 3d N = 4 Theories (2015, background): “[41, 42](9): first taking a subring of gauge-invariant operators in AH , and then imposing the moment-map constraints”
Graded Calabi Yau algebras of dimension 3 (2006, background): “This is indeed the standard resolution for preprojective algebras of non-Dynkin quivers (for extended Dynkin quivers this was shown in [7]; for general quivers this can be deduced from the Hochschild cohomology calculated in [6]).”
Calabi-Yau algebras (2006, unknown): “ends x→u, and that annihilates the other 2n+ 1 generators of this free algebra. Now, let H:= δ ·τ+ t·τ2. This is an even potential; one computes ∂H ∂t = τ 2,and ∂H ∂τ = δ.From this, using (2.8.4) and [CBEG], Lemma 7.1.1(ii), we ﬁnd ξH = adτ+δ∂t −τ2∂τ. Further, according to (1.5.9), in the basisθi = ”
Hypertoric category O (2010, unknown): “ociated graded map grµλ : Sym(k) −→∼ grZ(U) →gr D ∼=C[T∗Cn] 13 is given by composition with the classical moment map T∗Cn →k∗. The ring U λ is known as a noncommutative Hamiltonian reduction of D by K[CBEG07]. The associated graded algebra grUλ = gr D K . µα(k) ∼= (gr D )K . grµα(k) ∼= C[T∗Cn]K . ”
Calabi-Yau Frobenius algebras (2007, methodology): “ribe the maps relating the Hochschild (co)homology of 20 Dynkin preprojective algebras with the extended Dynkin case; in the latter case, the Hochschild cohomology groups were described over k = C in [CBEG07] and later over Z in [Sch07]. The new theorems proved here are: Theorem 3.2.7, which compute”
Poisson structures on moduli spaces of representations (2011, unknown): “It is shown in [5] that the necklace Lie algebra structure on KQ/[KQ, KQ] descends to a Lie algebra structure on Π λ /[Π λ , Π λ ].”
Quiver varieties and Lusztig's algebra (2004, background): “[5, 12]), define a (non-commutative) Poisson bracket directly on LΩ as follows: {f , f } = ∑”
Quantizations of conical symplectic resolutions I: local and global structure (2012, background): “the residual action of the component group. We observe that this geometric operation of symplectic reduction is closely related to an algebraic one introduced by Crawley-Boevey, Etingof, and Ginzburg [CBEG07]. Let Yξ := A A·hη(x) −ξ(x) |x∈gi. Proposition 3.11 If A= ΓS(D) and Ared = ΓS(Dred), then A”
Simple finite-dimensional double algebras (2016, result): “In particular, the notion of a double Poisson algebra introduced in [1] and [10] fits this ideology (however, this approach is different from the one in [6]).”
Calabi-Yau algebras and superpotentials (2010, background): “More precisely iδ : Ω 1 lA→ Ω 1 lA⊗ Ω 1 lA is defined as follows: for a ∈ A one has iδ(a) = 0 iδ(Da) = δ(a) Following [5] we put (10.1) ιδ(ω) = (−1) |iδ(ω) ′||iδ(ω) ′′|iδ(ω) ′′iδ(ω) ′ Definition 10.1.1.”