Double Poisson algebras

By Michel van den Bergh. 2004

217 citations recorded by the source corpus.

In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and Ginzburg. Our (quasi-)Poisson brackets induce classical (quasi-)Poisson brackets on representation spaces. As an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68c1192dfab827fc6d0b3e2379e6d61a3a76ced7

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geometry of Multiplicative Preprojective Algebra (2007, methodology): “The proof needs a long calculation (see [36]).”
Poisson structures on moduli spaces of representations (2011, unknown): “Note that what are here called H 0 -Poisson structures were called " noncommutative Poisson structures " in [7] and " Poisson structures " in [13] and [10] is an H 0 -Poisson structure.”
Geometry and braiding of Stokes data; Fission and wild character varieties (2011, background): “gues of multiplicative quiver varieties for groups besides general linear groups, by considering the reductions Br := GA r H//G in general, e.g. if H = H 1 × H 2.) Remark 4.4. Note that van den Bergh [59, 60] constructs some “noncommutative quasi-Hamiltonian spaces” such that after choosing a linear”
Simple finite-dimensional double algebras (2016, background): “Thus this double algebra is simple if and only if dimV = 1, α 6= 0.”
Quasi-Poisson structures on representation spaces of surfaces (2012, methodology): “) β∗p(α −1) p•αν•∗ ⊗[ν∗•α•pβp∗] = X p∈α∩β εp(α,β) [β∗pαp•ν•∗]⊗ [ν∗•α•pβp∗]. Lemma7.2. The double bracket {{−,−}}s is quasi-Poisson. Proof. In the proof we will use the following notation suggested in [VdB]. Given a map {{−,−}} : A⊗2 → A⊗2 and u∈ A, v∈ A⊗2, we set {{u,v}}L= nn u,v(1) oo ⊗v(2), {{u,v}”