Derived Representation Schemes and Cyclic Homology

Berest, Yuri; Khachatryan, George A.; Ramadoss, Ajay C.

doi:10.48550/arxiv.1112.1449

Cited by 3 publications

(1 citation statement)

References 29 publications

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“…Building on the work of Bergman [5] and Cohn [18] (see [3,4] for excellent expositions and insightful generalizations), it is well known that the functor (2.9) is representable: we have an adjunction (see [4,Proposition 3])…”

Section: 3mentioning

confidence: 99%

On the noncommutative Poisson geometry of certain wild character varieties

Fairon¹,

Fernández²

2021

Preprint

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To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson varieties attached to colored quivers whose representation theory is controlled by fission algebras: noncommutative algebras generalizing the multiplicative preprojective algebras of Crawley-Boevey and Shaw. Previously, Van den Bergh exploited the Kontsevich-Rosenberg principle to prove that the natural Poisson structure of any (non-colored) multiplicative quiver variety is induced by an H 0 -Poisson structure on the underlying multiplicative preprojective algebra. Moreover, he noticed that this noncommutative structure comes from a Hamiltonian double quasi-Poisson algebra constructed from the quiver; this offers a noncommutative analogue of quasi-Hamiltonian reduction. In this article we conjecture that, via the Kontsevich-Rosenberg principle, the natural Poisson structure on each colored multiplicative quiver variety is induced by an H 0 -Poisson structure on the underlying fission algebra which, in turn, is obtained from a Hamiltonian double quasi-Poisson algebra attached to the colored quiver. We study some consequences of this conjecture and we prove it in two significant cases: the monochromatic interval and the monochromatic triangle. Contents MAXIME FAIRON AND DAVID FERN ÁNDEZ 5.4. Conditions obtained from Case 3 29 5.5. Conditions obtained from Case 4 38 5.6. Checking the conditions for the double bracket of Theorem 4.5 40 Appendix A. Remaining expressions for the double bracket on B(∆) 43 A.1. The brackets of the form { {w ij , v kl } } 43 A.2. The brackets of the form { {v kl , w ij } } 43 A.3. The brackets of the form { {w ij , w kl } } 43 References 44

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