Derived representation schemes and cyclic homology

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

doi:10.1016/j.aim.2013.06.020

Cited by 33 publications

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“…in [3]. Following the works [3,14], we see that if a DG algebra admits a shifted bi-symplectic structure, then its DG representation schemes have a shifted symplectic structure. We then apply it to the Koszul Calabi-Yau algebra case.…”

mentioning

confidence: 74%

“…The following proposition was obtained in [3] (see also [9]), and therefore we will only sketch its proof. Given an associative algebra A, denote by (CC • (A), b) and (CH • (A), b) the Connes cyclic complex and the Hochschild chain complex of A respectively.…”

Section: Noncommutative Geometry For Koszul Algebras Supposementioning

confidence: 98%

“…al. [3] suggested that one should instead consider the derived representation schemes of A. In this case, one replaces A with its cofibrant resolution, say QA, in the category of differential graded algebras, and then considers the DG representation schemes of QA, which are then smooth in the DG sense.…”

Section: Derived Representation Schemesmentioning

confidence: 99%

“…Formally, their result is stated as follows. According to [3], L(A) V is called the derived representation scheme (or DRep for short) of A in V , and is also denoted by DRep V (A); its homology H • (DRep V (A)) is called the representation homology of A, and is sometimes denoted by H • (A, V ).…”

Section: Derived Representation Schemesmentioning

confidence: 99%

“…In §8, we briefly discuss some relationships of the current paper with the series of papers by Berest and his collaborators [1,2,3,4], where the derived representation schemes of associative algebras were introduced and studied.…”

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confidence: 99%

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Calabi-Yau algebras and the shifted noncommutative symplectic structure

Chen

Eshmatov

2020

Advances in Mathematics

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In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [14], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [28].1 2 XIAOJUN CHEN AND FARKHOD ESHMATOV justify this question, let us remind a version of noncommutative symplectic structure introduced by Crawley-Boevey, Etingof and Ginzburg in [14], which they called the bi-symplectic structure in the paper. For an associative algebra, a bi-symplectic structure on it is a closed 2-form in its Karoubi-de Rham complex that induces an isomorphism between the space of noncommutative vector fields (more precisely, the space of double derivations) and the space of noncommutative 1-forms. In [38, Appendix], Van den Bergh showed that any bi-symplectic structure naturally gives rise to a double Poisson structure, which is completely analogous to the classical case. However, the reverse is in general not true. Nevertheless, it is still interesting to ask if this is true in the special case of Calabi-Yau algebras.The second motivation of the paper comes from the 2012 paper [28] of Pantev, Toën, Vaquié and Vezzosi, where they introduced the notion of shifted symplectic structure for derived stacks. This not only generalizes the classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on the various moduli spaces that mathematicians are now studying. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincaré duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to noncommutative spaces, such as Calabi-Yau categories.Note that Calabi-Yau algebras are highly related to Calabi-Yau categories. For example, a theorem of Keller (see [21, Lemma 4.1]) says that the bounded derived category of a Calabi-Yau algebra is a Calabi-Yau category. Applying the idea of [28], we would expect that the noncommutative Poincaré duality of a Calabi-Yau algebra shall also play a role in the corresponding shifted noncommutative symplectic structure if it exists.The main results of the current paper may be summarized as follows. Let A be a Calabi-Yau algebra of dimension n. Assume A is also Koszul, and denote its Koszul dual coalegbra by A ¡ . LetR = Ω(Ã ¡ ) be the cobar construction of the co-unitalizationÃ ¡ of A ¡ . In this paper, we show thatR, rather than Ω(A ¡ ) as studied in [1,9], has a (2 − n)-shifted bi-symplectic structure. Such bi-symplectic structure comes from the volume form of the noncommutative Poincaré duality of A, and naturally induces a (2 − n)-shifted symplectic structure on the DG representation schemes Rep V (R), for all vector spaces V (see Theorem 4.6). By taking the corresponding trace maps we obtain a commutative di...

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confidence: 74%

Section: Noncommutative Geometry For Koszul Algebras Supposementioning

confidence: 98%

Section: Derived Representation Schemesmentioning

confidence: 99%

Section: Derived Representation Schemesmentioning

confidence: 99%

mentioning

confidence: 99%

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Calabi-Yau algebras and the shifted noncommutative symplectic structure

Chen

Eshmatov

2020

Advances in Mathematics

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Dual Hodge decompositions and derived Poisson brackets

Berest

Ramadoss

Zhang

2017

Sel. Math. New Ser.

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Abstract. We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov [19]. We give a topological interpretation of such Lie Hodge decompositions in terms of S 1 -equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas-Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas-Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.

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Derived representation schemes and Nakajima quiver varieties

D'Alesio

2021

Sel. Math. New Ser.

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We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $$S^4$$ S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.

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Derived representation schemes and cyclic homology

Abstract: Abstract. We describe the derived functor DRep V (A) of the affine representation scheme Rep V (A) parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V . We construct the characteristic maps Tr V (A)n : HCn (

Cited by 33 publications

References 31 publications

Calabi-Yau algebras and the shifted noncommutative symplectic structure

Calabi-Yau algebras and the shifted noncommutative symplectic structure

Dual Hodge decompositions and derived Poisson brackets

Derived representation schemes and Nakajima quiver varieties

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