Stable representation homology and Koszul duality

Berest, Yuri; Ramadoss, Ajay C.

doi:10.1515/crelle-2014-0001

Cited by 17 publications

(43 citation statements)

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“…The problem, however, is that the 'injective resolution' RBG is 'too big' and complicated, and it is hard to understand RLoc G (X) (in particular, to compute π i [RLoc G (X)] for i > 0) even in simplest examples. Kapranov's definition of the derived moduli space of G-local systems was refined and generalized within the framework of derived algebraic geometry by Toën, Vezzosi, Pridham and others (see, e.g., [50,52,53,36,38,39,40] and [51] for a general overview) 4 ; however, explicit computations seem still to be missing.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Vanishing theorems for representation homology and the derived cotangent complex

Berest

Ramadoss

Yeung

2019

Algebr. Geom. Topol.

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Let G be a reductive affine algebraic group defined over a field k of characteristic zero. In this paper, we study the cotangent complex of the derived G-representation scheme DRep G (X) of a pointed connected topological space X. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G (X) to the representation homology HR * (X, G) := π * O[DRep G (X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in R 3 and generalized lens spaces. In particular, for any f.g. virtually free group Γ, we show that HR i (BΓ, G) = 0 for all i > 0. For a closed Riemann surface Σg of genus g ≥ 1, we have HR i (Σg, G) = 0 for all i > dim G. The sharp vanishing bounds for Σg depend actually on the genus: we conjecture that if g = 1, then HR i (Σg , G) = 0 for i > rank G, and if g ≥ 2, then HR i (Σg, G) = 0 for i > dim Z(G) , where Z(G) is the center of G. We prove these bounds locally on the smooth locus of the representation scheme Rep G [π 1 (Σg )] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined K-theoretic virtual fundamental class for DRep G (X) in the sense of Ciocan-Fontanine and Kapranov [12]. We give a new "Tor formula" for this class in terms of functor homology.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Vanishing theorems for representation homology and the derived cotangent complex

Berest

Ramadoss

Yeung

2019

Algebr. Geom. Topol.

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“…As we remarked before, the vertical maps are both isomorphisms, while the two horizontal maps are neither surjective nor injective in general (for see [4] more details).…”

Section: Identification Of Gl-invariant 1-forms and Vectorsmentioning

confidence: 97%

“…then it is surjective. However, for an arbitrary DG algebra, Berest and Ramadoss showed in [4] that this is in general not true; there are some homological obstructions for Tr to be so.…”

Section: Gl-invariants and The Trace Mapmentioning

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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“…As explained in[5, Appendix A], this map may be called the cyclic de Rham differential. Its kernel is [R,R] (see loc.…”

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confidence: 99%

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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Stable representation homology and Koszul duality

Abstract: This paper is a sequel to [BKR], where we study the derived affine scheme DRep

Cited by 17 publications

References 11 publications

Vanishing theorems for representation homology and the derived cotangent complex

Vanishing theorems for representation homology and the derived cotangent complex

Calabi-Yau algebras and the shifted noncommutative symplectic structure

Dual Hodge decompositions and derived Poisson brackets

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