arXiv subjects

Yuri Berest

Publications and source records attributed to Yuri Berest.

At least 19 recordsLinked to original sources

arXiv2026

Ganea decompositions of classifying spaces

We study homotopy decompositions of the classifying spaces $BG$ of compact connected Lie groups obtained by (relative) fiber-cofiber construction. Given a pair of Borel fibrations $ F \to E \to BG $ and $F' \to E' \to BG $, this construction yields a tower (telescope) of spaces $ X_{m}(F,F') $ over $BG$ indexed by $ \mathbb{Z}_+ $ that converges in the sense that $\text{hocolim} \,(X_{m})\,$ is weakly homotopy equivalent to $BG$. We determine cohomological conditions on the fibrations that produce the spaces $X_{m}(F,F')$ with properties similar to those of the spaces of quasi-invariants of Weyl groups constructed by the first and third authors. We prove that, under these conditions, the resulting homotopy decompositions of $BG$ are sharp (over $\mathbb{Q}$), the spaces $X_{m}(F,F')$ are rationally formal and Cohen-Macaulay, their cohomology rings being finite rank free modules over $H^*(BG, \mathbb{Q})$. We construct many examples which include the fundamental (maximal torus) fibration $ G/T \to BT \to BG $ as well as the universal fibration $\, E_{\rm com}G_{\bf 1} \to B_{\rm com}G_{\bf 1} \to BG \,$ for the classifying space $B_{\rm com}G$ of commuting elements in $G$ introduced by Adem and G\'{o}mez, as the first fibration in the pair. In most cases, we give an explicit presentation for the (equivariant) cohomology rings in terms of characteristic classes and compute the (equivariant) $K$-theory of the spaces involved. The paper contains an Appendix, where we re-examine the topological fiber-cofiber construction in an abstract setting, proving an $\infty$-categorical extension of the classical Ganea Theorem.

math.AT↗

arXiv2025

Quasi-flag manifolds and moment graphs

We introduce and study a new class of topological $G$-spaces generalizing the classical flag manifolds $G/T$ of compact connected Lie groups. These spaces, which we call the $m$-quasi-flag manifolds $ F_m = F_m(G,T) $, are topological realizations of the algebras $ Q_k(W) $ of $k$-quasi-invariant polynomials of the Weyl group $ W $ in the sense that their (even-dimensional) $G$-equivariant cohomology $ H_G(F_m, {\mathbb C}) $ is naturally isomorphic to $ Q_k(W) $, where $ m $ is a $W$-invariant integer-valued multiplicity function on the system of roots of $W$ and $ k = \frac{m}{2}$ or $ \frac{m+1}{2}$ depending on whether $m$ is even or odd. Many topological properties and algebraic structures related to the flag manifolds can be extended to quasi-flag manifolds. We compute the cohomology of quasi-flag manifolds by constructing their rational algebraic models in terms of coaffine stacks -- a certain kind of derived stacks introduced by B.To\"en and J. Lurie to provide an algebro-geometric framework for rational homotopy theory. Besides cohomology, we also compute the equivariant K-theory of quasi-flag manifolds and extend some of our cohomological results to the multiplicative setting. On the topological side, our approach is strongly influenced by the classical work on homotopy decompositions of classifying spaces of compact Lie groups; however, the diagrams that we use in our decompositions do not arise from collections of subgroups of $G$ but rather from moment graphs -- combinatorial objects introduced in a different area of topology called the GKM theory.

math.AT↗

arXiv2023

Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman)

This is the first in a series of papers, where we introduce and study topological spaces that realize the algebras of quasi-invariants of finite reflection groups. Our result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the ring of invariant polynomials a Weyl group $W$ as a cohomology ring of the classifying space $BG$ of the associated Lie group $G$. In the present paper, we state our realization problem for the algebras of quasi-invariants of Weyl groups and give its solution in the rank one case (for $G = SU(2)$). We call the resulting $G$-spaces $ F_m(G,T) $ the $m$-quasi-flag manifolds and their Borel homotopy quotients $ X_m(G,T) $ the spaces of $m$-quasi-invariants. We compute the equivariant $K$-theory and the equivariant (complex analytic) elliptic cohomology of these spaces and identify them with exponential and elliptic quasi-invariants of $W$. We also extend our construction of spaces quasi-invariants to a certain class of finite loop spaces $ \Omega B $ of homotopy type of $ S^3 $ originally introduced by D. L. Rector. We study the cochain spectra $ C^*(X_m,k) $ associated to the spaces of quasi-invariants and show that these are Gorenstein commutative ring spectra in the sense of Dwyer, Greenlees and Iyengar.

math.AT↗

arXiv2022

Symmetric Homology is Representation Homology

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology theory is naturally equivalent to the (one-dimensional) representation homology theory introduced by the authors (jointly with G. Khachatryan) in \cite{BKR}. Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including the main conjecture of \cite{AF07} on topological interpretation of symmetric homology of polynomial algebras.

math.AT↗

arXiv2022

Derived Character Maps of Groups Representations

In this paper, we construct and study derived character maps of finite-dimensional representations of $\infty$-groups. As models for $\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch). We define cyclic, symmetric and representation homology for `group algebras' over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring $k$ is a field of characteristic zero. We also study the behavior of the derived character maps of $n$-dimensional representations in the stable limit as $ n\to \infty$, in which case we show that they `converge' to become isomorphisms.

math.AT↗

arXiv2020

Representation homology of simply connected spaces

Let $G$ be an affine algebraic group defined over field $k$ of characteristic zero. We study the derived moduli space of G-local systems on a pointed connected CW complex X trivialized at the basepoint of $X$. This derived moduli space is represented by an affine DG scheme RLoc$_G(X,*)$: we call the (co)homology of the structure sheaf of RLoc$_G(X,*)$ the representation homology of $X$ in $G$ and denote it by HR$_*(X,G)$. The HR$_0(X,G)$ is isomorphic to the coordinate ring of the representation variety Rep$_G[π_1(X)]$ of the fundamental group of $X$ in $G$ -- a well-known algebro-geometric invariant of $X$ with many applications in topology. The case when X is simply connected seems much less studied: in this case, the HR$_0(X,G)$ is trivial but the higher representation homology is still an interesting rational invariant of $X$ depending on the algebraic group $G$. In this paper, we use rational homotopy theory to compute the HR$_*(X,G)$ for an arbitrary simply connected space $X$ (of finite rational type) in terms of its Quillen and Sullivan algebraic models. When $G$ is reductive, we also compute the $G$-invariant part of representation homology, HR$_*(X,G)^G$, and study the question when HR$_*(X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be closely related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by B. Feigin and P. Hanlon in the 1980s and proved by S. Fishel, I. Grojnowski and C. Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces $X$ for which HR$_*(X,G)^G$ is a graded symmetric algebra for any complex reductive group $G$.

math.AT↗

arXiv2020

Representation homology of topological spaces

In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, $S^1$-equivariant homology of the free loop space and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in $\R^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $\R^3$.

math.AT↗

arXiv2020

Deformed Calogero--Moser operators and ideals of rational Cherednik algebras

We consider a class of hyperplane arrangements $\mathcal A$ in ${\mathbb C}^n$ that generalise the locus configurations of \cite{CFV}. To such an arrangement we associate a second order partial differential operator of Calogero-Moser type, and prove that this operator is completely integrable (in the sense that its centraliser in $\mathcal{D}({\mathbb C}^n\setminus\mathcal A)$ contains a maximal commutative subalgebra of Krull dimension $n$). Our approach is based on the study of shift operators and associated ideals in the spherical Cherednik algebra that may be of independent interest. The examples include all known families of deformed (rational) Calogero-Moser systems that appeared in the literature; we also construct some new examples, including a BC-type analogues of completely integrable operators recently found by D. Gaiotto and M. Rap\v{c}\'ak in \cite{GR}. We describe these examples in a general framework of rational Cherednik algebras close in spirit to \cite{BEG} and \cite{BC}.

math-ph↗

arXiv2020

Hodge decomposition of string topology

Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in our earlier work.

math.QA↗

arXiv2019

Cyclotomic Expansion of Generalized Jones Polynomials

In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in $S^3$ carries a natural action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on 3 parameters $q, t_1, t_2$. As a consequence, for a knot $K$ satisfying this conjecture, we defined a three-variable polynomial invariant $J^K_n(q,t_1,t_2)$ generalizing the classical colored Jones polynomials $J^K_n(q)$. In this paper, we give explicit formulas and provide a quantum group interpretation for the generalized Jones polynomials $J^K_n(q,t_1,t_2)$. Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by K.\ Habiro: as in the classical case, they imply the integrality of $J^K_n(q,t_1,t_2)$ and, in fact, make sense for an arbitrary knot $K$ independent of whether or not it satisfies our earlier conjecture. When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of $J^K_n(q,t_1)$ are determined by Macdonald orthogonal polynomials of type $A_1$.

math.QA↗

arXiv2018

Vanishing theorems for representation homology and the derived cotangent complex

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 $ {\rm 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 $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := π_*{\mathcal O}[{\rm 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 $ {\mathbb R}^3 $ and generalized lens spaces. In particular, for any f.g. virtually free group $ Γ$, we show that $\, {\rm HR}_i({\rm B}Γ, G) = 0 \,$ for all $ i > 0 $. For a closed Riemann surface $Σ_g $ of genus $ g \ge 1 $, we have $\, {\rm HR}_i(Σ_g, G) = 0 \,$ for all $ i > \dim G $. The sharp vanishing bounds for $ Σ_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $\, {\rm HR}_i(Σ_g, G) = 0 \,$ for $ i > {\rm rank}\,G $, and if $ g \ge 2 $, then $\, {\rm HR}_i(Σ_g, G) = 0 \,$ for $ i > \dim\,{\mathcal Z}(G) \,$, where $ {\mathcal Z}(G) $ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme $ {\rm Rep}_G[π_1(Σ_g)]\,$ 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 $ {\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.

math.AT↗

arXiv2016

Perverse Sheaves and Knot Contact Homology

In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in $ {\mathbb R}^3 $, we define a differential graded (DG) $k$-category $ \tilde{\mathscr A} $ with finitely many objects, whose quasi-equivalence class is a topological invariant of $ L $. In the case when $L$ is a knot, the endomorphism algebra of a distinguished object of $ \tilde{\mathscr A} $ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng and Sullivan in [EENS13a]. The input of our construction is a natural action of the braid group $B_n$ on the category of perverse sheaves on a two-dimensional disk with singularities at $n$ marked points, studied by Gelfand, MacPherson and Vilonen in [GMV96]. As an application, we show that the category of finite-dimensional representations of the link $k$-category $ \tilde{A} = H_0(\tilde{\mathscr A}) $ defined as the $0$th homology of our DG category $ \tilde{\mathscr A} $ is equivalent to the category of perverse sheaves on $ {\mathbb R}^3 $ which are singular along the link $ L $. We also obtain several generalizations of the category $ \tilde{\mathscr A} $ by extending the Gelfand-MacPherson-Vilonen braid action.

math.AT↗

arXiv2016

Affine cubic surfaces and character varieties of knots

It is known that the fundamental group homomorphism $π_1(T^2) \to π_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the $\mathrm{SL}_2$-character varieties of the corresponding fundamental groups. In our earlier work [BS16], we proposed a conjecture that the classical restriction map admits a canonical 2-parameter deformation into a smooth cubic surface. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [BH95] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel-Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel-Hilden conjecture holds is closed under taking connect sums and certain knot coverings.

math.GT↗

arXiv2016

Dual Hodge decompositions and derived Poisson brackets

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. 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.

math.QA↗

arXiv2015

Chern-Simons forms and higher character maps of Lie representations

This paper is a sequel to our earlier work [BFPRW], where we study the derived representation scheme DRep_{g}(A) parametrizing the representations of a Lie algebra A in a finite-dimensional reductive Lie algebra g. In [BFPRW], we defined two canonical maps Tr_{g}(A): HC^{(r)}(A) \to \H[\DRep_{g}(A)]^G and Φ_{g}(A): H[\DRep_{g}(A)]^G \to H[\DRep_{h}(A)]^W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons classes of a canonical g-torsor associated to the pair (A, g). Our construction is inspired by (and, in a sense, dual to) the classical construction of `additive regulator maps' due to Beilinson and Feigin. As a consequence, we show that, if A is an abelian Lie algebra, the composite map Phi_{g}(A) Tr_{g}(A) is represented by a canonical differential operator acting on differential forms on Sym(A) and depending only on the Cartan data (h, W, P), where P is a W-invariant polynomial on h. We derive a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [BFPRW].

math.RT↗

arXiv2015

Representation Homology, Lie Algebra Cohomology and Derived Harish-Chandra Homomorphism

We study the derived representation scheme DRep_n(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep_n(A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gl_n^*(C) defined over a Koszul dual coalgebra of A. We extend this isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRep_g(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic to the Chevalley-Eilenberg homology of the Lie coalgebra g^*(C), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φ_g(a) : DRep_g(a)^G -> DRep_h(a)^W, which is a homological extension of the classical restriction homomorphism. We call Φ_g(a) a derived Harish-Chandra homomorphism. We conjecture that, for a two-dimensional abelian Lie algebra a, the derived Harish-Chandra homomorphism is a quasi-isomorphism, and provide some evidence for this conjecture. For any complex Lie algebra g, we compute the Euler characteristic of DRep_g(a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep_h(a)^W.This yields an interesting combinatorial identity, which we prove for gl_n and sl_n (for all n). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proved by S.Fishel, I.Grojnowski and C.Teleman. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

math.RT↗

arXiv2014

Double affine Hecke algebras and generalized Jones polynomials

In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct 3-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev in \cite{RT90}. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

math.QA↗

arXiv2014

Stable representation homology and Koszul duality

This paper is a sequel to [BKR], where we studied the derived affine scheme DRep_n(A) of the classical representation scheme Rep_n(A) for an associative k-algebra A. In [BKR], we have constructed canonical trace maps Tr_n(A): HC(A) -> H[DRep_n(A)]^GL extending the usual characters of representations to higher cyclic homology. This raises a question whether a well known theorem of Procesi [P] holds in the derived setting: namely, is the algebra homomorphism Sym[Tr_n(A)]: Sym[HC(A)] -> H[DRep_n(A)]^GL defined by Tr_n(A) surjective ? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense DG subalgebra DRep_\infty(A)^Tr of the topological DG algebra DRep_\infty(A)^{GL_\infty}. It turns out that on passing to the inverse limit (as n -> \infty), the family of maps Sym[Tr_n(A)] "stabilizes" to an isomorphism Sym[\bar{HC}(A)] = H[DRep_\infty(A)^Tr]. The derived version of Procesi's theorem does therefore hold in the limit. However, for a fixed (finite) n, there exist homological obstructions to the surjectivity of Sym[Tr_n(A)], and we show on simple examples that these obstructions do not vanish in general. We compare our result with the classical theorem of Loday-Quillen and Tsygan on stable homology of matrix Lie algebras. We show that the relative Chevalley-Eilenberg complex C(gl_\infty(A), gl_\infty(k); k) equipped with the natural coalgebra structure is Koszul dual to the DG algebra DRep_\infty(A)^Tr. We also extend our main results to bigraded DG algebras, in which case we show that DRep_{\infty}(A)^Tr = DRep_{\infty}(A)^GL_{\infty}. As an application, we compute the (bigraded) Euler characteristics of DRep_\infty(A)^GL_{\infty} and \bar{HC}(A) and derive some interesting combinatorial identities.

math.KT↗