arXiv subjects

Yuri Berest

Publications and source records attributed to Yuri Berest.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2014

Dixmier Groups and Borel Subgroups

Let G be the group of symplectic (unimodular) automorphisms of the free associative algebra on two generators. A theorem of G.Wilson and the first author asserts that G acts transitively the Calogero-Moser spaces C_n for all n. We generalize this theorem in two ways: first, we prove that the action of G on C_n is doubly transitive, meaning that G acts transitively on the configuration space of (ordered) pairs of points in C_n; second, we prove that the diagonal action of G on the product of (any number of) copies of C_n is transitive provided the corresponding n's are pairwise distinct. In the second part of the paper, we study the isotropy subgroups G_n of G in C_n. We equip each G_n with the structure of an ind-algebraic group and classify the Borel subgroups of these ind-algebraic groups for all n. Our classification shows that every Borel subgroup of G (= G_0) is conjugate to the subgroup B of triangular (elementary) automorphisms; on the other hand, for n > 0, the conjugacy classes of Borel subgroups of G_n are parametrized by certain orbits of B in C_n. Our main result is that the conjugacy classes of non-abelian Borel subgroups of G_n correspond precisely to the B-orbits of the C^*-fixed points in C_n and thus, are in bijection with the partitions of n. We also prove an infinite-dimensional analogue of the classical theorem of R.Steinberg, characterizing the (non-abelian) Borel subgroups of G_n in purely group-theoretic terms. Together with our classification this last theorem implies that the G_n are pairwise non-isomorphic as abstract groups. Our study of the groups G_n is motivated by the fact that these are the automorphism groups of non-isomorphic simple algebras Morita equivalent to the Weyl algebra A_1(C). From this perspective, our results generalize well-known theorems of J.Dixmier and L.Makar-Limanov about the automorphism group of A_1(C).

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arXiv2013

Derived Representation Schemes and Cyclic Homology

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: HC_n(A) \to H_n[DRep_V(A)], extending the canonical trace Tr_V(A): HC_0(A) \to k[Rep_V(A)] to the higher cyclic homology of the algebra A, and describe a related derived version of the representation functor introduced recently by M. Van den Bergh. We study various operations on the homology of DRep_V(A) induced by known operations on cyclic and Hochschild homology of A.

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arXiv2013

Derived Representation Schemes and Noncommutative Geometry

Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be thought of as a substitute or "approximation" for Spec(A). The idea is that every property or noncommutative geometric structure on A should induce a corresponding geometric property or structure on $Rep_n(A)$ for all n. In recent years, many interesting structures in noncommutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (e.g., A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, for some n, the scheme $Rep_n(A)$ fails to have the corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor $Rep_n$ is not "exact" and should be replaced by its derived functor $DRep_n$ (in the sense of non-abelian homological algebra). The higher homology of $DRep_n(A)$, which we call representation homology, obstructs $Rep_n(A)$ from having the desired property and thus measures the failure of the Kontsevich-Rosenberg "approximation." In this paper, which is mostly a survey, we prove several results confirming this intuition. We also give a number of examples and explicit computations illustrating the theory developed in [BKR] and [BR].

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arXiv2012

Dunkl Operators and Quasi-invariants of Complex Reflection Groups

These are lecture notes of a minicourse given by the first author at the Summer School on Quantization at the University of Notre Dame in June 2011. The notes were written up and expanded by the second author who took the liberty of adding a few interesting results and proofs from the literature. In a broad sense, our goal is to give an introduction to representation theory of rational Cherednik algebras and some of its recent applications. More specifically, we focus on the two concepts featuring in the title (Dunkl operators and quasi-invariants) and explain the relation between them. The course was originally designed for graduate students and nonexperts in representation theory. In these notes, we tried to preserve an informal style, even at the expense of making imprecise claims and sacrificing rigor.

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arXiv2012

Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras

Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard (commutative) Poisson structures on all representation spaces $ \Rep_V(A) $. It turns out that such a weak Poisson structure on $A$ is a Lie algebra bracket on the 0-th cyclic homology $ \HC_0(A) $ satisfying some extra conditions; it was thus called in an {\it $ H_0$-Poisson structure}. This paper studies a higher homological extension of this construction. In our more general setting, we show that noncommutative Poisson structures in the above sense behave nicely with respect to homotopy (in the sense that homotopy equivalent NC Poisson structures on $A$ induce (via the derived representation functor) homotopy equivalent Poisson algebra structures on the derved representation schemes $\DRep_V(A) $). For an ordinary algebra $A$, a noncommutative Poisson structure on a semifree (more generally, cofibrant) resolution of $A$ yields a graded (super) Lie algebra structure on the full cyclic homology $ \HC_\bullet(A) $ extending Crawley-Boevey's $\H_0$-Poisson structure on $ \HC_0(A) $. We call such structures {\it derived Poisson structures} on $A$. We also show that derived Poisson structures do arise in nature: the cobar construction $Ω(C)$ of an $(-n)$-cyclic coassociative DG coalgebra (in particular, of the linear dual of a finite dimensional $n$-cyclic DG algebra) $C$ carries a $(2-n)$-double Poisson bracket in the sense of Van den Bergh. This in turn induces a corresponding noncommutative $(2-n)$-Poisson structure on $Ω(C)$. When (the semifree) DG algebra $Ω(C)$ resolves an honest algebra $A$, $A$ acquires a derived $(2-n)$-Poisson structure.

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arXiv2010

Trees, Amalgams and Calogero-Moser Spaces

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra $ \c < x, y > $ on the Calogero-Moser varieties $ \CC_n $ defined in \cite{BW}. Our results generalize well-known theorems of Dixmier and Makar-Limanov on automorphisms of $ A_1 $, answering an old question of Stafford (see \cite{St}). Finally, we propose a natural extension of the Dixmier Conjecture for $ A_1 $ to the class of Morita equivalent algebras.

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arXiv2010

A Simple Construction of Derived Representation Schemes

We present a simple algebraic construction of the (non-abelian) derived functors DRep_n(A) of the representation scheme Rep_n(A), parametrizing the n-dimensional representations of an associative algebra A. We construct a related derived version of the representation functor introduced recently by M. Van den Bergh and, as an application, compute the derived tangent spaces TDRep_n(A) to Rep_n(A). We prove that our construction of DRep_n(A) agrees with an earlier construction of derived action spaces, due to I. Ciocan-Fontanine and M. Kapranov; however, our approach, proofs and motivation are quite different. This paper is mainly a research announcement; detailed proofs and applications will appear elsewhere.

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arXiv2010

The Picard Group of a Noncommutative Algebraic Torus

We compute the Picard group $ Pic(A_q) $ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $ R(A_q) $ of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to $ A_q $. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective $A_q$-modules to irreducible representations of the double affine Hecke algebras (DAHA) $ H_{t, q^{-1/2}}(S_n) $ at $ t = 1 $. We show that, under this correspondence, the action of $ Pic(A_q) $ on $ R(A_q) $ agrees with the action of $ SL_2(Z) $ on $ H_{t, q^{-1/2}}(S_n) $ constructed by I.Cherednik. We compare our results with smooth and analytic cases. In particular, when $ |q| \not= 1 $, we find that $ Pic(A_q) $ is isomorphic to the group of auto-equivalences $ Auteq(D^b(X))/Z $ of the bounded derived category of coherent sheaves on the elliptic curve $ X = C*/Z $ modulo translations.

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arXiv2010

Ideals of Rings of Differential Operators on Algebraic Curves (With an Appendix by George Wilson)

Let X be a complex smooth affine irreducible curve, and let D = D(X) be the ring of global differential operators on X. In this paper, we give a geometric classification of left ideals in $ D $ and study the natural action of the Picard group of D on the space J(D) of isomorphism classes of such ideals. We recall that, up to isomorphism in the Grothendieck group K_0(D), the ideals of D are classified by the Picard group of X: there is a natural fibration γ: J(D) \to Pic(X), whose fibres are the stable isomorphism classes of ideals of D (see \cite{BW}). In this paper, we refine this classification by describing the fibres of γin terms of finite-dimensional algebraic varieties C_n(X, I), which we call the (generalized) Calogero-Moser spaces. We define these varieties as representation varieties of deformed preprojective algebras over a certain extension of the ring of regular functions on $ X $. As in the classical case (see \cite{Wi}), we prove that C_n(X, I) are smooth affine irreducible varieties of dimension 2n. Our results generalize the description of left ideals of the first Weyl algebra A_1(C) in \cite{BW1, BW2}; however, our methods are quite different.

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arXiv2010

Quasi-Invariants of Complex Reflection Groups

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the coordinate ring of some (singular) affine variety X_k. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) to this setting: in particular, we show that the variety X_k and the module Q_k are Cohen-Macaulay, and the rings of differential operators on X_k and Q_k are simple rings, Morita equivalent to the Weyl algebra A_n(C), where n = dim X_k . Our approach relies on representation theory of complex Cherednik algebras and is parallel to that of [BEG]. As a by-product, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

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arXiv2010

Cherednik algebras and differential operators on quasi-invariants

We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. We further study an algebra Q of quasi-invariant polynomials on h introduced by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset C[h]. We prove that the algebra D(Q) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra, by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective (Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).

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arXiv2008

Differential operators on an affine curve: ideal classes and Picard groups

Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration γ: R \to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of γinto Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define γin the framework of the Grassmannian description of R due to Cannings and Holland.

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arXiv2008

Calogero-Moser Spaces over Algebraic Curves

In these notes, we give a survey of the main results of [BC] and [BW]. Our aim is to generalize the geometric classification of (one-sided) ideals of the first Weyl algebra $ A_1(C) $ (see [BW1, BW2]) to the ring $ D(X) $ of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of D(X) up to stable isomorphism, in terms of the Picard group of X; then, we refine this classification by describing each stable isomorphism class as a disjoint union of (certain quotients of) generalized Calogero-Moser spaces C_n(X, I). The latter are defined as representation varieties of deformed preprojective algebras over a one-point extension of the ring of regular functions on X by the line bundle I. As in the classical case, C_n(X, I) turn out to be smooth irreducible varieties of dimension 2n.

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arXiv2008

Heat Kernel Coefficients for Two-Dimensional Schrodinger Operators

In this note, we compute the Hadamard coefficients of (algebraically) integrable Schrodinger operators in two dimensions. These operators first appeared in [BL] and [B] in connection with Huygens' principle, and our result completes, in a sense, the investigation initiated in those papers.

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arXiv2007

$A_{\infty}$-modules and Calogero-Moser Spaces

We re-examine the bijective correspondence between the set of isomorphism classes of ideals of the first Weyl algebra and associated quiver varieties (Calogero-Moser spaces) \cite{BW1, BW2}. We give a new explicit construction of this correspondence based on the notion of $\A$-envelope of a rank one torsion-free $A_1$-module. Though perhaps less geometric than other methods, our approach is much simpler and seems more natural from the point of view of deformation theory.

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arXiv2007

Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ Π_λ(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on $\A$-modules over $ A_1 $ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $ \C^2/Γ$, where $ Γ$ is a finite cyclic subgroup of $ \SL(2, \C) $.

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arXiv2006

Mad Subalgebras of Rings of Differential Operators on Curves

We study the maximal abelian ad-nilpotent (mad) subalgebras of the domains D Morita equivalent to the first Weyl algebra. We give a complete description both of the individual mad subalgebras and of the space of all such. A surprising consequence is that this last space is independent of D. Our results generalize some classic theorems of Dixmier about the Weyl algebra.

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