arXiv subjects

Pavel Etingof

Publications and source records attributed to Pavel Etingof.

At least 163 records · Page 9Linked to original sources

arXiv2007

Noncommutative complete intersections and matrix integrals

We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep_n(A), of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of each nonvanishing cyclic and Hochschild homology group of an RCI algebra. The proof involves a noncommutative cyclic Koszul complex, K(A), and a matrix integral similar to the one arising in quiver gauge theory.

math.AG↗

arXiv2006

New deformations of group algebras of Coxeter groups, II

In our previous paper math.QA/0409261, we defined a deformation of the group algebra of the group of even elements of a Coxeter group W, and showed that it is flat for all values of parameters if and only if all the rank 3 parabolic subgroups of W are infinite. In this paper, we study what happens in the general case. Then the deformation is flat only for some values of parameters, and the set of all such values is called the flatness locus. The main result of the paper is an explicit description of the this flatness locus as a scheme over Z. More specifically, we show that this scheme is the intersection of the flatness loci for the subalgebras corresponding to parabolic subgroups of rank 3. The latter are determined by solving the rigid multiplicative Deligne-Simpson problem. We also define additive analogs of our algebras and study their properties.

math.QA↗

arXiv2006

Monodromy of trigonometric KZ equations

The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo quantum groups. This result was generalized by the second author to simple Lie superalgebras of type A-G. In this paper, we generalize the Drinfeld-Kohno theorem to the case of the trigonometric Knizhnik-Zamolodchikov equations for simple Lie superalgebras of type A-G. The equations contain a classical r-matrix on the Lie superalgebra, and the answer expresses through the quantum R-matrix of the corresponding quantum group. The proof is based on the quantization theory for Lie bialgebras developed by the first author and D. Kazhdan.

math.QA↗

arXiv2006

An algebraic extension of the MacMahon Master Theorem

We present a new algebraic extension of the classical MacMahon Master Theorem. The basis of our extension is the Koszul duality for non-quadratic algebras defined by Berger. Combinatorial implications are also discussed.

math.CO↗

arXiv2006

Noncommutative Geometry and Quiver algebras

We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T^*B, which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, P=P(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space P/[P,P] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where P is the preprojective algebra associated with a quiver of non-Dynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of P in terms of the Lie algebra P/[P,P].

math.AG↗

arXiv2006

On central extensions of preprojective algebras

We determine the structure of the center and the trace space of the centrally extended preprojective algebra of an ADE quiver, introduced by the first and the third authors in math/0503393. It turns out that this structure has a mysterious relationship to the structure of the maximal nilpotent subalgebra of the corresponding simple Lie algebra.

math.RT↗

arXiv2006

Generalized double affine Hecke algebras of rank 1 and quantized Del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z_l\ltimes Z^2, where l is 2,3,4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D4 is the Cherednik algebra of type C^\check C_1, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.

math.QA↗

arXiv2006

Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups

We introduce a central extension of the preprojective algebra of a finite Dynkin quiver (depending on a regular weight for the corresponding root system), whose natural deformed version is flat (unlike that for the preprojective algebra). We calculate the Hilbert polynomial of the central extension, and show that it is a Frobenius algebra. As a corollary, we obtain the Hilbert series of the usual deformed preprojective algebra in which the deformation parameters are variables, and show that this algebra is Gorenstein (although it is not a flat module over the ring of parameters). The proofs are based on the fact that our central extension for the weight ρis the image of the quantum Heisenberg algebra in the fusion category of representations of quantum SL(2) under a tensor functor into bimodules over a semisimple algebra. Finally, we explain how our algebras are connected to cyclotomic Hecke algebras of complex reflection groups of rank 2, and in particular show that the dimension of the latter for generic parameters is equal to the order of the group, as conjectured by Broue, Malle, and Rouquier.

math.RT↗

arXiv2006

Generalized double affine Hecke algebras of higher rank

We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA is the algebra defined by Sahi in q-alg/9710032, which is a generalization of the Cherednik algebra of type BCn. We prove the formal PBW theorem for GDAHA, and parametrize its irreducible representations in the case when D is affine (i.e. extended D4, E6, E7, E8) and q=1. We formulate a series of conjectures regarding algebraic properties of GDAHA. We expect that, similarly to how GDAHA of rank 1 provide quantizations of del Pezzo surfaces (as shown in math.QA/0406480), GDAHA of higher rank provide quantizations of deformations of Hilbert schemes of these surfaces. The proofs are based on the study of the rational version of GDAHA (which is closely related to the algebras studied in math.QA/0401038), and differential equations of Knizhnik-Zamolodchikov type.

math.QA↗

arXiv2006

Quantization, orbifold cohomology, and Cherednik algebras

We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes A^{\otimes n}$ in terms of the Hochschild homology of the associative algebra $A$ (over $\Bbb C$). It allows us to compute the Hochschild (co)homology of $\Bbb C[W]\ltimes A^{\otimes n}$ where $A$ is the $q$-Weyl algebra or any its degeneration and $W$ is the Weyl group of type $A_{n-1}$ or $B_n$. For a deformation quantization $A_+$ of an affine symplectic variety $X$ we show that the Hochschild homology of $S^n A$, $A=A_+[\hbar^{-1}]$ is additively isomorphic to the Chen-Ruan orbifold cohomology of $S^nX$ with coefficients in $\Bbb C((\hbar))$. We prove that for $X$ satisfying $H^1(X,\Bbb C)=0$ (or $A\in VB(d)$) the deformation of $S^nX$ ($\Bbb C[S_n]\ltimes A^{\otimes n}$) which does not come from deformations of $X$ ($A$) exists if and only if $\dim X=2$ ($d=2$). In particular if $A$ is $q$-Weyl algebra (its trigonometric or rational degeneration) then the corresponding nontrivial deformations yield the double affine Hecke algebras of type $A_{n-1}$ (its trigonometric or rational versions) introduced by Cherednik.

math.QA↗

arXiv2005

Koszulity and the Hilbert series of preprojective algebras

The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra of Q over any field k is Koszul, and has Hilbert series 1/(1-Ct+t^2), where C is the adjacency matrix of the double of Q. (This result, in somewhat less general formulations, was previously obtained by Martinez-Villa and Malkin-Ostrik-Vybornov). We also prove a similar result for the partial preprojective algebra of any connected quiver Q, associated to a subset J of the set I of vertices of Q (by definition, this is the quotient of the path algebra of the double by the preprojective algebra relations imposed only at vertices not contained in J). Namely, we show that if J is not empty then this algebra is Koszul, and its Hilbert series is 1/(1-Ct+D_Jt^2), where D_J is the diagonal matrix with (D_J)_{ii}=0 if i is in J and (D_J)_{ii}=1 otherwise. Finally, we show that both results are valid in a slightly more general framework of modified preprojective algebras.

math.RA↗

arXiv2005

Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

math.RT↗

arXiv2005

Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential

We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's conjecture connecting certain "instanton counting" partition function for an arbitrary simple group G with the prepotential of the Toda integrable system associated with the affine Lie algebra whose affine Dynkin diagram is dual to that of the affinization of the Lie algebra of G (for G=SL(n) this conjecture was proved earlier in the works of Nekrasov-Okounkov and Nakajima-Yoshioka). Our proof is totally different and it is based on the results of the paper math.AG/0401409 by the first author.

math.AG↗

arXiv2005

Continuous Hecke algebras

We study a generalization of graded Hecke algebras introduced by Drinfeld in 1986, in which the role of a finite group G is played by a reductive algebraic group. This includes continuous generalizations of symplectic reflection algebras and of rational Cherednik algebras. We prove direct and converse PBW theorems, and also study versions of our algebras in which the group G is replaced with its Lie algebra. We also begin to develop the representation theory of continuous Hecke algebras, which unifies representation theories of real reductive groups, Drinfeld-Lusztig degenerate affine Hecke algebras, and symplectic reflection (in particular, Cherednik) algebras. The paper is dedicated to V. Drinfeld's 50-th birthday.

math.QA↗

arXiv2005

On a Generalized Two-Fluid Hele-Shaw Flow

We propose a method of construction of exact solutions of free boundary problems corresponding to Hele-Shaw flows in presence of an external field. Such a field may arise, in particular, due to electrokinetic phenomena. Both a general approach and a number of examples are presented.

math-ph↗

arXiv2004

Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology

We prove the additive version of the conjecture proposed by Ginzburg and Kaledin. This conjecture states that if X/G is an orbifold modeled on a quotient of a smooth affine symplectic variety X (over C) by a finite group G\subset Aut(X) and A is a G-stable quantum algebra of functions on X then the graded vector space HH(A^G) of the Hochschild cohomology of the algebra A^G of invariants is isomorphic to the graded vector space H_{CR}(X/G)((h)) of the Chen-Ruan (stringy) cohomology of the orbifold X/G.

math.QA↗

arXiv2004

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with grA in RG(p), where grA is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_\pm(p), A(q), or H(32), constructed in math.QA/0310253, math.QA/0402159 and math.QA/0403096. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.

math.QA↗

arXiv2004

On rack cohomology

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given by Carter, Jelsovsky, Kamada, and Saito, are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced by Carter, Elhamdadi, and Saito. We also give a group-theoretical interpretation of the second cohomology group for racks.

math.QA↗