Symplectic Reflection Algebras Research Articles

Poisson Trace Orders

Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

Deformations of symplectic singularities and orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

Drinfeld Hecke algebras for symmetric groups in positive characteristic

We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, Lusztig’s graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting via its natural (reducible) reflection representation.

Degree-One Rational Cherednik Algebras for the Symmetric Group

Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of $\mathfrak{gl}_n$-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the $\mathfrak{sl}_n$-type rational Cherednik algebras $H_{0,c}$.

LD-stability for Goldie rings

The lower transcendence degree , introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras , the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD -stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras , finite W -algebras of type A , generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups . We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.

Ideals Generated by Traces or by Supertraces in the Symplectic Reflection Algebra H1,V(I2(2m + 1)) II

The algebra ℋ:=H1,v(I2(2m + 1)) of observables of the Calogero model based on the root system I2(2m + 1) has an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces. In the preceding paper we found all values of the parameter ν for which either the space of traces contains a degenerate nonzero trace trν or the space of supertraces contains a degenerate nonzero supertrace strν and, as a consequence, the algebra ℋ has two-sided ideals: one consisting of all vectors in the kernel of the form or another consisting of all vectors in the kernel of the form . We noticed that if , where z ∈ ℤ\\(2m + 1)ℤ, then there exist both a degenerate trace and a degenerate supertrace on ℋ. Here we prove that the ideals determined by these degenerate forms coincide.

Etingof’s conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing. We provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof’s conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac–Moody actions and wall-crossing functors.

Symplectic reflection algebras in positive characteristic as Ore extensions

We investigate PBW deformations of where G is the cyclic group of order p and the ground field k has characteristic p. The algebras are a version of symplectic reflection algebras that only exist in positive characteristic. They also admit a presentation as Ore extensions over and the combinatorics of the derivation used in this presentation is related to André and Eulerian polynomials. We find the center of and classify the simple modules. We also study a version of in which G is replaced by an elementary abelian p-group Gr.

Erratum to: Traces and Supertraces on Symplectic Reflection Algebras

Erratum to: Traces and Supertraces on Symplectic Reflection Algebras

Derived Equivalences for Symplectic Reflection Algebras

AbstractIn this paper we study derived equivalences for symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over these algebras and categories of coherent sheaves over quantizations of $\mathbb{Q}$-factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications to the generalized Bernstein inequality and to perversity of wall crossing functors.

Congenial algebras: Extensions and examples

We study the congeniality property of algebras, as defined by Bao, He, and Zhang, to establish a version of Auslander’s theorem for various families of filtered algebras. It is shown that the property is preserved under homomorphic images and tensor products under some mild conditions. Examples of congenial algebras in this paper include enveloping algebras of Lie superalgebras, iterated differential operator rings, quantized Weyl algebras, down-up algebras, and symplectic reflection algebras.

Barbasch-Sahi algebras and Dirac cohomology

We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over $\mathbb{C}$ with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic $0$ with an orthogonal module. Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called Vogan's conjecture for connected semisimple Lie groups, analogous results for graded affine Hecke algebras and for symplectic reflection algebras and Drinfeld Hecke algebras.

Traces and Supertraces on Symplectic Reflection Algebras

The symplectic reflection algebra H1,ν (G) has a T(G)-dimensional space of traces, and if it is regarded as a superalgebra with a natural parity, then it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and are independent of the parameter ν. We present values of T(G) and S(G) for the groups generated by the root systems and for the groups G = Γ ≀ SN, where Γ is a finite subgroup of Sp(2,ℂ).

Deformations of algebras defined by tilting bundles

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to C⁎-equivariant deformations over schemes with a good C⁎-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the C⁎-equivariant case.We apply our results to rational surface singularities, generalising the deformed preprojective algebras of Crawley-Boevey and Holland, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras.

Double extensions of Lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended.We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2.

The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H1,η(Γ≀SN)

Symplectic reflection algebra H1η(G) has a T(G)-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and do not depend on the parameter η.In this paper, the values T(G) and S(G) are explicitly calculated for the groups G = Γ≀SN, where Γ is a finite subgroup of Sp(2,ℂ).

Deformation Quantization and Superconformal Symmetry in Three Dimensions

We investigate the structure of certain protected operator algebras that arise in three-dimensional {mathcal{N}=4} superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly—they are spherical subalgebras of symplectic reflection algebras—but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general {mathcal{N}=4} SCFTs.

Bernstein inequality and holonomic modules

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

Ideals generated by traces or by supertraces in the symplectic reflection algebra H1,ν(I2(2m + 1))

For each complex number ν, an associative symplectic reflection algebra ℋ := H1,ν (I2(2m+1)), based on the group generated by root system I2(2m+1), has an m-dimensional space of traces and an (m+1)-dimensional space of supertraces. A (super)trace sp is said to be degenerate if the corresponding bilinear (super)symmetric form Bsp(x,y) = sp(xy) is degenerate. We find all values of the parameter ν for which either the space of traces contains a degenerate nonzero trace or the space of supertraces contains a degenerate nonzero supertrace and, as a consequence, the algebra ℋ has a two-sided ideal of null-vectors. The analogous results for the case \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${H_{1,{v_1},{v_2}}}({I_2}(2m))$$\\end{document} are also presented.

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory, due to A. Perucca.