The algebras of Kleinian type are finite dimensional semisimple rational algebras $A$ such that the group of units of an order in $A$ is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings $RG$, with $R$ an order in a number field and $G$ a finite group, such that $RG^*$ is virtually a direct product of free-by-free groups.
Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger.
The notion of a Harish-Chandra bimodule, i.e. finitely generated $U(\mathfrak{g})$-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [Ivan Losev, Dimensions of irreducible modules over W-algebras and Goldie ranks. arXiv:1209.1083]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, i.e. the algebra itself, for a deformation of Kleinian singularity of type $A$ is unitarizable. We obtain a partial classification of unitarizable regular bimodules.
For a certain class of (nonunital) subalgebras of deformed preprojective algebra of affine type we describe their centers as certain deformation of Kleinian singularity and find their PI-degree. These results can be applied to algebras generated by elements with given spectrum and scalar sum.
We introduce a new class of algebras over discrete valuation rings, called Kleinian 4-rings, which generalize the group algebra of the Kleinian 4-group. For these algebras we describe the lattices and their cohomologies. In the case of regular lattices we obtain an explicit form of cocycles defining the cohomology classes.
We describe the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces are related to regular subsets of the corresponding complexified Cartan algebras.
Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $Γ\subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}} = eH_{\mathbf{k}} e$ the spherical subalgebra of $H_{\mathbf{k}}$. We show that for suitable ${\mathbf{k}}$ there is a filtered $\Z^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\mathrm{qgr} \simeq U_{\bf k}$-mod, (2) there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$. Here $ \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$ is the category of coherent sheaves on the $Γ$-Hilbert scheme. and for a gr…
In this paper we prove an analogue of a recent result of Gordon and Stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the n-th symmetric power of C^2 with the geometry of the Hilbert scheme of n points in C^2 through the formalism of Z-algebras. Our work produces, for every regular noncommutative deformation O^λof a Kleinian singularity X=C^2/Γ, as defined by Crawley-Boevey and Holland, a filtered Z-algebra which is Morita equivalent to O^λ, such that the associated graded Z-algebra is Morita equivalent to the minimal resolution of…
For any $(m,n)$-periodic higher spin six-vertex configuration $\mathscr{L}$, we construct a one-parameter family $Δ_ξ$ of pseudo-unitarizable representations of the corresponding noncommutative fiber product $\mathcal{A}(\mathscr{L})$ by difference operators acting on the space of sections of a complex line bundle $L_ξ$ over the face lattice $F$. The indefinite inner product is given explicitly in terms of a combinatorial sign function defined on $F$. We prove that each simple integral weight $\mathcal{A}(\mathscr{L})$-module (previously classified by the author, see arXiv:1612.08125) occurs a…
For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/Γ$ as a Nakajima quiver variety for the framed McKay quiver of $Γ$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, a…
For a finite subgroup $Γ\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/Γ]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $Γ$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/Γ$; we present a…
We relate completions of infinitesimal Hecke algebras of sl2 to noncommutative deformations of Kleinian singularities of type D of Crawley-Boevey and Holland. As a consequence, we show an analogue of the inequality of Bernstein and simplicity of generic maximal primitive quotients of these algebras. We also establish Skryabin type equivalence for these algebras.
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type $A$ the former is obtained by Frenkel-Kac construction from the latter.
The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.
The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.
We define Poisson structures on certain transversal slices to conjugacy classes in complex simple algebraic groups introduced in arXiv:0809.0205. These slices are associated to the elements of the Weyl group, and the Poisson structures on them are analogous to the Poisson structures introduced by J. de Boer, T. Tjin and A. Premet in papers arXiv:hep-th/9211109 and http://www.maths.man.ac.uk/DeptWeb/Homepages/aap/Reprints/Transverse.ps on the Slodowy slices in complex simple Lie algebras. The quantum deformations of these Poisson structures are known as W-algebras of finite type. As an applicat…
Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $Δ_R(A)$ of $A$ has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient category. If $R$ has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that $D_{sg}(R)$ determines $Δ_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander algebra. …
We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two Kleinian singularities of type $A$. Special examples are closely related to Lie-Heisenberg algebras, the affine Lie algebra $A_1^{(1)}$, and a finite W-algebra associated to $\mathfrak{sl}_4$. The algebras depend on two scalars and two polynomials that must satisfy the Mazorchuk-Turowska Equation (MTE), which we re-interpret as a quantization of the ice rule (lo…
The centralizer algebra of the action of the unitary group on the real tensor powers of its natural module, is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for U(n), and with the decomposition of the above tensor powers into irreducible modules, is considered.
A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak{J}$.
This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori-Matsumoto involution, this is equivalent also to the classification of the spherical unitary dual for type E. This work completes the classification of the Whittaker Iwahori-spherical unitary dual, or equivalently, the spherical unitary dual, of split linear algebraic p-adic groups.
In this paper the algebra of invariants for the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We prove that the algebra of invariants is finitely generated.
In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this construction, we obtain new formulas, alternative to Weingarten calculus, for the integral of a polynomial over the unitary group with respect to the Haar measure. In particular, we obtain a closed formula for the law of moments of the first k rows of the unitary group.
We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.
