Primitive Quotients Research Articles

The noncommutative geometry of matrix polynomial algebras

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension.

Derivations of a family of quantum second Weyl algebras

In view of a well-known theorem of Dixmier, its is natural to consider primitive quotients of Uq+(g) as quantum analogues of Weyl algebras. In this work, we study primitive quotients of Uq+(G2) and compute their Lie algebra of derivations. In particular, we show that, in some cases, all derivations are inner showing that the corresponding primitive quotients of Uq+(G2) should be considered as deformations of Weyl algebras.

Fixed rings of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, such as Weyl algebras, primitive quotients of U(sl2), and multiparameter quantized Weyl algebras. In this work, we study invariants of TGWAs under diagonal automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply our results particularly to k-finitistic TGWAs of type (A1)n and A2, and study properties of their fixed rings, including the noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring. As an auxiliary result, in order to study invariants of tensor products of TGWAs, we prove that the class of regular, μ-consistent TGWAs is closed under tensor products.

Simple [formula omitted]-graded domains of Gelfand–Kirillov dimension two

Let k be an algebraically closed field and A a Z-graded finitely generated simple k-algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z-graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of U(sl2).

Classification of twisted generalized Weyl algebras over polynomial rings

Let R be a polynomial ring in m variables over a field of characteristic zero. We classify all rank n twisted generalized Weyl algebras over R, up to Zn-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of U(g) where g=gln, sln, or sp2n, algebras related to U(slˆ2) and a finite W-algebra associated to sl4. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case n=2, which can be solved using methods from previous work by the authors [17], [15]. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation.

Word problems in Elliott monoids

Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is partial, but for every AF algebra B whose Murray–von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid E(B). Let M1 be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid E(M1) yield a natural word problem for every AF algebra B with singly generated E(B), because B is automatically a quotient of M1. Given two formulas ϕ and ψ in the language of involutive monoids, the problem asks to decide whether ϕ and ψ code the same equivalence of projections of B. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of M1 is solvable in polynomial time, and so is the word problem of the Behncke–Leptin algebras An,k, and of the Effros–Shen algebras Fθ, for θ∈[0,1]∖Q a real algebraic number, or θ=1/e. We construct a quotient of M1 having a Gödel incomplete word problem, and show that no primitive quotient of M1 is Gödel incomplete.

Brauer relations for finite groups in the ring of semisimplified modular representations

Let G be a finite group and p be a prime. We study the kernel of the map, between the Burnside ring of G and the Grothendieck ring of Fp[G]-modules, taking a G-set to its associated permutation module. We are able, for all finite groups, to classify the primitive quotient of the kernel; that is for each G, the kernel modulo elements coming from the kernel for proper subquotients of G. We are able to identify exactly which groups have non-trivial primitive quotient and we give generators for the primitive quotient in the soluble case.

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

To each multiquiver $\Gamma$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras $\mathcal{A}(\Gamma)$ carry a canonical representation by differential operators and that $\mathcal{A}(\Gamma)$ is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of $\Gamma$ for when this representation is faithful or locally surjective. By forgetting some of the structure of $\Gamma$ one obtains a Dynkin diagram, $D(\Gamma)$. We show that the generalized Cartan matrix of $\mathcal{A}(\Gamma)$ coincides with the one corresponding to $D(\Gamma)$ and that $\mathcal{A}(\Gamma)$ contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient $U/J$ of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if $J$ is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types $A$ and $C$ are closely related to $\mathcal{A}(\Gamma)$ for specific $\Gamma$. We also prove one result in the affine case.

Endomorphisms of quantum generalized Weyl algebras

We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra \({U_q({\mathfrak{sl}}_2)}\) and certain minimal primitive quotients of the positive part of \({U_q({\mathfrak{so}}_5)}\).

Endomorphisms of Quantum Generalized Weyl Algebras

We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.

GOLDIE RANK OF PRIMITIVE QUOTIENTS VIA LATTICE POINT ENUMERATION

AbstractLet k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

Antipodal distance-transitive covers with primitive quotient of diameter two

As a consequence of a famous theorem by Derek Smith, an unknown distance-transitive graph is either primitive of diameter at least two and valency at least three or is antipodal, bipartite, or both. In the imprimitive cases an unknown graph must have a primitive core of diameter at least two and valency at least three. It seems that the known list of primitive graphs is complete. Here, starting from an earlier work by Brouwer and Van Bon, we find every distance-transitive antipodal cover whose primitive quotient is one of the known distance-transitive graphs of diameter two and valency at least three.

Enveloping algebras of Slodowy slices and Goldie rank

Let \( U\left( {\mathfrak{g},e} \right) \) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra \( \mathfrak{g} = {\text{Lie}}(G) \) and let I be a primitive ideal of the enveloping algebra \( U\left( \mathfrak{g} \right) \) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that \( I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) \) for some finite dimensional irreducible \( U\left( {\mathfrak{g},e} \right) \)-module V, where Q e stands for the generalised Gelfand–Graev \( \mathfrak{g} \)-module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient \( {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} \) always divides dim V. For \( \mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n} \), we use a theorem of Joseph on Goldie fields of primitive quotients of \( U\left( \mathfrak{g} \right) \) to establish the equality \( {\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V \). We show that this equality continues to hold for \( \mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n} \) provided that the Goldie field of \( {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} \) is isomorphic to a Weyl skew-field and use this result to disprove Joseph’s version of the Gelfand–Kirillov conjecture formulated in the mid-1970s.

On Harmonic Elements for Semi-simple Lie Algebras

Let g be a semi-simple complex Lie algebra and g=n−⊕h⊕n its triangular decomposition. Let U(g), resp. Uq(g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math.116, 127–177). On the one hand, we give specialization results concerning harmonic elements, central elements of Uq(g), and the decomposition of Joseph and Letzter (cited above). For g=sln+1, we describe the specialization of quantum harmonic space in the N-filtered algebra U(sln+1) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys.35, 359–374). This enables us to study a Joseph–Letzter decomposition in the algebra U(sln+1). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C0 of integral points of a convex cone.

On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Let g be a semisimple or affine Lie algebra and U q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U q (g) module V relative to a parabolic subalgebra p⊂g is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication. For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A 4) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of ‘linear’ factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.

SUR LES INVARIANTS DE Bλ SOUS L'ACTION DE SOUS-GROUPES FINIS D'AUTOMORPHISMES: CONJECTURE DE GELFAND-KIRILLOV ET HOMOLOGIE DE HOCHSCHILD

Let B λ be a minimal primitive quotient of U(sl 2) and G a finite subgroup of automorphisms of B λ. We prove that the algebras of invariants BG λ, satisfy the Gelfand-Kirillov conjecture, that is Frac (BG λ) ≃ D 1(C) (where D 1(C) denotes the 1st Weyl field). Then we compute the 0th Hochschild homology group of these algebras.

A relation between Hochschild homology and cohomology for Gorenstein rings

Let “$HH$” stand for Hochschild (co)homology. In this note we show that for many rings $A$ there exists $d\in \mathbb {N}$ such that for an arbitrary $A$-bimodule $N$ we have $HH^i(N)=HH_{d-i}(N)$. Such a result may be viewed as an analog of Poincaré duality. Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo.

The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras

The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras

On the enright functor in the highest weight category

LetG be a complex semisimple Lie group,B its Borel subgroup andX a flag variety ofG. We define a functor on the category ofB-equivariantD X-modules that corresponds, under the global section functor, to the Enright functor on the highest weight category. We use this to lift Enright functor to the mixed version of the highest weight category. As an application we obtain that the socle and the cosocle filtration of a primitive quotient of the enveloping algebra coincide.

Unitary Representations of Lie Groups and Operators of Finite Rank

Besides being a principal object of study in harmonic analysis, the convolution algebra of Ll-functions on a locally compact group is traditionally used for investigating (unitary) group representations by methods of associative algebra. With the development of the theory of C*-algebras this role was more and more taken over by the group C*-algebras. A considerable amount of information concerning group C*-algebras, in particular of Lie groups, has accumulated. This article may be regarded as part of a program to use the obtained insights and methods in order to derive results in terms of more classical objects, as Ll-functions or, in the context of Lie groups, even smooth functions, briefly, to prove regularity theorems. To make this more concrete, let me describe two such regularity questions not to be treated in this paper. The topology on the unitary dual G of a given locally compact group G can be characterized as follows. A set A in G is closed if and only if for each ir E G -.. A there exists an f E C* (G) such that 7r(f) =, 0 and p(f) = 0 for all p E A. One may ask whether one can always find an f E L1 (G) with these properties. It turns out that for some groups this is possible, for instance for groups with polynomially growing Haar measure; for others it is not, for instance for noncompact semisimple Lie groups. More information on this topic can be found in [2]. Secondly, as Pukanszky has shown, [40], each primitive quotient C* (G)/P of the group C*-algebra of a connected Lie group G has a unique faithful trace trp. A priori it is not clear that there is an Ll-function f on G such that 0 < trp(f) < oo. But Charbonnel has proved [5], [6], that one can find such an f even in D(G) = Cc (G). In this article we shall solve the following problem in the affirmative. Let ir be a continuous irreducible unitary representation of a connected Lie group H, and suppose that ir(C*(H)) contains the compact operators on the representation space As; i.e., the norm closure of ir (L1 (H)) contains the compact operators. Is it true that ir (Li(H)) contains an operator of rank one? Actually, we shall do somewhat better. We shall construct a smooth function f on H such that ir(f) is an operator of rank one and such that f r