Given an affine algebraic variety V and a quantization A of its coordinate ring, it is conjectured that the primitive ideal space of A can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological quotient maps are given in the case of quantum toric varieties.
We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.
This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are bre…
We investigate a class of algebras that provides multiparameter versions of both quantum symplectic space and quantum Euclidean $2n$-space. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a class of algebras introduced by Oh. We describe the structure of the prime and primitive ideals of these algebras. Other structural results include normal separation and catenarity.
We prove analogues of some of Kostant's theorems for infinitesimal Cherednik algebras of gl_n. As a consequence, it follows that in positive characteristic the Azumaya and smooth loci of the center of these algebras coincide.
This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure o…
Explicit generating sets are found for all primitive ideals in the generic quantized coordinate rings of the 3x3 special and general linear groups over an arbitrary algebraically closed field. (Previously, generators were only known up to certain localizations.) The generating sets form polynormal regular sequences, from which it follows that all primitive factor algebras of these quantized coordinate rings are Auslander-Gorenstein and Cohen-Macaulay.
We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.
We investigate the algebra $F_q(N)$ introduced by Faddeev, Reshetikhin and Takhadjian. In case $q$ is a primitive root of unity the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini-Kac-Procesi Conjecture is proved for this case. In the case of $q$ generic, the primitive ideals are described. A related algebra studied by Oh is also treated.
Let $A$ denote the commutative polynomial ring in $n$ variables, over an algebraically closed field $k$, and let $R$ denote the standard multiparameter quantization of $A$ determined by a multiplicatively antisymmetric $n\times n$ matrix $(q_{ij})$. In this paper we prove, when -1 cannot be multiplicatively generated by the $q_{ij}$, that the primitive spectrum of $R$ is a topological quotient of $k^n$. Under the same hypothesis, we further prove that the prime spectrum of $R$ is a topological quotient of the prime spectrum of $A$.
Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the defining relations for $U_q(\mathfrak{g})$ can be slightly modified in such a way that the resulting algebra $U_q(\mathfrak{g};\mathcal{S})$ allows a homomorphism onto (an extension of) the algebra $\mathrm{Pol}(\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q})$ of functions on the quantum flag manifold $\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q}$ corresponding to $\mathcal…
Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ vari…
Let g be a complex simple Lie algebra of type B_2 and q be a non-zero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of the positive part U^+(g) of the enveloping algebra of g, whose Gelfand-Kirillov dimension is equal to 2, are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U_q^+(g) of the quantized enveloping algebra of g and then we study the simple factor algebras of U_…
We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric grou…
A class of Poisson algebras considered as a Poisson version of the multiparameter quantized coordinate rings of symplectic and Euclidean $2n$-spaces is constructed and the prime Poisson ideals and the symplectic ideals of these Poisson algebras are described. As a result, it is shown that the multiparameter quantized symplectic and Euclidean $2n$-spaces are topological quotients of their classical spaces.
Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson prime and Poisson primitive spectra of R onto the prime and primitive spectra of A. The…
Universal measuring coalgebras provide an enrichment of the category of algebras over the category of coalgebras. By considering the special case of the tensor algebra on a vector space V, the category of linear spaces itself becomes enriched over coalgebras, and the universal measuring coalgebra is the dual coalgebra of the tensor algebra T(V tensor V*). Given a braiding R on V the universal measuring coalgebra P_R(V) which preserves the grading is naturally dual to the Fadeev-Takhtadjhan-Reshitikin bialgebra A(R) and therefore provides a representation of the quantized universal enveloping a…
We study the algebra $U_ζ$ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra $\frak {g=sl}_2$ modulo the two-sided ideal generated by $K^l-1$. We show that $U_ζ$ is a smash product of the quantum deformation of the restricted universal enveloping algebra $\bold u_ζ$ of $\frak g$ and the ordinary universal enveloping algebra $U$ of $\frak g$, and we compute the primitive (= prime) ideals of $\Uz$. Next we describe a decomposition of $\bold u_ζ$ into the simple $U$- submodules, which leads to an explicit formula for the center and the indecomposa…
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.
We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund tra…
Joseph and Hodges-Levasseur (in the A case) described the spectra of all quantum function algebras R_q[G] on simple algebraic groups in terms of the centers of certain localizations of quotients of R_q[G] by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. We determine the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it we deduce a more explicit description of all prime ideal…
Let $\Oq(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\Oq(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the $\Oq(B)$-equivariant $\Oq(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.
De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E_6 and E_6^{(1)} map isomorphically onto distinguished subalgebras of the Faddeev-Reshetikhin-Takhtajan universal bialgebra associated to the braiding on the quantum half-spin representation of Uq(so_{10}). We identify the quotients as those obtained by facto…
A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra $R$ on which an algebraic torus $H$ acts rationally, by Poisson automorphisms, such that $R$ has only finitely many prime Poisson $H$-stable ideals. In this setting, an additional characterization of the Poisson primitive ideals of $R$ is obtained -- they are precisely the prime Poisson ideals maximal in their $H$-strata (where two prime Poisson ideals are in the same $H$-stratum if the intersections of their $H$-orbits coincide). Further, the Zariski topology on the space of Poisson primitive ide…
This paper investigates the Poisson geometry of cluster algebras and the corresponding ideal theory of quantum cluster algebras. We then show how our approach can be applied to the ring theory of quantized coordinate rings. We give a new construction for the Dixmier map constructed by Yakimov from the space of symplectic leaves on $\CC[G]$ to the space of primitive ideals on $\CC_q[G]$ and give further evidence that this map is a homeomorphism.
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of D_q with moment parameter λ. We show that A^λ_d(Q) is a flat formal deformation of Lusztig's quiver varieties, and their multiplicative count…
Poisson brackets (P.b) are the natural initial terms for the deformation quantization of commutative algebras. There is an open problem whether any Poisson bracket on the polynomial algebra of $n$ variables can be quantized. It is known (Poincare-Birkhoff-Witt theorem) that any linear P.b. for all $n$ can be quantized. On the other hand, it is easy to show that in case $n=2$ any P.b. is quantizable as well. Quadratic P.b. appear as the unitial terms for the quantization of polynomial algebras as quadratic algebras. The problem of the quantization of quadratic P.b. is also open. In the paper we…
We construct homomorphic images of $su(n,n)^{\mathbb C}$ in Weyl Algebras ${\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\mathcal H}_{2nr}$, $su(n,n)^{\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their l…
