Milen Yakimov

General methods for constructing bispectral operators

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.

Invariant prime ideals in quantizations of nilpotent Lie algebras

De Concini, Kac and Procesi defined a family of subalgebras U^w_+ of a quantized universal enveloping algebra U_q(g), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras U(n_+ \cap Ad_w(n_-)) where n_\pm are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U^w_+ as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U^w_+ and show that the corresponding poset is isomorphic to W^{\leq w}, where H is the group of group-like elements of U_q(g). Moreover, for each H-prime of U^w_+ we construct a generating set in terms of Demazure modules related to fundamental representations.

Bispectral Algebras of Commuting Ordinary Differential Operators

Abstract:We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N. It combines and unifies the ideas of Duistermaat–Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.

Poisson structures on affine spaces and flag varieties. II

The standard Poisson structures on the flag varieties G/P of a complex reductive algebraic group G are investigated. It is shown that the orbits of symplectic leaves in G/P under a fixed maximal torus of G are smooth irreducible locally closed subvarieties of G/P, isomorphic to intersections of dual Schubert cells in the full flag variety G/B of G, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson-Dirac submanifolds. For a parabolic subgroup P with abelian unipotent radical (in which case G/P is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor L of P on G/P are complete Poisson subvarieties which are quotients of L, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on G/P vanishes at all special base points for the L-orbits on G/P constructed by Richardson, Rohrle, and Steinberg.

On the spectra of quantum groups

Introduction Previous results on spectra of quantum function algebras A description of the centers of Josephs localizations Primitive ideals of R q [G] and a Dixmier map for R q [G] Separation of variables for the algebras S i?½ w A classification of the normal and prime elements of the De Concini-Kac-Procesi algebras Module structure of R w over their subalgebras generated by Josephs normal elements A classification of maximal ideals of R q [G] and a question of Goodearl and Zhang Chain properties and homological applications Bibliography

Quantum cluster algebra structures on quantum nilpotent algebras

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein–Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts. The proofs rely on Chatters’ notion of noncommutative unique factorization domains. Toric frames are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand–Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the homogeneous prime elements in chains of noncommutative UFDs and their alterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and categorification of cluster algebras. It is expected that this combinatorial model of sequences of homogeneous prime elements will have applications to the unified categorification of quantum nilpotent algebras. Received by the editor 23 December 2013. 2010 Mathematics Subject Classification. Primary 16T20; Secondary 13F60, 17B37, 14M15.

Quantum cluster algebras and quantum nilpotent algebras

Significance Cluster algebras are used to study in a unified fashion phenomena from many areas of mathematics. In this paper, we present a new approach to cluster algebras based on noncommutative ring theory. It deals with large, axiomatically defined classes of algebras and does not require initial combinatorial data. Because of this, it has a broad range of applications to open problems on constructing cluster algebra structures on coordinate rings and their quantum counterparts. A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras.

The Launois–Lenagan conjecture

Abstract In this note we prove the Launois–Lenagan conjecture on the classification of the automorphism groups of the algebras of quantum matrices R q [ M n ] of square shape for all positive integers n , base fields K , and deformation parameters q ∈ K ⁎ which are not roots of unity.

A Proof of the Goodearl–Lenagan Polynormality Conjecture

The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi and Lusztig, are large classes of iterated skew polynomial rings with rich ring theoretic structure. In this paper, we prove in an explicit way that all torus invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special case of the algebras of quantum matrices, this construction yields explicit polynormal generating sets consisting of quantum minors for all of their torus invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan polynormality conjecture. Furthermore we prove that Spec U^w_-(g) is normally separated for all simple Lie algebras g and Weyl group elements w, and deduce from it that all algebras U^w_-(g) are catenary.

A classification of H-primes of quantum partial flag varieties

We classify the invariant prime ideals of a quantum partial flag variety under the action of the related maximal torus. As a result we construct a bijection between them and the torus orbits of symplectic leaves of the standard Poisson structure on the corresponding flag variety. It was previously shown by K. Goodearl and the author that the latter are precisely the Lusztig strata of the partial flag variety.