On Quantum Flag Algebras

Examples of quantum cluster algebras associated to partial flag varieties

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiß, Leclerc and Schröer, whose work generalizes that...

The Gelfand-kirillov Dimensions of Algebras Arising from Representation Theory

This note is to study a variety of graded algebras that arise from the induced representations for semisimple algebraic groups and quantum groups. These algebras will play an important role in a study of the cohomology groups of line bundles over the flag varieties. This short note concentrates on the calculation of the Gelfand-Kirillov dimensions of these algebras.

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Generalized W∞ Higher-Spin Algebras and Symbolic Calculus on Flag Manifolds

We study a new class of infinite-dimensional Lie algebras W∞(N+, N−) generalizing the standard W∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a grouptheoretic framework. Here we interpret W∞(N+, N−) either as an infinite continuation of the pseudo-unitary symmetry U(N+, N−), or as a “higher-U(N+, N−)-spin extension” of the diffeomorphism algebra diff(N+, N−) of the N = N++N− tor...

Double-partition Quantum Cluster Algebras

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but sub-classes have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines L. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of...

A Mackey Decomposition Theorem and Cohomology for Quantum Groups at Roots of 1

Introduction This is a continuation of the investigation in [L3] into the module structure of those modules for quantum enveloping algebras U analogous to the coho-mology groups of line bundles on the flag varieties for semisimple algebraic groups. In [L3], we used the admissible categories for Hopf algebras and their induction theory and proved a modified Borel-Bott-Weil theorem for R l(w) Ind