Related papers: Canonical bases and quantum coordinate ring

We introduce the notion of integrable modules over $\imath$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is…

We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of…

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function…

In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the…

We construct canonical bases in tensor products of several lowest and highest weight integrable modules, generalizing Lusztig's work.

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis.…

The level l Fock space admits canonical bases G_e and G_\infty. They correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We establish that the transition matrices relating these two bases are unitriangular with…

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…

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal…

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We…

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…

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an…

The $\imath$quantum groups have two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. The isoclasses of perverse sheaves provide…

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q -> 1 each subalgebra specializes to the enveloping algebra U(k), where k is a fixed…

We identify the canonical basis of the quantum adjoint representation of a quantized enveloping algebra with a basis that we defined before the theory of canonical bases was available.

We construct a monomial basis of a quantum affine algebra of simply-laced type, associated to the PBW basis of Beck-Nakajima. We show that there exists a simple algorithm of computing canonical basis in terms of the monomial basis. We…

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $…

We realize the enveloping algebra of the positive part of a symmetrizable Kac-Moody algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.

We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable…

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as…