Related papers: On bases of quantized enveloping algebras
We compare two known methods of extending a complex, unital, commutative normed algebra so as to include solutions to sets of monic polynomials over the original algebra. (One of these is a generalisation of a construction from the thesis…
In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian…
In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, have been studied at roots of unity. In this context, the quantum Heisenberg enveloping…
We construct canonical bases in tensor products of several lowest and highest weight integrable modules, generalizing Lusztig's work.
We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.
This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.
Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half…
Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra…
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the…
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…
In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.
We give a construction of a basis of the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated with a Dynkin quiver in terms of parity sheaves.
In this article, we provide an explicit description of the Lipschitz saturation $A^*_{B,R}$ of a subalgebra $A\subseteq R[[t^{\gamma_1},\ldots, t^{\gamma_n}]]$ over a ring $R$ in terms of the numerical semigroup $\Gamma =…
We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a…
The present paper mainly considers the representation type of the enveloping algebra of monomial algebra. Let $A$ be a monomial algebra and $A^e= A\otimes_{\mathrm{l}\!\mathrm{k}} A^{\mathrm{op}}$ its enveloping algebra. It is shown that…
Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of…
We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig's conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type…
Combinatorial and topological aspects of monoids with an absorbing element and their associated algebras are considered. Phd thesis.
We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz.…
We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the…