scholarly journals Factorizable Module Algebras

2018 ◽  
Vol 2019 (21) ◽  
pp. 6711-6764
Author(s):  
Arkady Berenstein ◽  
Karl Schmidt
Keyword(s):  
Large Class ◽  
Reductive Groups ◽  
Parabolic Subgroups ◽  
Module Algebra ◽  
Lie Bialgebra ◽  
Affine Spaces ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Module Algebras ◽  
Basic Affine

Abstract The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G, their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_{q}(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_{q}(\mathfrak{g}^{\ast })$ of the dual Lie bialgebra $\mathfrak{g}^{\ast }$ of $\mathfrak{g}$.

scholarly journals Universal K-matrix for quantum symmetric pairs

2019 ◽  
Vol 2019 (747) ◽  
pp. 299-353 ◽  
Author(s):  
Martina Balagović ◽  
Stefan Kolb
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Symmetric Pair ◽  
Moody Algebra ◽  
R Matrix ◽  
Reflection Equation ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
K Matrix

Abstract Let {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras {{B_{\mathbf{c},\mathbf{s}}}} of {{U_{q}(\mathfrak{g})}} have a universal K-matrix if {{\mathfrak{g}}} is of finite type. By a universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of {{U_{q}(\mathfrak{g})}} which commutes with {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable {{U_{q}(\mathfrak{g})}} -modules in category {{\mathcal{O}}} . The construction of the universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for {{U_{q}(\mathfrak{g})}} . Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

scholarly journals Canonical bases and standard monomials

1998 ◽  
Vol 41 (3) ◽  
pp. 611-623
Author(s):  
R. J. Marsh
Keyword(s):  
Lie Algebra ◽  
Canonical Basis ◽  
Monomial Basis ◽  
Fundamental Weight ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Standard Monomials ◽  
Standard Monomial ◽  
Quantized Enveloping Algebra

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

scholarly journals Differential operators on quantized flag manifolds at roots of unity, II

2014 ◽  
Vol 214 ◽  
pp. 1-52
Author(s):  
Toshiyuki Tanisaki
Keyword(s):  
Differential Operators ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Roots Of Unity ◽  
Central Character ◽  
Derived Equivalence ◽  
Bernstein Type ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Category Of Modules

AbstractWe formulate a Beilinson-Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twistedD-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350124
Author(s):  
YONG WANG ◽  
GUANGQUAN GUO
Keyword(s):  
Type Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Morita Context ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Maschke Type Theorem ◽  
Module Algebras ◽  
Product Algebra ◽  
Separable Extensions

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

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