scholarly journals Canonical bases and higher representation theory

2014 ◽  
Vol 151 (1) ◽  
pp. 121-166 ◽  
Author(s):  
Ben Webster
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Finite Type ◽  
Canonical Basis ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Natural Categories ◽  
Grothendieck Groups ◽  
Quantized Universal Enveloping Algebra

AbstractThis paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

scholarly journals Canonical bases arising from quantum symmetric pairs of Kac–Moody type

2021 ◽  
Vol 157 (7) ◽  
pp. 1507-1537
Author(s):  
Huanchen Bao ◽  
Weiqiang Wang
Keyword(s):  
Quantum Group ◽  
Finite Type ◽  
Canonical Basis ◽  
Integral Form ◽  
New Approach ◽  
Canonical Bases ◽  
Highest Weight ◽  
Technical Assumption ◽  
Rank One ◽  
Divided Powers

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

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 An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

2003 ◽  
Vol 6 ◽  
pp. 105-118 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Canonical Basis ◽  
Irreducible Module ◽  
Semisimple Lie Algebra ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

scholarly journals Highest-weight vectors in tensor products of Verma modules for U_q(sl_2)

2018 ◽  
Vol 36 (4) ◽  
pp. 107-119 ◽  
Author(s):  
Elizabeth Creath ◽  
Dijana Jakelic
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Alternative Proof ◽  
Verma Modules ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Structure Constants ◽  
Quantized Universal Enveloping Algebra

We obtain an explicit basis for the subspace spanned by highest-weight vectors in a tensor product of two highest-weight modules for the quantized universal enveloping algebra of sl_2. The structure constants provide a generalization of the Clebsh-Gordan coefficients. As a byproduct, we give an alternative proof for the decomposition of these tensor products as direct sums of indecomposable modules and supply generators for all highest weight summands.

scholarly journals Homomorphisms between standard modules over finite-type KLR algebras

2017 ◽  
Vol 153 (3) ◽  
pp. 621-646 ◽  
Author(s):  
Alexander S. Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
Positive Part ◽  
Enveloping Algebra ◽  
Klr Algebras ◽  
Quantized Enveloping Algebra ◽  
Standard Module

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

scholarly journals On Generalized Minors and Quiver Representations

2018 ◽  
Vol 2020 (3) ◽  
pp. 914-956 ◽  
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Finite Type ◽  
Dynkin Diagram ◽  
Cluster Algebra ◽  
Coordinate Ring ◽  
Coxeter Element ◽  
Quiver Representations ◽  
Group Representations ◽  
Weight Group ◽  
Highest Weight ◽  
Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

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 Factorization of the canonical bases for higher-level Fock spaces

2011 ◽  
Vol 55 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Susumu Ariki ◽  
Nicolas Jacon ◽  
Cédric Lecouvey
Keyword(s):  
Fock Space ◽  
Fock Spaces ◽  
Transition Matrices ◽  
Canonical Bases ◽  
Highest Weight ◽  
Highest Weight Modules ◽  
Graphic Module ◽  
Module Structures ◽  
Weight Modules

AbstractThe level l Fock space admits canonical bases $\mathcal{G}_{e}$ and $\smash{\mathcal{G}_{\infty}}$. They correspond to $\smash{\mathcal{U}_{v}(\widehat{\mathfrak{sl}}_{e})}$ and $\mathcal{U}_{v}({\mathfrak{sl}}_{\infty})$-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

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