Canonical bases of modified quantum algebras for type A2

2018 ◽  
Vol 17 (06) ◽  
pp. 1850113
Author(s):  
Weideng Cui
Keyword(s):  
Canonical Basis ◽  
Quantum Algebra ◽  
Explicit Description ◽  
Quantum Algebras ◽  
Canonical Bases ◽  
Type Formula

The modified quantum algebra [Formula: see text] associated to a quantum algebra [Formula: see text] was introduced by Lusztig. [Formula: see text] has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of [Formula: see text] for type [Formula: see text].

Quantum algebra embeddings: deforming functionals and algebraic approach

1994 ◽  
Vol 72 (7-8) ◽  
pp. 519-526 ◽  
Author(s):  
J. Van der Jeugt
Keyword(s):  
Hypergeometric Function ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Quantum Algebra ◽  
Physical Models ◽  
Quantum Algebras ◽  
Basic Hypergeometric Function

The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the q-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g., [Formula: see text] and [Formula: see text]. For the last example, we give the q-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function [Formula: see text].

APPLICATIONS OF COMPUTER ALGEBRA IN QUANTUM GROUPS

1994 ◽  
Vol 05 (04) ◽  
pp. 701-706
Author(s):  
W.-H. STEEB
Keyword(s):  
Quantum Groups ◽  
Computer Algebra ◽  
Kronecker Product ◽  
Quantum Algebra ◽  
Theoretical Physics ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Helpful Tool

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

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 ON CANONICAL BASES AND INDUCTION OF -GRAPHS

2018 ◽  
Vol 239 ◽  
pp. 1-41
Author(s):  
JOHANNES HAHN
Keyword(s):  
Coxeter Groups ◽  
Canonical Basis ◽  
Hecke Algebras ◽  
Free Module ◽  
Base Change ◽  
Manuscripta Math ◽  
Laurent Polynomials ◽  
Graph Algebras ◽  
Canonical Bases ◽  
Change Matrix

A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed “standard basis” through a triangular base change matrix with polynomial entries whose constant terms equal the identity matrix. Among the better known examples of canonical bases are the Kazhdan–Lusztig basis of Iwahori–Hecke algebras (see Kazhdan and Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184), Lusztig’s canonical basis of quantum groups (see Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3(2) (1990), 447–498) and the Howlett–Yin basis of induced $W$-graph modules (see Howlett and Yin, Inducing W-graphs I, Math. Z. 244(2) (2003), 415–431; Inducing W-graphs II, Manuscripta Math. 115(4) (2004), 495–511). This paper has two major theoretical goals: first to show that having bases is superfluous in the sense that canonicalization can be generalized to nonfree modules. This construction is functorial in the appropriate sense. The second goal is to show that Howlett–Yin induction of $W$-graphs is well-behaved a functor between module categories of $W$-graph algebras that satisfies various properties one hopes for when a functor is called “induction,” for example transitivity and a Mackey theorem.

SYMMETRY, INTEGRABILITY AND DEFORMATIONS OF LONG-RANGE INTERACTING HAMILTONIANS

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2903-2908 ◽  
Author(s):  
ANGEL BALLESTEROS
Keyword(s):  
Long Range ◽  
Hamiltonian Systems ◽  
Quantum Algebra ◽  
Complete Integrability ◽  
Quantum Algebras ◽  
R Matrix ◽  
Completely Integrable ◽  
Integrable Deformation ◽  
Coalgebra Structure ◽  
Integrable Deformations

The notion of coalgebra symmetry in Hamiltonian systems is analysed. It is shown how the complete integrability of some long-range interacting Hamiltonians can be extracted from their associated coalgebra structure with no use of a quantum R-matrix. Within this framework, integrable deformations can be considered as direct consequences of the introduction of coalgebra deformations (quantum algebras). As an example, the Gaudin magnet is derived from a sl(2) coalgebra, and a completely integrable deformation of this Hamiltonian is obtained through a twisted gl(2) quantum algebra.

scholarly journals The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

2018 ◽  
Vol 2019 (20) ◽  
pp. 6179-6215 ◽  
Author(s):  
Jie Du ◽  
Qiang Fu
Keyword(s):  
Canonical Basis ◽  
Alternative Algebra ◽  
Integral Form ◽  
Loop Algebra ◽  
Algebra Structure ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Weyl Duality ◽  
Integral Quantum

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

scholarly journals Quantization of canonical bases and the quantum symplectic double

2021 ◽  
Author(s):  
Dylan G. L. Allegretti
Keyword(s):  
Canonical Basis ◽  
Canonical Bases ◽  
Regular Functions

AbstractWe describe a natural q-deformation of Fock and Goncharov’s canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type A. We then describe an extension of this construction involving a cluster variety called the symplectic double.

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