scholarly journals Defining relations of quantum symmetric pair coideal subalgebras

2021 ◽  
Vol 9 ◽  
Author(s):  
Stefan Kolb ◽  
Milen Yakimov
Keyword(s):  
Chebyshev Polynomials ◽  
Hermite Polynomials ◽  
Symmetric Pair ◽  
Star Products ◽  
Graded Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
New Family

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$ -graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

scholarly journals Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

2020 ◽  
pp. 2140016
Author(s):  
W. Riley Casper ◽  
Stefan Kolb ◽  
Milen Yakimov
Keyword(s):  
Recurrence Relations ◽  
Star Product ◽  
Hermite Polynomials ◽  
Star Products ◽  
Defining Relations ◽  
Algebraic Properties ◽  
Product Method ◽  
Usual Quantum ◽  
Multivariate Orthogonal Polynomials ◽  
Nichols Algebras

We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

scholarly journals On a Class of Humbert-Hermite Polynomials

2019 ◽  
Author(s):  
M. A Pathan ◽  
Waseem Khan
Keyword(s):  
Chebyshev Polynomials ◽  
Hermite Polynomials ◽  
Ultraspherical Polynomials

A unified presentation of a class of Humbert’s polynomials in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinnsy, Horadam-Pethe, Djordjević , Gould, Milovanović and Djordjević, Pathan and Khan polynomials and many not so called ’named’ polynomials has inspired the present paper and the authors define here generalized Humbert-Hermite polynomials of two variables. Several expansions of Humbert- Hermite polynomials, Hermite-Gegenbaurer (or ultraspherical) polynomials and Hermite- Chebyshev polynomials are proved.

A note on two-variable Chebyshev polynomials

2017 ◽  
Vol 24 (3) ◽  
pp. 339-349 ◽  
Author(s):  
Clemente Cesarano ◽  
Claudio Fornaro
Keyword(s):  
Chebyshev Polynomials ◽  
Hermite Polynomials ◽  
Integral Representations

AbstractIn this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.

scholarly journals PERTURBATIVE SYMMETRIES ON NONCOMMUTATIVE SPACES

2004 ◽  
Vol 19 (32) ◽  
pp. 5693-5706 ◽  
Author(s):  
CHRISTIAN BLOHMANN
Keyword(s):  
Field Theory ◽  
Lie Algebras ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Plane ◽  
Star Products ◽  
Quantum Field ◽  
Enveloping Algebras ◽  
Noncommutative Spaces ◽  
Noncommutative Quantum

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semisimple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. The result can be used to construct invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.

scholarly journals Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

2015 ◽  
Vol 26 (08) ◽  
pp. 1550053 ◽  
Author(s):  
Christopher Sadowski
Keyword(s):  
Lie Algebras ◽  
Operator Algebras ◽  
Intertwining Operators ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
Standard Formula ◽  
Exact Sequences

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

scholarly journals Filtrations on block subalgebras of reduced enveloping algebras

2021 ◽  
pp. 2350001
Author(s):  
Andrei Ionov ◽  
Dylan Pentland
Keyword(s):  
Positive Characteristic ◽  
Adjoint Representation ◽  
Universal Enveloping Algebra ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Nilpotent Cone ◽  
Algebra Of Functions ◽  
Pbw Filtration

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincaré-Birkhoff-Witt (PBW) filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra [Formula: see text] of the restricted universal enveloping algebra [Formula: see text] and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighborhood of [Formula: see text] in the nilpotent cone and the coinvariants algebra corresponding to [Formula: see text]. In the case of [Formula: see text] in characteristic [Formula: see text] we determine the associated graded algebras of these filtrations on block subalgebras of [Formula: see text]. We also apply this to determine the structure of the adjoint representation of [Formula: see text].

scholarly journals Multi-Dimensional Chebyshev Polynomials: A Non-Conventional Approach

2019 ◽  
Vol 10 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Clemente Cesarano
Keyword(s):  
Chebyshev Polynomials ◽  
Polynomial Interpolation ◽  
Hermite Polynomials ◽  
Integral Transforms ◽  
Multi Index ◽  
Special Cases ◽  
The Laplace Transform ◽  
Translation Operators ◽  
Sturm Liouville ◽  
Symbolic Approach

Abstract Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

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