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The compact presentation for the alternating central extension of the $q$-Onsager algebra
TL;DR: The compact presentation of the alternating central extension of the O_q-Onsager algebra as discussed by the authors was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q.
Abstract: The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal O_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$.
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TL;DR: This article showed that the alternating central extension of the O_q-Onsager algebra is related to the algebra $\mathcal A_q$ in the same way that $U^+_q+q+Q$ is related with alternating central extensions of the enveloping algebra U_q.
Abstract: The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $\mathcal A_q$ for $O_q$. Soon afterwards, Baseilhac and Shigechi gave a presentation of $\mathcal A_q$ by generators and relations. We show that these generators give a PBW basis for $\mathcal A_q$. Using this PBW basis, we show that the algebra $\mathcal A_q$ is isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F$ is the ground field and $\lbrace z_n \rbrace_{n=1}^\infty $ are mutually commuting indeterminates. Recall the positive part $U^+_q$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. Our results show that $O_q$ is related to $\mathcal A_q$ in the same way that $U^+_q$ is related to the alternating central extension of $U^+_q$. For this reason, we propose to call $\mathcal A_q$ the alternating central extension of $O_q$.
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TL;DR: In this paper, it was shown that the standard tensor product factorization of the alternating generators of the O_q-Onsager algebra is an algebra isomorphism of algebras.
Abstract: The $q$-Onsager algebra $O_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. The alternating central extension $\mathcal O_q$ has a presentation involving the alternating generators $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$ and a large number of relations. Let $\langle \mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal O_q$ generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebra isomorphism $O_q \to \langle \mathcal W_0, \mathcal W_1 \rangle$ that sends $W_0\mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. It is known that the center $\mathcal Z$ of $\mathcal O_q$ is isomorphic to a polynomial algebra in countably many variables.
It is known that the multiplication map $\langle \mathcal W_0, \mathcal W_1 \rangle \otimes \mathcal Z \to \mathcal O_q$, $ w \otimes z \mapsto wz$ is an isomorphism of algebras. We call this isomorphism the standard tensor product factorization of $\mathcal O_q$. In the study of $\mathcal O_q$ there are two natural points of view: we can start with the alternating generators, or we can start with the standard tensor product factorization. It is not obvious how these two points of view are related. The goal of the paper is to describe this relationship. We give seven main results; the principal one is an attractive factorization of the generating function for some algebraically independent elements that generate $\mathcal Z$.
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TL;DR: In this article, the authors introduce a non-commutative, associative, semi-simple C-algebra T e T(x) whose structure reflects the combinatorial structure of Y.
Abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple \Bbb {C}-algebra T e T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y.
In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter.
We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”.
We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur.
We close with some conjectures and open problems.
428 citations
TL;DR: In this paper, a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras is developed.
245 citations
01 Jan 1999
TL;DR: In this paper, the authors consider a pair of linear transformations A : V → V and A :V → V satisfying the following four conditions: (i) A and V are both diagonalizable on V, (ii) A is both diagonalisable on V and V is both diagonalizable on A, and (iii) there is no subspace W of V such that both AW ⊆ W, AW⊆ AW, AW ∆ W, other than W = 0 and W = V.
Abstract: Inspired by the theory of P -and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V satisfying the following four conditions. (i) A and A are both diagonalizable on V . (ii) There exists an ordering V0, V1, . . . , Vd of the eigenspaces of A such that AVi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d), where V−1 = 0, Vd+1 = 0. (iii) There exists an ordering V ∗ 0 , V ∗ 1 , . . . , V ∗ δ of the eigenspaces of A ∗ such that AV ∗ i ⊆ V ∗ i−1 + V ∗ i + V ∗ i+1 (0 ≤ i ≤ δ), where V ∗ −1 = 0, V ∗ δ+1 = 0. (iv) There is no subspace W of V such that both AW ⊆ W , AW ⊆ W , other than W = 0 and W = V . We call such a pair a TD pair. Referring to the above TD pair, we show d = δ. We show that for 0 ≤ i ≤ d, the eigenspaces Vi and V ∗ i have the same dimension. Denoting this common dimension by ρi, we show the sequence ρ0, ρ1, . . . , ρd is symmetric and unimodal, i.e. ρi−1 ≤ ρi for 1 ≤ i ≤ d/2 and ρi = ρd−i for 0 ≤ i ≤ d. We show that there exists a sequence of scalars β, γ, γ, , taken from F such that both 0 = [A,AA − βAAA+AA − γ(AA +AA)− A], 0 = [A, AA− βAAA +AA − γ(AA+AA)− A], where [r, s] means rs − sr. The sequence is unique if d ≥ 3. Let θi (resp. θ ∗ i ) denote the eigenvalue of A (resp. A) associated with Vi (resp. V ∗ i ), for 0 ≤ i ≤ d. We show the expressions θi−2 − θi+1 θi−1 − θi , θ i−2 − θ ∗ i+1 θ i−1 − θ ∗ i both equal β + 1, for 2 ≤ i ≤ d − 1. We hope these results will ultimately lead to a complete classification of the TD pairs.
195 citations
01 Apr 2001
TL;DR: The Tridiagonal algebra as discussed by the authors is an algebra on two generators which is defined as follows: a field is a field, and a sequence of scalars taken from a field can be represented by two symbols A and A. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by A. In the first part of this paper, we survey what is known about irreducible finite di-mensional T-modules.
Abstract: We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let β, γ, γ � , , � denote a sequence of scalars taken from K. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by two symbols A, Asubject to the relations (A, A 2 A � − βAAA + AA 2 − γ(AA � + AA) − A � ) = 0, (A � , A �2 A − βAAA � + AA �2 − γ � (AA + AA � ) − � A) = 0, In the first part of this paper, we survey what is known about irreducible finite di- mensional T-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible T-module based on the Askey-Wilson polynomials. This module is on the vector space K(x) consisting of all polynomials in an indeterminant x that have coefficients inK. Denoting by A the linear transformation on K(x) which is multiplication by x, and denoting by Aan
187 citations
TL;DR: In this paper, the transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q-Onsager algebra.
Abstract: The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2N. The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al. In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2N. Numerical checks performed for small values of N support the analysis.
169 citations