An action of the free product Z2⋆Z2⋆Z2 on the q-Onsager algebra and its current algebra
TL;DR: In this paper, the Lusztig automorphisms of the q-Onsager algebra O q (sl ˆ 2 ) were used to obtain an action of the free product Z 2 ⋆ Z 2 (Z 2 ) on O q as a group of auto/antiauto-morphisms.
About: This article is published in Nuclear Physics.The article was published on 2018-08-29 and is currently open access. It has received 13 citations till now. The article focuses on the topics: Current algebra & Automorphism.
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TL;DR: In this paper, the alternating PBW basis for Uq+ is introduced, which is related to the alternating q-shuffle algebra associated with affine sl2, and is used for the first time in the positive part of Uq(sl^2).
Abstract: The positive part Uq+ of Uq(sl^2) has a presentation with two generators A, B that satisfy the cubic q-Serre relations. We introduce a PBW basis for Uq+, said to be alternating. Each element of this PBW basis commutes with exactly one of A, B, qAB − q−1BA. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a q-shuffle algebra associated with affine sl2. We show how the alternating PBW basis is related to the PBW basis for Uq+ found by Damiani in 1993.The positive part Uq+ of Uq(sl^2) has a presentation with two generators A, B that satisfy the cubic q-Serre relations. We introduce a PBW basis for Uq+, said to be alternating. Each element of this PBW basis commutes with exactly one of A, B, qAB − q−1BA. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a q-shuffle algebra associated with affine sl2. We show how the alternating PBW basis is related to the PBW basis for Uq+ found by Damiani in 1993.
18 citations
Cites background from "An action of the free product Z2⋆Z2..."
...The papers [2], [13], [14], [15], [16], [18] might be helpful in this direction....
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TL;DR: In this article, the positive part U q + of the quantum group U q ( sl ˆ 2 ) has a presentation with two generators A, B that satisfy the cubic q-Serre relations.
16 citations
TL;DR: In this paper, the alternating PBW basis for the positive part of $U + q (widehat{\mathfrak{sl}}_2) has been introduced, where the elements of each type mutually commute.
Abstract: The positive part $U^+_q$ of $U_q(\widehat{\mathfrak{sl}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U^+_q$, said to be alternating. Each element of this PBW basis commutes with exactly one of $A$, $B$, $qAB-q^{-1}BA$. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a $q$-shuffle algebra associated with affine $\mathfrak{sl}_2$. We show how the alternating PBW basis is related to the PBW basis for $U^+_q$ found by Damiani in 1993.
12 citations
TL;DR: In this article, Scrimshaw et al. gave a proof of the analog conjecture for the Onsager algebra O q and a proof for a homomorphic image of O q called the universal Askey-Wilson algebra.
10 citations
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TL;DR: In this paper, the alternating generators of the positive part of the quantum group $U_q(\widehat{\mathfrak{sl}}_2) have been used to obtain a central extension of the algebra $U^+_q.
Abstract: This paper is about the positive part $U^+_q$ of the quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. Recently we introduced a type of element in $U^+_q$, said to be alternating. Each alternating element commutes with exactly one of $A$, $B$, $qBA-q^{-1}AB$, $qAB-q^{-1}BA$; this gives four types of alternating elements. There are infinitely many alternating elements of each type, and these mutually commute. In the present paper we use the alternating elements to obtain a central extension $\mathcal U^+_q$ of $U^+_q$. We define $\mathcal U^+_q$ by generators and relations. These generators, said to be alternating, are in bijection with the alternating elements of $U^+_q$. We display a surjective algebra homomorphism $\mathcal U^+_q \to U^+_q$ that sends each alternating generator of $\mathcal U^+_q$ to the corresponding alternating element in $U^+_q$. We adjust this homomorphism to obtain an algebra isomorphism $\mathcal U_q^+ \to U^+_q \otimes \mathbb F \lbrack z_1, z_2,\ldots\rbrack$ where $\mathbb F$ is the ground field and $\lbrace z_n\rbrace_{n=1}^\infty$ are mutually commuting indeterminates. We show that the alternating generators form a PBW basis for $\mathcal U_q^+$. We discuss how $\mathcal U^+_q$ is related to the work of Baseilhac, Koizumi, Shigechi concerning the $q$-Onsager algebra and integrable lattice models.
7 citations
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TL;DR: In this article, the eigenwert problem involved in the corresponding computation for a long strip crystal of finite width, joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum.
Abstract: The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$
5,081 citations
"An action of the free product Z2⋆Z2..." refers methods in this paper
...The algebra Oq is a q-deformation of the Onsager algebra from mathematical physics [17], [21, Remark 9....
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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 article, the concept of a Leonard system was introduced, and it was shown that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,ϱ,ϱ* taken from K such that both
335 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
"An action of the free product Z2⋆Z2..." refers background in this paper
...The algebra Oq appears in the theory of quantum groups, as a coideal subalgebra of Uq(ŝl2) [3, 11, 16]....
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TL;DR: In this article, an ordered pair of linear transformations (i.e., a Leonard pair on a field and a vector space over a field with finite positive dimension) is considered.
Abstract: Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal
(ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal
We call such a pair a Leonard pair on $V$ We give an overview of the theory of Leonard pairs
212 citations