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# The alternating central extension of the $q$-Onsager algebra

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 $O_q$.

Topics: Current algebra (54%),
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Pascal Baseilhac1Institutions (1)
01 Jun 2021-Nuclear Physics
Abstract: An infinite dimensional algebra denoted A ¯ q that is isomorphic to a central extension of U q + - the positive part of U q ( s l 2 ˆ ) - has been recently proposed by Paul Terwilliger. It provides an ‘alternating’ Poincare-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of A ¯ q in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for A ¯ q are constructed; (b) explicit isomorphisms from A ¯ q to certain Drinfeld type ‘alternating’ subalgebras of U q ( g l 2 ˆ ) are obtained; (c) the image in U q + of all the generators of A ¯ q in terms of Damiani's root vectors is obtained. A new tensor product decomposition for U q ( s l 2 ˆ ) in terms of Drinfeld type ‘alternating’ subalgebras follows. The specialization q → 1 of A ¯ q is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra.

Topics: Tensor product (52%),

9 Citations

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Abstract: This paper concerns the positive part $U^+_q$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an algebra $\mathcal U^+_q$ called the alternating central extension of $U^+_q$. We presented $\mathcal U^+_q$ 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 $\mathcal U^+_q$ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal U^+_q$.

Topics: Quantum group (51%)

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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 $U_q(\widehat{\mathfrak{sl}}_2)$.

Topics: Current algebra (56%)

2 Citations

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Abstract: A unified framework for the Chevalley and equitable presentation of $U_q(sl_2)$ is introduced. It is given in terms of a system of Freidel-Maillet type equations satisfied by a pair of quantum K-operators ${\cal K}^\pm$, whose entries are expressed in terms of either Chevalley or equitable generators. The Hopf algebra structure is reconsidered in light of this presentation, and interwining relations for K-operators are obtained. A K-operator solving a spectral parameter dependent Freidel-Maillet equation is also considered. Specializations to $U_q(sl_2)$ admit a decomposition in terms of ${\cal K}^\pm$. Explicit examples of K-matrices are constructed.

Topics: Type (model theory) (56%), Hopf algebra (55%)

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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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Journal Article
Lars Onsager1Institutions (1)
01 Feb 1944-Physical Review
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 ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$

Topics: , Quaternion algebra (51%)

4,709 Citations

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George M. Bergman1Institutions (1)
Topics: Lemma (mathematics) (85%)

1,198 Citations

Open accessJournal Article
Paul Terwilliger1Institutions (1)
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.

Topics: Association scheme (58%), , Intersection number (53%) ... read more

399 Citations

Open accessBook
Roger W. Carter1Institutions (1)
01 Jan 2005-
Abstract: 1. Basic concepts 2. Representations of soluble and nilpotent Lie algebras 3. Cartan subalgebras 4. The Cartan decomposition 5. The root systems and the Weyl group 6. The Cartan matrix and the Dynkin diagram 7. The existence and uniqueness theorems 8. The simple Lie algebras 9. Some universal constructions 10. Irreducible modules for semisimple Lie algebras 11. Further properties of the universal enveloping algebra 12. Character and dimension formulae 13. Fundamental modules for simple Lie algebras 14. Generalized Cartan matrices and Kac-Moody algebras 15. The classification of generalised Cartan matrices 16 The invariant form, root system and Weyl group 17. Kac-Moody algebras of affine type 18. Realisations of affine Kac-Moody algebras 19. Some representations of symmetrisable Kac-Moody algebras 20. Representations of affine Kac-Moody algebras 21. Borcherds Lie algebras Appendix.

Topics: Cartan matrix (80%), Kac–Moody algebra (77%), Affine Lie algebra (76%) ... read more

251 Citations

Journal Article
Marc Rosso1Institutions (1)
Abstract: Let U q + be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix We show that U q + is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z n This method gives supersymetric as well as multiparametric versions of U q + in a uniform way (for a suitable choice of the Hopf bimodule) We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U q sl(2) and a suitable irreducible finite dimensional representation