Some algebra related to $P$-and $Q$-polynomial association schemes

Abstract

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^{\ast }$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair is the same thing as a tridiagonal pair such that for each transformation all eigenspaces have dimension one.

26 pages

References in corpus (1)

• sl(N) Onsager's Algebra and Integrability

Cited by in corpus (37)

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• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
• The alternating central extension of the $q$-Onsager algebra
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• On relative $t$-designs in polynomial association schemes
• A bilinear form relating two Leonard systems
• Generalized q-Onsager algebras and boundary affine Toda field theories
• Generalized q-Onsager Algebras and Dynamical K-matrices
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra
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• A conjecture concerning the $q$-Onsager algebra
• A classification of sharp tridiagonal pairs
• Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
• Analogues of Lusztig's higher order relations for the q-Onsager algebra
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• The Tetrahedron algebra, the Onsager algebra, and the ${\mathfrak{sl}}_2$ loop algebra
• Evaluation modules for the $q$-tetrahedron algebra
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Raising/lowering maps and modules for the quantum affine algebra $U_q(s{l}_{2}hat)$
• The algebra $U_q({\mathfrak{sl}}_2)$ in disguise
• On a Certain Subalgebra of $U_q({\widehat{\mathfrak{sl}}}_2)$ Related to the Degenerate $q$-Onsager Algebra
• The Erdős-Ko-Rado basis for a Leonard system
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The determinant of $A{A}^{\ast }-A^{\ast }A$ for a Leonard pair $A,A^{\ast }$
• Double Lowering Operators on Polynomial
• Two commuting operators associated with a tridiagonal pair
• Thin Hessenberg Pairs
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