THE AUGMENTED TRIDIAGONAL ALGEBRA

Tatsuro ITO; Paul TERWILLIGER

doi:10.2206/kyushujm.64.81

Kyushu Journal of Mathematics

Online ISSN : 1883-2032
Print ISSN : 1340-6116
ISSN-L : 1340-6116

THE AUGMENTED TRIDIAGONAL ALGEBRA

Tatsuro ITO, Paul TERWILLIGER

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Keywords: P-and Q-polynomial association scheme, Terwilliger algebra, tridiagonal pair, Leonard pair, tridiagonal algebra, q-Onsager algebra

JOURNAL FREE ACCESS

2009 Volume 64 Issue 1 Pages 81-144

DOI https://doi.org/10.2206/kyushujm.64.81

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Abstract

Motivated by investigations of the tridiagonal pairs of linear transformations, we introduce the augmented tridiagonal algebra Τ_q. This is an infinite-dimensional associative C-algebra with 1. We classify the finite-dimensional irreducible representations of Τ_q.All such representations are explicitly constructed via embeddings of Τ_q into the U_q(sl₂)-loop algebra. As an application, tridiagonal pairs over C are classified in the case where q is not a root of unity.

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