The augmented tridiagonal algebra

Details The augmented tridiagonal algebra The augmented tridiagonal algebra

2009-04-20

arxiv.org/abs/0904.2889v1

Mathematics

Quantum Algebra

67 pages

Scientific paper

Motivated by investigations of the tridiagonal pairs of linear transformations, we introduce the augmented tridiagonal algebra ${\mathcal T}_q$. This is an infinite-dimensional associative ${\mathbb C}$-algebra with 1. We classify the finite-dimensional irreducible representations of ${\mathcal T}_q$. All such representations are explicitly constructed via embeddings of ${\mathcal T}_q$ into the $U_q(sl_2)$-loop algebra. As an application, tridiagonal pairs over ${\mathbb C}$ are classified in the case where $q$ is not a root of unity.

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