The augmented tridiagonal algebra

Abstract

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 $U_q(s{l}_2)$-loop algebra. As an application, tridiagonal pairs over $\mathbb{C}$ are classified in the case where $q$ is not a root of unity.

67 pages

References in corpus (1)

• The Drinfel'd polynomial of a tridiagonal pair

Cited by in corpus (5)

• Towards a classification of the tridiagonal pairs
• Mock Tridiagonal Systems
• Tridiagonal pairs of $q$-Racah type
• The structure of a tridiagonal pair
• On the shape of a tridiagonal pair