The $q$-tetrahedron algebra and its finite dimensional irreducible modules

Tatsuro Ito; Paul Terwilliger

paper

The $q$ -tetrahedron algebra and its finite dimensional irreducible modules

arXiv:math/0602199

Abstract

Recently B. Hartwig and the second author found a presentation for the three-point $s l_{2}$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $⊠$ by generators and relations, and displayed an isomorphism from $⊠$ to the three-point $s l_{2}$ loop algebra. We introduce a quantum analog of $⊠$ which we call $⊠_{q}$ . We define $⊠_{q}$ via generators and relations. We show how $⊠_{q}$ is related to the quantum group $U_{q} (s l_{2})$ , the $U_{q} (s l_{2})$ loop algebra, and the positive part of $U_{q} (s l_{2}^)$ . We describe the finite dimensional irreducible $⊠_{q}$ -modules under the assumption that $q$ is not a root of 1, and the underlying field is algebraically closed.

arXiv comment25 pages

The q-tetrahedron algebra and its finite dimensional irreducible modules

Abstract

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The $q$ -tetrahedron algebra and its finite dimensional irreducible modules