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

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

2006-02-09

arxiv.org/abs/math/0602199v1

Mathematics

Quantum Algebra

25 pages

Scientific paper

Recently B. Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $\boxtimes$ by generators and relations, and displayed an isomorphism from $\boxtimes$ to the three-point $sl_2$ loop algebra. We introduce a quantum analog of $\boxtimes$ which we call $\boxtimes_q$. We define $\boxtimes_q$ via generators and relations. We show how $\boxtimes_q$ is related to the quantum group $U_q(sl_2)$, the $U_q(sl_2)$ loop algebra, and the positive part of $U_q(\hat{sl_2})$. We describe the finite dimensional irreducible $\boxtimes_q$-modules under the assumption that $q$ is not a root of 1, and the underlying field is algebraically closed.

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