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

TL;DR

This paper introduces a quantum analog of a certain algebra related to $sl_2$, explores its structure, and classifies its finite dimensional irreducible modules when q is not a root of unity.

Contribution

The paper defines the algebra $oxtimes_q$, relates it to quantum groups, and classifies its finite dimensional irreducible modules.

Findings

$oxtimes_q$ is related to $U_q(sl_2)$ and its loop algebra.

Finite dimensional irreducible $oxtimes_q$-modules are classified for generic q.

The algebra $oxtimes_q$ provides a quantum analog of a classical algebra related to $sl_2$.

Abstract

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.