The q-tetrahedron algebra and its finite dimensional irreducible modules

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Recently, Hartwig and the second author found a presentation for the three-point 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 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 Uq(2), the Uq(2) loop algebra, and the positive part of [image omitted]. 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.