The q-Tetrahedron Algebra and Its Finite Dimensional Irreducible Modules

Recently, Hartwig and the second author found a presentation for the three-point 𝔰𝔩2 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 𝔰𝔩2 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(𝔰𝔩2), the Uq(𝔰𝔩2) loop algebra, and the positive part of . 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.

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