## Finite-dimensional Modules for the Quantum Affine Algebra Uq(g) and Its Borel Subalgebra |

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action affine algebra Uq(g affine Kac-Moody algebra affine type basis consisting Benkart and Terwilliger Borel subalgebra Cartan matrix choose consisting of weight define Definition 3.7 dimensional irreducible modules direct sum dissertation eigenspaces eigenvalues equitable presentation extends uniquely fcf1 acts finite finite-dimensional irreducible g)-module finite-dimensional irreducible modules finite-dimensional irreducible t/,(g)-module finite-dimensional irreducible U-°-module finite-dimensional irreducible Uq(g)-module following hold following relations integers invariant under U>0 irreducible t/-°-module l2)-module structure Lemma Let g Lie algebra linear transformations Zi module structure modules for Uq(g nonzero and invariant notation of Definition Note positive integer Proposition 2.27 quantum affine algebra quantum group result follows root of unity set of weights span(v3 structure is irreducible suffices to show Suppose t/,(g)-module of type Theorem 3.2 tridiagonal pairs U-°-module of type Ui(a Ui(s unital associative F-algebra universal enveloping algebra Uq(g)-module with set vector of type vector space weight spaces weight vector Weyl group Wi(s