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Quantum Cluster Algebras At Roots Of Unity, Poisson-Lie Groups, And Discriminants, Kurt Malcolm Trampel Iii

LSU Doctoral Dissertations

This dissertation studies quantum algebras at roots of unity in regards to cluster structure and Poisson structure. Moreover, quantum cluster algebras at roots of unity are rigorously defined. The discriminants of these algebras are described, in terms of frozen cluster variables for quantum cluster algebras and Poisson primes for specializations of quantum algebras. The discriminant is a useful invariant for representation theoretic and algebraic study, whose laborious computation deters direct evaluation. The discriminants of quantum Schubert cells at roots of unity will be computed from the two distinct approaches. These methods can be applied to many other quantum algebras.