Mathematical Sciences: Enveloping Superalgebras, Quantum Groups, and Their Prime Ideals

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This award is concerned with research into the internal structure and representation theory of certain associative algebras that are of current interest within mathematical physics and Lie representation theory, namely quantum groups and enveloping algebras of Lie superalgebras. The quantum groups to be studied are the quantized coordinate rings of algebraic groups; the Lie superalgebras which will be considered are the classical simple and solvable ones. The focus throughout will be on prime and primitive ideal theory, and the principle methods involve new induction techniques for ring extensions. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.