Mathematical Sciences: Lie Algebras, Lie Superalgebras, and Quantum Groups

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This award concerns research on the theory of Lie algebras. One of the principal investigators will work on the classification of simple and semisimple Lie superalgebras of prime characteristic. He will also work on vertex operators, specifically on explicit construction of certain modules, as well as on certain Hopf algebras. A second principal investigator will study representations of quantum groups and associated solutions of the quantum Yang-Baxter equations. She plans work on the case of quantum enveloping algebras of affine Kac-Moody Lie algebras at roots of unity. She also plans to describe minimal cyclic representations, and in particular to show that the minimum dimension equals the dimension of the minimal orbit of the corresponding Lie group. A third principal investigator will study various aspects of the representation theory of Lie superalgebras. He will work on generic modules over classical Lie superalgebras and twisted D-modules; finite-dimensional atypical modules of classical Lie superalgebras; representations of Cartan-type classical Lie superalgebras; representations of semisimple Lie superalgebras; and representations of Kac-Moody Lie superalgebras. This research is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of spherical motion.