Lie Superalgebra Representations: Algebraic and Geometric Methods

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Recently there have been several essential developments in the representation theory of Lie superalgebras. One of them is Serganova's reduction of the Kac problem on characters of atypical finite-dimensional gl(m+ne)-modules to the semi-simplicity conjecture for a functor analogous to the Vogan functor. Another one is Kac and Wakimoto's discovery of a striking relationship between number theory and integrable representations of affine Lie superalgebras. Finally, there is the theory of triangular decomposition of arbitrary finite-dimensional Lie superalgebras and the related theory of generic irreducible highest weight modules developed by Serganova and the principal investigator. The research supported by this award is on five different topics related naturally to those developments: geometry of finite-dimensional generic modules over arbitrary finite-dimensional Lie superalgebras; arbitrary generic modules over finite-dimensional Lie superalgebras; special (both non-generic and atypical) modules over classical Lie superalgebras; generic integrable modules over Kac-Moody superalgebras; and quantum superalgebras. In most of the cases, it is planned to use a combination of algebraic and geometric methods. 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 sperical motion.