Finite-dimensional Hopf algebras and their representations

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This proposal will study the structure and representations of certain Hopf algebras, especially those Hopf algebras constructed from finite groups. A major topic concerns the Frobenius-Schur indicators of the representations. These indicators are invariants of the tensor category of representations of the Hopf algebra, and can be extended to more general objects. They should give information not only on the representations of a Hopf algebra, but they also serve as a tool in proving more general facts about Hopf algebras themselves. The methods for many of the problems involve the theory of finite groups.Hopf algebras are special algebraic objects which arose in topology and algebraic groups in the 1940's and 1950's. However since then they have appeared in other parts of mathematics, such as geometry (knot theory), and in several parts of mathematical physics (conformal field theory and statistical mechanics). In many cases Hopf algebras can be used to give invariants of these structures which help to understand and classify them. Thus they are of general interest.