Mathematical Sciences: Actions of Finite Groups and Finite Dimensional Hopf Algebras on Rings

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LORENZ ABSTRACT This award supports the research of Professor Martin Lorenz in the area of noncommutative ring theory with particular emphasis on the structure of rings of invariants arising from actions of finite groups and finite dimensional Hopf algebras. The main objectives will be the description of the Grothendieck groups G_0 and K_0 of these rings, the continued development of the representation theory of finite dimensional Hopf algebras, and the detailed analysis of invariant rings of multiplicative group actions on Laurent polynomial rings. Invariant theory and representation theory of groups are classical algebraic themes that are ubiquitous in pure mathematics and are indispensable in parts of applied mathematics and physics as well. Both are formally subsumed in the theory of Hopf algebras, more recent algebraic structures with a wide range of applications. In recognition of their relevance in physics, certain types of Hopf algebras are referred to as quantum groups. Hopf algebras further naturally arise in knot theory, an area of particular interest to molecular chemists as well as physicists.