Invariant Theory of Finite Groups and Finite-Dimensional Hopf Algebras

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This award supports the research of Professor Martin Lorenz in the area of algebra with particular emphasis on the structure of rings of invariants arising from actions of finite groups and finite dimensional Hopf algebras. The main objectives are the description of the Grothendieck groups of these rings, the Cohen-Macaulay property of invariant rings under finite group actions, rationality questions for certain fields of invariants, and the continued development of the representation theory of finite dimensional Hopf algebras. The subject and the methods employed are mainly algebraic (commutative and noncommutative), but there are strong connections with algebraic K-theory and algebraic geometry. This project aims at deepening our present knowledge of the invariant rings of specific, especially relevant types of finite group actions while also extending the boundaries of both invariant and representation theory of finite groups by approaching these fields, whenever appropriate, from the more general perspective of Hopf algebras.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, notably coding theory, and in theoretical physics as well. Both theories 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.