Research Projects

Bredon introduced equivariant cohomology for topological spaces with finite group action G, a concept known as Borel cohomology. However, it is difficult to compute for simple objects. The Mackey functor, a concept from Green and Dress, connects various theories, including representation theory of finite groups, K-theory for spaces, and G-algebra. A Green functor for G over a commutative ring R is a Mackey functor A with an extra multiplicative structure compatible with the multiplicative structure. This project aims to introduce Green functors that correspond to non-associative algebras like Lie algebras, Leibniz algebras, and dendriform algebras. The primary focus is to study the cohomology of different types of non-associative algebras via Mackey functor theory. The study will begin with the cohomology of non-associative Green functors for cyclic groups of prime orders, followed by arbitrary finite groups and a formal deformation theory. The project also plans to study the Rota-Baxter operator on non-associative Green functors and rational homotopy theory for non-associative Green functors based on M. Livernet's study on Leibniz algebras. The project's results could provide new insights into the theory of non-associative algebras and may indicate a new connection to related areas of mathematics.