Invariants of HOPF actions on path algebras of quivers

The work of this thesis focuses primarily on non-commutative algebras and actions of Hopf algebras. Specifically, we study the possible H-module algebra structures which can be imposed on path algebras of quivers, for a variety of Hopf algebras, H, and then given a possible action, classify the invariant ring.

A Hopf algebra is a bialgebra (H, μ, η, ∆, ε) together with an antipode S : H → Hop which is compatible with the counit, ε, of H. A quiver is a directed graph, and the path algebra kQ of a quiver Q is a vector space where all the paths of the quiver form a basis, and multiplication is given by concatenation of paths whenever possible, and zero otherwise. In their paper, [9], Kinser and Walton classify Hopf actions of a specific family of Hopf algebras called a Taft algebras, T(n), on path algebras of loopless, finite, Schurian quivers. In this thesis, we extend their result to path algebras of any finite quiver and classify the invariant subring, kQT(n), in the case where the group like element g ∈ T(n) acts transitively on Q0.

In the future, we hope that the ideas presented in this work extend to a classification of quantum groups, such as uq(sl2), acting on path algebras of finite quivers.