DOI

10.17077/etd.ftz33rv3

Document Type

Dissertation

Date of Degree

Summer 2018

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Kinser, Ryan

First Committee Member

Bleher, Frauke

Second Committee Member

Camillo, Victor

Third Committee Member

Frohman, Charles

Fourth Committee Member

Iovanov, Miodrag

Abstract

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.

Keywords

Actions, Hopf Algebras, Invariants, Path Algebras, Quivers, Taft Algebras

Pages

vii, 85 pages

Bibliography

Includes bibliographical references (pages 84-85).

Copyright

Copyright © 2018 Ana Berrizbeitia

Included in

Mathematics Commons

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