## Degree

Doctor of Philosophy

## Program

Mathematics

## Supervisor

Dr. David Riley

## Abstract

When an algebra is endowed with the additional structure of an action or a grading, one can often make striking conclusions about the algebra based on the properties of the structure-induced subspaces. For example, if *A* is an associative *G*-graded algebra such that the homogeneous component *A _{1}* satisfies an identity of degree

*d*, then Bergen and Cohen showed that

*A*is itself a PI-algebra. Bahturin, Giambruno and Riley later used combinatorial methods to show that the degree of the identity satisfied by

*A*is bounded above by a function of

*d*and |

*G|*. Utilizing a similar approach, we prove an analogue of this result which applies to associative algebras whose induced Lie or Jordan algebras are

*G*-graded.

Group-gradings and actions by a group of automorphisms are examples of Hopf algebras acting on *H*-algebras. If *H* is finite-dimensional, semisimple, commutative, and splits over its base field, then it is known that *A* is an *H*-algebra precisely when the *H*-action on *A* induces a certain group-grading of *A*. We extend this duality to incorporate other natural *H*-actions. To this end, we introduce the notion of an oriented *H*-algebra. For example, if *A* has an action by a group of both automorphisms and anti-automorphisms, then *A* is not an *H*-algebra, but* A* is an oriented *H*-algebra. The vector space gradings associated to oriented *H*-algebra actions are not generally group-gradings, or even set-gradings. However, when* A* is a Lie algebra, the grading is a quasigroup-grading, and, when *A* is an associative algebra, the grading is what we call a Lie-Jordan-grading.

Lastly, we call certain *H*-polynomials in the free associative *H*-algebra essential, and show that, if an (associative) *H*-algebra *A* satisfies an essential *H*-identity of degree *d*, then *A* satisfies an ordinary identity of bounded degree. Furthermore, in the case when *H* is *m-*dimensional, semisimple and commutative, we prove that, if A^{H} satisfies an ordinary identity of degree *d*, then *A* satisfies an essential *H*-identity of degree *dm*. From this we are able to recover several well-known results as special cases.

## Recommended Citation

Plyley, Chris, "Polynomial Identities on Algebras with Actions" (2014). *Electronic Thesis and Dissertation Repository*. 2399.

https://ir.lib.uwo.ca/etd/2399