Polynomial Identities on Algebras with Actions
 Publication Year:
 2014

 Bepress 227

 Bepress 49
 Repository URL:
 https://ir.lib.uwo.ca/etd/2399; https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=3773&context=etd
 Author(s):
 Tags:
 Noncommutative Algebra; Polynomial Identity Algebras; Graded Algebras; Hopf Algebras; Antiautomorphisms; Algebra
article description
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 structureinduced subspaces. For example, if A is an associative Ggraded algebra such that the homogeneous component A1 satisfies an identity of degree d, then Bergen and Cohen showed that A is itself a PIalgebra. 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 Ggraded.Groupgradings and actions by a group of automorphisms are examples of Hopf algebras acting on Halgebras. If H is finitedimensional, semisimple, commutative, and splits over its base field, then it is known that A is an Halgebra precisely when the Haction on A induces a certain groupgrading of A. We extend this duality to incorporate other natural Hactions. To this end, we introduce the notion of an oriented Halgebra. For example, if A has an action by a group of both automorphisms and antiautomorphisms, then A is not an Halgebra, but A is an oriented Halgebra. The vector space gradings associated to oriented Halgebra actions are not generally groupgradings, or even setgradings. However, when A is a Lie algebra, the grading is a quasigroupgrading, and, when A is an associative algebra, the grading is what we call a LieJordangrading.Lastly, we call certain Hpolynomials in the free associative Halgebra essential, and show that, if an (associative) Halgebra A satisfies an essential Hidentity of degree d, then A satisfies an ordinary identity of bounded degree. Furthermore, in the case when H is mdimensional, semisimple and commutative, we prove that, if AH satisfies an ordinary identity of degree d, then A satisfies an essential Hidentity of degree dm. From this we are able to recover several wellknown results as special cases.