Polynomial Identities on Algebras with Actions
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 A1 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 AH 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.