{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/35267"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/35267","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Plyley, Chris"],"institution":"The University of Western Ontario","degree_name":"Ph D","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["David Riley"],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-08-05","date_published":"2014-08-05","updated_at":"2026-07-27T21:56:20Z","subjects":["Noncommutative Algebra","Polynomial Identity Algebras","Graded Algebras","Hopf Algebras","Anti-automorphisms"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/35267","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["David Riley"]},{"key":"dc:creator","label":"Author","values":["Plyley, Chris"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T20:36:24Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T20:36:24Z"]},{"key":"dc:date.issued","label":"Date","values":["2014-08-05"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph D"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Noncommutative Algebra","Polynomial Identity Algebras","Graded Algebras","Hopf Algebras","Anti-automorphisms"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/35267"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Polynomial Identities on Algebras with Actions"]}]}],"canonical_facts":{"dc:contributor.advisor":["David Riley"],"dc:creator":["Plyley, Chris"],"dc:date.accessioned":["2025-07-10T20:36:24Z"],"dc:date.available":["2025-07-10T20:36:24Z"],"dc:date.issued":["2014-08-05"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.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."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/35267"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["Noncommutative Algebra","Polynomial Identity Algebras","Graded Algebras","Hopf Algebras","Anti-automorphisms"],"dc:title":["Polynomial Identities on Algebras with Actions"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Ph D"]},"updated_at":"2026-07-27T21:56:20Z"}