{"id":{"repo_id":"liege","oai_identifier":"oai:orbi.ulg.ac.be:2268/179471"},"canonical_url":"https://search.dev.ndltd.org/etd/liege/oai:orbi.ulg.ac.be:2268/179471","repository":{"repo_id":"liege","name":"Université de Liège","base_url":"https://orbi.uliege.be/oai/request"},"display":{"title":"Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.","abstract":"This dissertation consists of two parts. The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras $\\bbO_{p,q}$ (resp. $\\bbO_{n}$) generalizing the algebra of octonion numbers $\\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras. Introduced by Morier-Genoud and Ovsienko, these algebras have a natural $\\bbZ_2^n$-grading ($p+q =n$), and they are characterized by a cubic form over the field $\\bbZ_2.$ We establish all the possible isomorphisms between the algebras $\\bbO_{p,q}$ preserving the structure of $\\bbZ_2^n$-graded algebra. The classification table of $\\bbO_{p,q}$ is quite similar to that of the real Clifford algebras $\\cC l_{p,q}$, the main difference is that the algebras $\\bbO_{n,0}$ and $\\bbO_{0,n}$ are exceptional. We also provide a periodicity for the algebras $\\bbO_n$ and $\\bbO_{p,q}$ analogous to the periodicity for the Clifford algebras $\\cC l_{n}$ and $\\cC l_{p,q}$. In the second part we consider superalgebras of Krichever-Novikov (K-N) type. Krichever and Novikov introduced a family of Lie algebras with two marked points generalizing the Witt algebra and its central extension called the Virasoro algebra. The K-N Lie (super)algebras for more than two marked points were studied by Schlichenmaier. In particular, he extended the explicit formula of $2$-cocycles due to Krichever and Novikov to multiple-point situation. We give an explicit construction of central extensions of Lie superalgebras of K-N type and we establish a $1$-cocycle with values in its dual space. In the case of Jordan superalgebras related to superalgebras of K-N type, we calculate a 1-cocycle with coefficients in the dual space.","abstract_html":"This dissertation consists of two parts. The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras <span class=\"etd-inline-math\">\\bbO<sub>p,q</sub></span> (resp. <span class=\"etd-inline-math\">\\bbO<sub>n</sub></span>) generalizing the algebra of octonion numbers $\\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras. Introduced by Morier-Genoud and Ovsienko, these algebras have a natural <span class=\"etd-inline-math\">\\bbZ<sub>2</sub><sup>n</sup></span>-grading ($p+q =n$), and they are characterized by a cubic form over the field <span class=\"etd-inline-math\">\\bbZ<sub>2</sub>.</span> We establish all the possible isomorphisms between the algebras <span class=\"etd-inline-math\">\\bbO<sub>p,q</sub></span> preserving the structure of <span class=\"etd-inline-math\">\\bbZ<sub>2</sub><sup>n</sup></span>-graded algebra. The classification table of <span class=\"etd-inline-math\">\\bbO<sub>p,q</sub></span> is quite similar to that of the real Clifford algebras <span class=\"etd-inline-math\">\\cC l<sub>p,q</sub></span>, the main difference is that the algebras <span class=\"etd-inline-math\">\\bbO<sub>n,0</sub></span> and <span class=\"etd-inline-math\">\\bbO<sub>0,n</sub></span> are exceptional. We also provide a periodicity for the algebras <span class=\"etd-inline-math\">\\bbO<sub>n</sub></span> and <span class=\"etd-inline-math\">\\bbO<sub>p,q</sub></span> analogous to the periodicity for the Clifford algebras <span class=\"etd-inline-math\">\\cC l<sub>n</sub></span> and <span class=\"etd-inline-math\">\\cC l<sub>p,q</sub></span>. In the second part we consider superalgebras of Krichever-Novikov (K-N) type. Krichever and Novikov introduced a family of Lie algebras with two marked points generalizing the Witt algebra and its central extension called the Virasoro algebra. The K-N Lie (super)algebras for more than two marked points were studied by Schlichenmaier. In particular, he extended the explicit formula of $2$-cocycles due to Krichever and Novikov to multiple-point situation. We give an explicit construction of central extensions of Lie superalgebras of K-N type and we establish a $1$-cocycle with values in its dual space. In the case of Jordan superalgebras related to superalgebras of K-N type, we calculate a 1-cocycle with coefficients in the dual space.","abstract_has_math":true,"creators":["Kreusch, Marie"],"institution":"ULiège - Université de Liège","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Lecomte, Pierre","Ovsienko, Valentin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-04-21","date_published":"2015-04-21","updated_at":"2026-07-24T02:49:29Z","subjects":["Octonion","Clifford algebra","binary cubic form","Twisted group algebra","nonassociative ans noncommutative algebra","graded algebra","Krichever-Novikov Lie superalgebra","non-trivial cocycle","Jordan superalgebra","Lie antialgebra","Physical, chemical, mathematical & earth Sciences","Mathematics","Physique, chimie, mathématiques & sciences de la terre","Mathématiques"],"languages":["en"],"rights":["open access","info:eu-repo/semantics/openAccess"],"rights_urls":["http://purl.org/coar/access_right/c_abf2"],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["info:hdl:2268/179471"],"render_values":[{"text":"info:hdl:2268/179471","href":null,"code":true}]}]},"links":{"outbound_url":"https://orbi.uliege.be/handle/2268/179471","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lecomte, Pierre","Ovsienko, Valentin"]},{"key":"dc:creator","label":"Author","values":["Kreusch, Marie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-04-21"]},{"key":"dc:publisher","label":"Institution","values":["ULiège - Université de Liège"]},{"key":"dc:type","label":"Dc Type","values":["doctoral thesis","http://purl.org/coar/resource_type/c_db06","info:eu-repo/semantics/doctoralThesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Octonion","Clifford algebra","binary cubic form","Twisted group algebra","nonassociative ans noncommutative algebra","graded algebra","Krichever-Novikov Lie superalgebra","non-trivial cocycle","Jordan superalgebra","Lie antialgebra","Physical, chemical, mathematical & earth Sciences","Mathematics","Physique, chimie, mathématiques & sciences de la terre","Mathématiques"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["open access","http://purl.org/coar/access_right/c_abf2","info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://orbi.uliege.be/handle/2268/179471","info:hdl:2268/179471","https://orbi.uliege.be/bitstream/2268/179471/1/ThesisOnline.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This dissertation consists of two parts. The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras $\\bbO_{p,q}$ (resp. $\\bbO_{n}$) generalizing the algebra of octonion numbers $\\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras. Introduced by Morier-Genoud and Ovsienko, these algebras have a natural $\\bbZ_2^n$-grading ($p+q =n$), and they are characterized by a cubic form over the field $\\bbZ_2.$ We establish all the possible isomorphisms between the algebras $\\bbO_{p,q}$ preserving the structure of $\\bbZ_2^n$-graded algebra. The classification table of $\\bbO_{p,q}$ is quite similar to that of the real Clifford algebras $\\cC l_{p,q}$, the main difference is that the algebras $\\bbO_{n,0}$ and $\\bbO_{0,n}$ are exceptional. We also provide a periodicity for the algebras $\\bbO_n$ and $\\bbO_{p,q}$ analogous to the periodicity for the Clifford algebras $\\cC l_{n}$ and $\\cC l_{p,q}$. In the second part we consider superalgebras of Krichever-Novikov (K-N) type. Krichever and Novikov introduced a family of Lie algebras with two marked points generalizing the Witt algebra and its central extension called the Virasoro algebra. The K-N Lie (super)algebras for more than two marked points were studied by Schlichenmaier. In particular, he extended the explicit formula of $2$-cocycles due to Krichever and Novikov to multiple-point situation. We give an explicit construction of central extensions of Lie superalgebras of K-N type and we establish a $1$-cocycle with values in its dual space. In the case of Jordan superalgebras related to superalgebras of K-N type, we calculate a 1-cocycle with coefficients in the dual space."]},{"key":"dc:format","label":"Dc Format","values":["139"]},{"key":"dc:title","label":"Title","values":["Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles."]}]}],"canonical_facts":{"dc:contributor":["Lecomte, Pierre","Ovsienko, Valentin"],"dc:creator":["Kreusch, Marie"],"dc:date":["2015-04-21"],"dc:description":["This dissertation consists of two parts. The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras $\\bbO_{p,q}$ (resp. $\\bbO_{n}$) generalizing the algebra of octonion numbers $\\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras. Introduced by Morier-Genoud and Ovsienko, these algebras have a natural $\\bbZ_2^n$-grading ($p+q =n$), and they are characterized by a cubic form over the field $\\bbZ_2.$ We establish all the possible isomorphisms between the algebras $\\bbO_{p,q}$ preserving the structure of $\\bbZ_2^n$-graded algebra. The classification table of $\\bbO_{p,q}$ is quite similar to that of the real Clifford algebras $\\cC l_{p,q}$, the main difference is that the algebras $\\bbO_{n,0}$ and $\\bbO_{0,n}$ are exceptional. We also provide a periodicity for the algebras $\\bbO_n$ and $\\bbO_{p,q}$ analogous to the periodicity for the Clifford algebras $\\cC l_{n}$ and $\\cC l_{p,q}$. In the second part we consider superalgebras of Krichever-Novikov (K-N) type. Krichever and Novikov introduced a family of Lie algebras with two marked points generalizing the Witt algebra and its central extension called the Virasoro algebra. The K-N Lie (super)algebras for more than two marked points were studied by Schlichenmaier. In particular, he extended the explicit formula of $2$-cocycles due to Krichever and Novikov to multiple-point situation. We give an explicit construction of central extensions of Lie superalgebras of K-N type and we establish a $1$-cocycle with values in its dual space. In the case of Jordan superalgebras related to superalgebras of K-N type, we calculate a 1-cocycle with coefficients in the dual space."],"dc:format":["139"],"dc:identifier":["https://orbi.uliege.be/handle/2268/179471","info:hdl:2268/179471","https://orbi.uliege.be/bitstream/2268/179471/1/ThesisOnline.pdf"],"dc:language":["en"],"dc:publisher":["ULiège - Université de Liège"],"dc:rights":["open access","http://purl.org/coar/access_right/c_abf2","info:eu-repo/semantics/openAccess"],"dc:subject":["Octonion","Clifford algebra","binary cubic form","Twisted group algebra","nonassociative ans noncommutative algebra","graded algebra","Krichever-Novikov Lie superalgebra","non-trivial cocycle","Jordan superalgebra","Lie antialgebra","Physical, chemical, mathematical & earth Sciences","Mathematics","Physique, chimie, mathématiques & sciences de la terre","Mathématiques"],"dc:title":["Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles."],"dc:type":["doctoral thesis","http://purl.org/coar/resource_type/c_db06","info:eu-repo/semantics/doctoralThesis"]},"updated_at":"2026-07-24T02:49:29Z"}