Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.

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 \bbZ₂ⁿ-grading ($p+q =n$), and they are characterized by a cubic form over the field \bbZ₂. We establish all the possible isomorphisms between the algebras \bbO_p,q preserving the structure of \bbZ₂ⁿ-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.

Grantor dc:publisher

ULiège - Université de Liège

Year dc:date

2015

Author dc:creator

• Kreusch, Marie

Contributors dc:contributor

• Lecomte, Pierre
• Ovsienko, Valentin

• 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

Statement dc:rights

• open access
• info:eu-repo/semantics/openAccess

Licence

• http://purl.org/coar/access_right/c_abf2

Language dc:language

en

Identifier

info:hdl:2268/179471

OAI identifier oai:identifier

oai:orbi.ulg.ac.be:2268/179471