2026-09-23T10:30:17Zhttps://orbi.uliege.be/oai/requestoai:orbi.ulg.ac.be:2268/1794712026-09-01T13:29:58Zcom_f00col_f03class_g03
doctoral thesis
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info:eu-repo/semantics/doctoralThesis
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Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.
Kreusch, Marie
Lecomte, Pierre
Ovsienko, Valentin
2015-04-21
139
https://orbi.uliege.be/handle/2268/179471
info:hdl:2268/179471
https://orbi.uliege.be/bitstream/2268/179471/1/ThesisOnline.pdf
en
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
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.
ULiège - Université de Liège