Reference : Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov su...
Dissertations and theses : Doctoral thesis
Physical, chemical, mathematical & earth Sciences : Mathematics
Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.
Kreusch, Marie mailto [Université de Liège - ULiège > Département de mathématique > Géométrie et théorie des algorithmes >]
Université de Liège, ​Liège, ​​Belgique
Docteur en Sciences
Lecomte, Pierre mailto
Ovsienko, Valentin mailto
Mathonet, Pierre mailto
Hansoul, Georges mailto
Morier-Genoud, Sophie mailto
Schlichenmaier, Martin mailto
[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
[en] 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.
Researchers ; Professionals

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