Reference : Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov su... |

Dissertations and theses : Doctoral thesis | |||

Physical, chemical, mathematical & earth Sciences : Mathematics | |||

http://hdl.handle.net/2268/179471 | |||

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

English | |

Kreusch, Marie [Université de Liège - ULiège > Département de mathématique > Géométrie et théorie des algorithmes >] | |

21-Apr-2015 | |

Université de Liège, Liège, Belgique | |

Docteur en Sciences | |

139 | |

Lecomte, Pierre | |

Ovsienko, Valentin | |

Mathonet, Pierre | |

Hansoul, Georges | |

Morier-Genoud, Sophie | |

Schlichenmaier, Martin | |

[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. | |

PAI - DYGEST | |

Researchers ; Professionals | |

http://hdl.handle.net/2268/179471 |

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