References of "Kreusch, Marie"
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See detailBott type periodicity for the higher octonions
Kreusch, Marie ULg

in Journal of Noncommutative Geometry (in press)

We study the series of complex nonassociative algebras $\bbO_n$ and real nonassociative algebras $\bbO_{p,q}$ introduced in~\cite{MGO2011}. These algebras generalize the classical algebras of octonions ... [more ▼]

We study the series of complex nonassociative algebras $\bbO_n$ and real nonassociative algebras $\bbO_{p,q}$ introduced in~\cite{MGO2011}. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $\bbO_{n}$ and $\bbO_{p,q}$ with $p+q=n$ have a natural $\Z_2^n$-grading, and they are characterized by cubic forms over the field $\Z_2$. We establish a periodicity for the algebras~$\bbO_{n}$ and $\bbO_{p,q}$ similar to that of the Clifford algebras $\mathrm{Cl}_{n}$ and~$\mathrm{Cl}_{p,q}$. [less ▲]

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See detailExplain Your Thesis in Three Minutes
Kreusch, Marie ULg

Article for general public (2016)

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See detailLes mathématiques à l’honneur à MT180
Favart, Evelyne ULg; Haesbroeck, Gentiane ULg; Kreusch, Marie ULg

Article for general public (2015)

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See detailClassification of the algebras $\mathbb{O}_{p,q}$
Kreusch, Marie ULg; Morier-Genoud, Sophie

in Communications in Algebra (2015), 43(9), 3799-3815

We study a series of real nonassociative algebras $O_{p,q}$ introduced in [5]. These algebras have a natural $Z^n_2$-grading, where $n = p + q$, and they are characterized by a cubic form over the field $Z ... [more ▼]

We study a series of real nonassociative algebras $O_{p,q}$ introduced in [5]. These algebras have a natural $Z^n_2$-grading, where $n = p + q$, and they are characterized by a cubic form over the field $Z_2$. We establish all the possible isomorphisms between the algebras $O_{p,q}$ preserving the structure of $Z^n_2$-graded algebra. The classification table of $O_{p,q}$ is quite similar to that of the real Clifford algebras $Cl_{p,q}$, the main difference is that the algebras $O_{n,0}$ and $O_{0,n}$ are exceptional. [less ▲]

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See detailHigher Octonions
Kreusch, Marie ULg

Conference (2015, June 11)

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See detailGraded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.
Kreusch, Marie ULg

Doctoral thesis (2015)

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 ... [more ▼]

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. [less ▲]

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See detailNoncommutative and nonassociative algebras
Kreusch, Marie ULg

Conference (2015, March 31)

We choose the abelian group ($\bbZ_2^n, +$) where $\bbZ_2 = \bbZ / 2 \bbZ$ and define a $\bbZ_2^n$-graded vector space \[ E = \bigoplus_{x \in \bbZ_2^n} E_x \] together with a multiplication $ \cdot :E ... [more ▼]

We choose the abelian group ($\bbZ_2^n, +$) where $\bbZ_2 = \bbZ / 2 \bbZ$ and define a $\bbZ_2^n$-graded vector space \[ E = \bigoplus_{x \in \bbZ_2^n} E_x \] together with a multiplication $ \cdot :E \times E \longrightarrow E$ respecting the grading \[ E_x \cdot E_y \subset E_{x+y} \quad \forall x,y \in \bbZ_2^n. \] This is called a $\bbZ_2^n$-graded algebra. We are interested in particular $\bbZ_2^n$-graded algebras where the product in noncommutative and nonassociative. This talk consists of two parts. The first one is the study of a series of $\bbZ_2^n$-graded algebras of finite dimension ($2^n$) where $n \geq 3$. This series of real noncommutative and nonassociative algebras, denoted $\bbO_{p,q}$ ($p+q=n$), generalizes the algebra of octonion numbers $\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras. The first \emph{question} is to classify these algebras up to isomorphisms. The classification table of $\bbO_{p,q}$ is quite similar to that of the real Clifford algebras $\cC l_{p,q}$. The second \emph{question} is to find a periodicity between these algebras. The periodicity for the algebras $\bbO_{p,q}$ is analogous to the periodicity for the Clifford algebras $\cC l_{p,q}$. In the second part we study $\bbZ_2$-graded algebras ($n=0$, ``superalgebras'') that can be of infinite dimension. We consider two kind of superalgebras $\cL_{g,N}$ and $\cJ_{g,N}$ that are noncommutative and nonassociative\footnote{The construction coming from spaces on a compact Riemann surface of genus $g$ with $N$ punctures}. Nevertheless, these superalgebras link together the classical Lie algebras and the classical commutative and associative algebras. The two last \emph{questions} are can we ``extend'' the algebras $\cL_{g,N}$ and $\cJ_{g,N}$? The first answer is yes (for $\cL_{g,N}$), while the second one is no (for $\cJ_{g,N}$). However, we can ``extend'' module $\cJ_{g,N}^*$. [less ▲]

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See detailAlgebras generalizing the Octonions
Kreusch, Marie ULg

Poster (2014, August 29)

A series of algebras, namely $O_p,q$, generalizing the algebra of the octonion numbers as the Clifford algebras generalizing the algebra of Quaternions numbers was introduced by Ovsienko and Morier-Genoud ... [more ▼]

A series of algebras, namely $O_p,q$, generalizing the algebra of the octonion numbers as the Clifford algebras generalizing the algebra of Quaternions numbers was introduced by Ovsienko and Morier-Genoud. We present a classification of these algebras (up to graded-isomorphism) and give a periodicity similar to the ones on the Clifford algebras. [less ▲]

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See detailTour d’horizon sur des algèbres généralisant les octonions
Kreusch, Marie ULg

Conference (2014, May 19)

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud $\&$ Ovsienko en ... [more ▼]

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud $\&$ Ovsienko en 2011. Ces algèbres, qui ne sont ni commutative, ni associative, peuvent être vues comme des algèbres twistées sur le groupe $(\mathbb{Z}_2)^n$ avec une fonction de twist cubique. \\ Les propriétés de périodicités de ces algèbres sont similaires à celles déjà bien connues sur les algèbres de Clifford. Ce résultat donnera lieu à une discussion sur les formes cubiques définies sur $(\mathbb{Z}_2)^n$ à valeurs dans $\mathbb{Z}_2$. [less ▲]

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See detailAu delà des octonions
Kreusch, Marie ULg

Scientific conference (2014, April 08)

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud et Ovsienko en 2011 ... [more ▼]

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud et Ovsienko en 2011. Ces algèbres, qui ne sont ni commutative, ni associative, peuvent être vues comme des algèbres twistées sur le groupe Z_2^n avec une fonction de twist cubique. Une classification de ces algèbres, semblable à la classification des algèbres de Clifford, sera exposée. De plus, celle-ci donnera lieu à une discussion sur les fonctions cubiques définies sur Z_2^n à valeurs dans Z_2. [less ▲]

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See detailAu delà des octonions
Kreusch, Marie ULg

Scientific conference (2013, December 16)

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud & Ovsienko en 2011 ... [more ▼]

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud & Ovsienko en 2011. Ces algèbres, qui ne sont ni commutative, ni associative, peuvent être vues comme des algèbres twistées sur le groupe $(Z_2)^n$ avec une fonction de twist cubique. Lors de ce séminaire, je replacerai dans un contexte général et historique ces algèbres pour les relier ensuite au problème d'Hurwitz-Radon. Par la suite, je parlerai de la classification de celles-ci qui est similaire à celle déjà connue sur les algèbres de Clifford. Enfin, j'aborderai certaines questions ouvertes. [less ▲]

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See detailAu delà des octonions
Kreusch, Marie ULg

Scientific conference (2013, September 26)

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud et Ovsienko en 2011 ... [more ▼]

Une nouvelle série d'algèbres réelles généralisant l'algèbre des octonions, tout comme les algèbres de Clifford prolongent l'algèbre des quaternions, a été introduite par Morier-Genoud et Ovsienko en 2011. Ces algèbres, qui ne sont ni commutative, ni associative, peuvent être vues comme des algèbres twistées sur le groupe (ℤ2)n avec une fonction de twist cubique. La classification de ces algèbres est similaire à celle déjà bien connue sur les algèbres de Clifford. En effet, il existe beaucoup de symétries concernant les algèbres de Clifford et on peut les retrouver en partie pour les algèbres généralisant les octonions. Cette classification sera exposée lors du séminaire avec les idées de certaines preuves. [less ▲]

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See detailUne généralisation des octonions
Kreusch, Marie ULg

Scientific conference (2013, July 05)

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See detailExtensions of superalgebras of Krichever-Novikov type
Kreusch, Marie ULg

in Letters in Mathematical Physics (2013), 103(11),

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we ... [more ▼]

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we calculate a 1-cocycle with coefficients in the dual space. [less ▲]

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See detailAu delà des nombres réels
Kreusch, Marie ULg

Poster (2013, June)

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See detailExtensions of Superalgebras of Krichever-Novikov type
Kreusch, Marie ULg

Poster (2013, April)

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we ... [more ▼]

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we calculate a 1-cocycle with coefficients in the dual space. [less ▲]

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See detailExtensions of superalgebras of Krichever-Novikov type
Kreusch, Marie ULg

Conference (2013, January 15)

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See detailLie antialgebras
Kreusch, Marie ULg

Conference (2011, June 28)

Lie antialgebras which is a $\Z_2$-graded commutative algebra (but not associative) was introduced in 2007 by Valentin Ovsienko. This notion takes place in the superspaces theory studied since years in ... [more ▼]

Lie antialgebras which is a $\Z_2$-graded commutative algebra (but not associative) was introduced in 2007 by Valentin Ovsienko. This notion takes place in the superspaces theory studied since years in geometry. This algebra was discovered in the context of symplectic geometry. In a way, Lie antialgebras unify in a special meaning associative and commutative algebras. Since this is quite a new subject a lot of things have to be done in the understanding of this structure. At first, I am going to explain the notion of superspaces. Then I will speak about the origins of this structure and present what has already been discovered about this new 'type' of algebra (universal algebra, representations, modules, relation to superalgebra,...). After, I am going to give some important examples of Lie antialgebras related to some known structures. Finally, I am going to present what I am searching for the moment and the questions that I am trying to answer. [less ▲]

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See detailLie antialgebras
Kreusch, Marie ULg

Conference (2011, June 20)

Lie antialgebras which is a $\Z_2$-graded commutative algebra (but not associative) was introduced in 2007 by Valentin Ovsienko. This notion takes place in the superspaces theory studied since years in ... [more ▼]

Lie antialgebras which is a $\Z_2$-graded commutative algebra (but not associative) was introduced in 2007 by Valentin Ovsienko. This notion takes place in the superspaces theory studied since years in geometry. This algebra was discovered in the context of symplectic geometry. In a way, Lie antialgebras unify in a special meaning associative and commutative algebras with Lie algebras. Since this is quite a new subject a lot of things have to be done in the understanding of this structure. At first, I am going to explain the notion of superspaces and in particular the one of Lie superalgebras and give some important examples. After I am going to introduce the topic of Lie antialgebras and also give some examples. And finally I am going to give a links between them (Lie antilagebra and Lie superalgebra). If I have time, I will probably speak a bit about extensions and relations with 2-cocycles in the cohomology theory. That's what I am interested in for the moment. [less ▲]

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