References of "Kreusch, Marie"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 1 to 19 of 19 1 Explain Your Thesis in Three MinutesKreusch, Marie Article for general public (2016)Detailed reference viewed: 52 (13 ULiège) Bott type periodicity for the higher octonionsKreusch, Marie in Journal of Noncommutative Geometry (2016), 9(4), 1041-1393We 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 ▲]Detailed reference viewed: 164 (53 ULiège) Les mathématiques à l’honneur à MT180Favart, Evelyne ; Haesbroeck, Gentiane ; Kreusch, Marie Article for general public (2015)Detailed reference viewed: 43 (18 ULiège) Classification of the algebras $\mathbb{O}_{p,q}$Kreusch, Marie ; Morier-Genoud, Sophiein Communications in Algebra (2015), 43(9), 3799-3815We 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}$are exceptional. [less ▲]Detailed reference viewed: 51 (24 ULiège) Higher OctonionsKreusch, Marie Conference (2015, June 11)Detailed reference viewed: 15 (2 ULiège) Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.Kreusch, Marie Doctoral thesis (2015)This dissertation consists of two parts. The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras$\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 ▲]Detailed reference viewed: 306 (39 ULiège) Noncommutative and nonassociative algebrasKreusch, Marie 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 ▲]Detailed reference viewed: 13 (1 ULiège) Algebras generalizing the OctonionsKreusch, Marie 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 ▲]Detailed reference viewed: 31 (9 ULiège) Tour d’horizon sur des algèbres généralisant les octonionsKreusch, Marie 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 ▲]Detailed reference viewed: 47 (16 ULiège) Au delà des octonionsKreusch, Marie 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 ▲]Detailed reference viewed: 25 (7 ULiège) Au delà des octonionsKreusch, Marie 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 ▲]Detailed reference viewed: 47 (2 ULiège) Au delà des octonionsKreusch, Marie 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 ▲]Detailed reference viewed: 15 (2 ULiège) Une généralisation des octonionsKreusch, Marie Scientific conference (2013, July 05)Voir la pièce jointeDetailed reference viewed: 19 (3 ULiège) Extensions of superalgebras of Krichever-Novikov typeKreusch, Marie in Letters in Mathematical Physics (2013), 103(11), 1171-1189An 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 ▲]Detailed reference viewed: 106 (38 ULiège) Au delà des nombres réelsKreusch, Marie Poster (2013, June)Detailed reference viewed: 60 (13 ULiège) Extensions of Superalgebras of Krichever-Novikov typeKreusch, Marie 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 ▲]Detailed reference viewed: 54 (19 ULiège) Extensions of superalgebras of Krichever-Novikov typeKreusch, Marie Conference (2013, January 15)Detailed reference viewed: 35 (12 ULiège) Lie antialgebrasKreusch, Marie 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 ▲]Detailed reference viewed: 23 (0 ULiège) Lie antialgebrasKreusch, Marie 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 ▲]Detailed reference viewed: 18 (0 ULiège) 1