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Triality
Post-edited
Authors : Elduque, Alberto (Author of the conference)
CIRM (Publisher )
17A75 - Composition algebras
17B60 - Lie (super)algebras associated with other structures (associative, Jordan, etc.)
20G15 - Linear algebraic groups over arbitrary fields
CIRM (Publisher )
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| geometric triality composition algebra symmetric composition algebra Okubo algebras algebraic triality Freudenthal magic square |
Abstract :
Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.
Keywords : Triality; composition algebra; symmetric; exceptional Lia algebras
MSC Codes : Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.
Keywords : Triality; composition algebra; symmetric; exceptional Lia algebras
17A75 - Composition algebras
17B60 - Lie (super)algebras associated with other structures (associative, Jordan, etc.)
20G15 - Linear algebraic groups over arbitrary fields
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 06/12/2019 Conference Date : 12/11/2019 Subseries : Research talks arXiv category : Group Theory ; Rings and Algebras Mathematical Area(s) : Algebra Format : MP4 (.mp4) - HD Video Time : 00:39:36 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-011-12_Elduque.mp4 
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Information on the Event
Event Title : Workshop on Differential Geometry and Nonassociative Algebras / Colloque en géométrie différentielle et algèbres non associativesEvent Organizers : Albuquerque, Helena ; Benayadi, Saïd ; Boucetta, Mohamed ; Makhlouf, Abdenacer Dates : 12/11/2019 - 16/11/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/2076.html
Citation Data
DOI : 10.24350/CIRM.V.19577803Cite this video as: Elduque, Alberto (2019). Triality. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19577803 URI : http://dx.doi.org/10.24350/CIRM.V.19577803 |
See Also
- [Multi angle] Closed $G_{2}$-structures / Author of the conference Fino, Anna.
- [Multi angle] Finite quantum geometry, exceptional quantum geometry and fundamental particles / Author of the conference Dubois-Violette, Michel.
- [Multi angle] On the geometry of noncommutative deformations / Author of the conference Boucetta, Mohamed.
- [Multi angle] Noncommutative localization in smooth Deformation quantization / Author of the conference Bordemann, Martin.
Bibliography
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- CHERNOUSOV, Vladimir, ELDUQUE, Alberto, KNUS, Max-Albert, et al. Algebraic groups of type D4, triality, and composition algebras. Documenta Math, 2013, vol. 18, p. 413-468. - http://emis.ams.org/journals/DMJDMV/vol-18/17.pdf
- CUNHA, Isabel et ELDUQUE, Alberto. An extended Freudenthal magic square in characteristic 3. Journal of Algebra, 2007, vol. 317, no 2, p. 471-509. - https://doi.org/10.1016/j.jalgebra.2007.07.028
- ELDUQUE, Alberto. Symmetric composition algebras. Journal of Algebra, 1997, vol. 196, no 1, p. 283-300. - https://doi.org/10.1006/jabr.1997.7071
- ELDUQUE, Alberto. The magic square and symmetric compositions. Revista Matemática Iberoamericana, 2004, vol. 20, no 2, p. 475-491. - https://projecteuclid.org/euclid.rmi/1087482023
- KNUS, Max-Albert, et al. The book of involutions. American Mathematical Soc., 1998. -
- KNUS, Max-Albert et TIGNOL, Jean-Pierre. Triality and algebraic groups of type 3D4. Doc. Math, 2015, p. 387-405. - http://emis.ams.org/journals/DMJDMV/vol-merkurjev/knus_tignol.pdf
- TITS, Jacques. Sur la trialité et certains groupes qui s'en déduisent. Publications Mathématiques de l'IHÉS, 1959, vol. 2, p. 13-60. - http://www.numdam.org/article/PMIHES_1959__2__13_0.pdf
- VAN DER BLIJ, Frederik et SPRINGER, Tonny Albert. Octaves and triality. Nieuw Arch. Wisk, 1960, vol. 3, no 8, p. 158-169. -
