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Authors : Elduque, Alberto (Author of the conference)
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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 :
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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https://videos.cirm-math.fr/2019-011-12_Elduque.mp4

Information on the Event

Event Title : Workshop on Differential Geometry and Nonassociative Algebras / Colloque en géométrie différentielle et algèbres non associatives
Event 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.19577803
Cite 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

Bibliography

CARTAN, Élie. Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. des Sci. Math., 1925, vol. 49, p. 361-374. - https://gallica.bnf.fr/ark:/12148/bpt6k486276p/f141.image.r=le%20principe%20de%20dualit%C3%A9#

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. -

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