Reducibly free nonspecial Jordan algebras

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

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Nilpotency of Alternative and Jordan Algebras

Siberian Mathematical Journal ◽

10.1134/s003744662101016x ◽

2021 ◽

Vol 62 (1) ◽

pp. 148-158

Author(s):

A. V. Popov

Keyword(s):

Jordan Algebras

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SYMMETRIES OF CERTAIN PHYSICAL THEORIES AND THE JORDAN ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92001617 ◽

1992 ◽

Vol 07 (15) ◽

pp. 3623-3637 ◽

Cited By ~ 1

Author(s):

R. FOOT ◽

G. C. JOSHI

Keyword(s):

Jordan Algebra ◽

Space Time ◽

Jordan Algebras ◽

Bosonic String ◽

The Other ◽

Structure Group ◽

Division Algebras ◽

Hermitian Matrices ◽

Higher Dimensional ◽

Algebraic Framework

It is shown that the sequence of Jordan algebras [Formula: see text], whose elements are the 3 × 3 Hermitian matrices over the division algebras ℝ, [Formula: see text], ℚ and [Formula: see text], can be associated with the bosonic string as well as the superstring. The construction reveals that the space–time symmetries of the first-quantized bosonic string and superstring actions can be related. The bosonic string and the superstring are associated with the exceptional Jordan algebra while the other Jordan algebras in the [Formula: see text] sequence can be related to parastring theories. We then proceed to further investigate a connection between the symmetries of supersymmetric Lagrangians and the transformations associated with the structure group of [Formula: see text]. The N = 1 on-shell supersymmetric Lagrangians in 3, 4 and 6-dimensions with a spin 0 field and a spin 1/2 field are incorporated within the Jordan-algebraic framework. We also make some remarks concerning a possible role for the division algebras in the construction of higher-dimensional extended objects.

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