Exceptional Periodicity and Magic Star Algebras. II : Gradings and HT-Algebras

Truini, Piero; Marrani, Alessio; Rios, Michael

doi:10.48550/arxiv.1910.07914

Cited by 1 publication

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“…It was later realized that the Magic Star projection had been actually envisaged almost ten years before by Mukai, which called it 'G 2 decomposition' [98] and stressed its relation to Legendre varieties. Truini's for mulation is based on pairs of Jordan algebras of degree three (endowed with an inner product and named Jordan pairs [14]) [97]; related algebraic structures were subsequently investigated in [99][100][101][102][103][104][105]. Also, the Magic Star projection and Jordan Pairs were exploited into a mathematical description of the fundamental interactions of elementary particles, as well as for an axiomatic formulation of a consistent theory of quantum gravity, firstly in [97], and then in [106] and in [103]; more recently, this led to the formulation of a quantum model for the Universe at its early stages, starting from an initial quantum state and driven by E 8 interactions [107].…”

Section: Introductionmentioning

confidence: 99%

Jordan algebraic interpretation of maximal parabolic subalgebras: exceptional Lie algebras

Dobrev¹,

Marrani²

2020

J. Phys. A: Math. Theor.

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With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras. In this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact real forms of finite-dimensional exceptional Lie algebras, in particular focusing on Jordan algebras of rank 2 and 3.

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