A generalization of the Kantor-Koecher-Tits construction

TL;DR

This paper extends the Kantor-Koecher-Tits construction to include Lie algebra extensions, enabling the realization of larger and exceptional Lie algebras from Jordan algebras, with broad implications for algebraic structures.

Contribution

It generalizes the construction to encompass extensions of Lie algebras and produces new realizations of exceptional Lie algebras from Jordan algebras over division algebras.

Findings

Generalized conformal realization of so(p+1,q+1) to so(p+n,q+n)

Constructed exceptional Lie algebras f4, e6, e7, e8 from 3x3 matrices over division algebras

Extended the construction to affine, hyperbolic, and further algebraic extensions

Abstract

The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p+1,q+1) generalizes to so(p+n,q+n), for arbitrary n, with a linearly realized subalgebra so(p,q). We also show that the construction applied to 3x3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.