A generalization of the Kantor-Koecher-Tits construction

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 3 ◊ 3 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 ane, hyperbolic and further extensions. 2000 MSC: 17Cxx This article aims to give a brief overview of results that have already been shown by the author in [7], where details and a more comprehensive list of references are provided. We will mostly consider algebras over the real numbers, even though we will complexify real Lie algebras in order to study properties of the corresponding Dynkin diagrams. However, algebras are assumed to be real if nothing else is stated. A Jordan algebra is a commutative algebra that satisfies the Jordan identity a 2 ‐ (b ‐ a) = (a 2 ‐ b) ‐ a (1) The symmetric part of the product in an associative but noncommutative algebra, 2(a ‐ b) = (ab + ba)