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A Generalization of the Kantor-Koecher-Tits Construction
並列摘要
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 affine, hyperbolic and further extensions.