Generalized conformal realizations of Kac-Moody algebras
arXiv:0711.0441 · doi:10.1063/1.3063628
Abstract
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of hermitian 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n=2. Moreover, we obtain their infinite-dimensional extensions for n greater or equal to 3. In the case of 2x2 matrices the resulting Lie algebras are of the form so(p+n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
arXiv comment24 pages, v2: abstract and introduction rewritten, contents rearranged, former section 4 moved to an appendix
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- Exceptional Lie algebras and M-theory
- Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras
- A generalization of the Kantor-Koecher-Tits construction