Generalized Conformal and Superconformal Group Actions and Jordan Algebras

TL;DR

The paper addresses generalized conformal symmetry by extending the usual quadratic invariance to spaces endowed with a degree $p$ norm via Kantor's $p$-angles on Jordan algebras. It constructs oscillator realizations of the generalized conformal groups and their Jordan triple system extensions, yielding both algebraic and differential-operator representations on the associated spaces and superspaces. A complete list of generalized conformal algebras for simple Jordan algebras and hermitian JTSs is provided, with parallel extensions to Jordan superalgebras and super JTSs following Kac's classification. The work employs the TKK construction to relate grade $+1$ generators to the underlying Jordan structures and derives explicit $(G,H)$ pairs, including exceptional cases. These results offer robust algebraic and analytic tools for applying generalized conformal and superconformal symmetry in higher-dimensional and supersymmetric physical contexts.

Abstract

We study the conformal groups of Jordan algebras along the lines suggested by Kantor. They provide a natural generalization of the concept of conformal transformations that leave 2-angles invariant to spaces where "p-angles" can be defined. We give an oscillator realization of the generalized conformal groups of Jordan algebras and Jordan triple systems(JTS). These results are extended to Jordan superalgebras and super JTS's. We give the conformal algebras of simple Jordan algebras, hermitian JTS's and the simple Jordan superalgebras as classified by Kac.