Generalized Conformal and Superconformal Group Actions and Jordan Algebras

TL;DR

This paper explores the extension of conformal groups to Jordan algebras and superalgebras, providing oscillator realizations and classifying conformal algebras for various Jordan structures.

Contribution

It introduces a generalized framework for conformal groups associated with Jordan algebras and superalgebras, including oscillator realizations and classification results.

Findings

Generalized conformal groups for Jordan algebras established

Oscillator realizations of these groups constructed

Classification of conformal algebras for simple Jordan structures provided

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.