Parametrization of a Conjugacy Class of the Special Linear Group

Abstract

A local description of the foliation of the group SL(n) into conjugacy classes, and also the foliation of 𝔰𝔩∗(n) into coadjoint orbits, requires introducing parameters on a conjugacy class (a coadjoint orbit). Under the assumption that the parameters are rational functions of natural coordinates (the matrix elements) on SL(n), the problem is reduced to solving a system of linear equations. This system arises from the requirement that the parameters be invariant with respect to translations along vector fields normal to the conjugacy class. Similarly, the problem of parametrization of coadjoint orbits in 𝔰𝔩∗(n) can be solved using the Cartan–Weyl basis for 𝔰𝔩(n). The adjoint action is the differential of the conjugation action. Consequently, the parameters on the conjugacy classes and the coadjoint orbits are related by a transformation determined by the mapping from the algebra (n) to the group SL(n). The groups SL(3), SL(4) are considered as examples. Bibliography: 13 titles.