Realizations of the Real Simple Lie Algebras: The Method of Construction

摘要

A method is presented for constructing skew-Hermitean realizations for any real semisimple Lie algebra g. The method is based on expansion of the Lie algebra g in the form of g=nsub(+)sup(b) x gsub(0)sup(b) x nsub(-)sup(b) which is a simple generalization of the triangular expansion, as well as on the representation g induced with the use of the representation sigma of the parabolic subalgebra gsub(0)sup(b) x nsub(-)sup(b). It is shown that each such representation could be transformed to realization (boson representation). It is proved that the method gives the realizations possessing two good properties which permit their application in the theory of representations. The realizations are skew-Hermitean and Schurean. As an example, the realizations for the algebra g1(3, R) are constructed in the explicit form. (Atomindex citation 16:069055)