A NONEXISTENCE THEOREM FOR REALIZATIONS OF SEMI-SIMPLE LIE ALGEBRAS

Realizations of Lie algebras in terms of real functions of n pairs of conjugate variables are considered. It is shown that the existence of a realization of a finite-dimensional, semi-simple Lie algebra restricts the rank of the algebra to a value less than or equal to n. An extension of this result to symmetry groups is proved. After some brief historical remarks, the concepts of transformation and symmetry groups in classical mechanics are introduced in Sect. 1 Section 2 deals with realizations of Lie algebras and is a preparation for an understanding of the main theorem which is dealt with in Sect. 3 In this Section the complex extension of the Lie algebra is discussed and used to prove the main theorem. Section 3 concludes with a corollary and some technical remarks. A brief summary and a discussion of the importance of the corollary in the theory of symmetry groups in classical mechanics is given in Sect. 4. A synopsis of the notation used follows at the end of the paper. © 1969 Società Italiana di Fisica.