Polar and Ol'shanski Decompositions 1. Symmetric Lie Algebras

The Cartan decomposition in a semisimple Lie group is a generalization of the polar decomposition of matrices. In this paper we consider an even more general setting in which one obtains an analogous decomposition. In the semisimple case, this decomposition was worked out in a seminal paper of G. I. Ol'shanski 13]. In this paper we give general necessary and suucient conditions for this decomposition to exist in arbitrary real nite dimensional Lie algebras and discuss various contexts and examples where this decomposition obtains, particularly examples related to contraction semigroups. In this section we review some elementary and standard properties of Lie algebras equipped with an involution. Let g be a Lie algebra equipped with a Lie algebra involution ; we call the pair (g;) a symmetric Lie algebra, or more brieey a symmetric algebra. We set g := fX 2 V : (X) = Xg; the eigenspaces for 1. The pair (g + ; g ?) is called the canonical decomposition of g. We have g = g + L g ? with projections from g to g given by (X) = 1 2 ? X (X). The next proposition summarizes straightforward equivalent formulations of a symmetric algebra in terms of the canonical decomposition and the involution ]. Proposition 1.1. Let (g;) be a symmetric algebra. (i) The canonical decomposition g = g + L g ? satisses g + ; g ] g and g ? ; g ] g. Conversely if a decomposition g = g + L g ? satisses these two conditions, then the corresponding involution (x + y) = x ? y for x 2 g + ; y 2 g ? is the unique involutive automorphism of g that yields the given decomposition. (ii) The mapping y 7 ! y ] = ] (y) is an involutive antiautomorphism. We have y = y ] , y 2 g ? and y ] = ?y , y 2 g +. Conversely given an involutive antiautomorphism y 7 ! y ] : g ! g, there exists a unique