BETWEEN (GL)OVER-CAP-(INFINITY) AND (SL)OVER-CAP-(N) AFFINE ALGEBRAS .1. GEOMETRICAL ACTIONS

We consider the central extended gl(infinity) Lie algebra and a set of its subalgebras parametrized by \q\ = 1, which coincides with the emb edding of the quantum tori Lie algebras (QTLA) in gl(infinity). For q( N) = 1 there exists an ideal, and a factor over this ideal is isomorph ic to an sl(N)(z) affine algebra. For a generic value q the correspond ing subalgebras are dense in gl(infinity). Thus, they interpolate betw een gl(infinity) and sl(N)(z). All these subalgebras are fixed points of automorphisms of gl(infinity). Using the automorphisms, we construc t geometrical actions for the subalgebras, starting from the Kirillov- Kostant form and the corresponding geometrical action for gl(infinity) .