Automorphisms and conjugacy of compact real forms of the classical infinite dimensional matrix Lie algebras

The object of this paper are the infinite dimensional matrix Lie algebras s 1(J, K), sp(J, K) and o(J, J, K), whose elements are matrices of infinite s ize with only finitely many non-zero entries in the field K of characterist ic zero. These Lie algebras are natural generalizations of the finite dimen sional simple Lie algebras s1(n, K), o(2n + 1, K), sp(n, K) and o(2n, K). T he automorphisms of the latter are well-known. The aim of this paper is to determine the automorphisms and the automorphism groups of the Lie algebras s1(J, K), sp(J, K) and o(J, J, K). The result of this specification enable s us furthermore to prove that all compact real forms of the Lie algebra s1 (J, C) are conjugate under automorphisms.