STRONG COMMUTATIVITY PRESERVING MAPS OF UPPER TRIANGULAR MATRIX LIE ALGEBRAS OVER A COMMUTATIVE RING

Let R be a commutative ring with identity 1, n >= 3, and let T-n(R) be the linear Lie algebra of all upper triangular n x n matrices over R. A linear map phi on T-n(R) is called to be strong commutativity preserving if [phi(x), phi(y)] = [x, y] for any x, y is an element of T-n(R). We show that an invertible linear map phi preserves strong commutativity on T-n(R) if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on T-n(R).