AUTOMORPHISMS AND DERIVATIONS OF CERTAIN SOLVABLE LIE ALGEBRAS OVER COMMUTATIVE RINGS

Let L be a finite-dimensional complex simple Lie algebra, L-Z be the Z-span of a Chevalley basis of L, and L-R = R circle times(Z) L-Z be a Chevalley algebra of type L over a commutative ring R with identity. Let B(R) be the solvable subalgebra of L-R spanned by the basis elements of the maximal toral subalgebra and the root vectors associated with positive roots. In this article, we prove that under some conditions for R, any automorphism of B(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of B(R) is uniquely decomposed as a sum of an inner derivation induced by root vectors and a diagonal derivation. Correspondingly, the automorphism group and the derivation algebra of B(R) are determined.