DERIVATIONS OF SKEW POLYNOMIAL-RINGS

Let F be a field of characteristic 0 and let lambda(i,j) is an element of F for 1 less than or equal to i,j less than or equal to n. Define R = F[<(X)over bar (1)>,<(X)over bar (2)>..., <(X)over bar (n)>] to be the skew polynomial ring with <(X)over bar (i)> <(X)over bar (j)> lam bda(i,j)<(X)over bar (j)><(X)over bar (i)> and let S = F[<(X)over bar (1)> <(X)over bar (2)>,...,<(X)over bar (n)>, <(X)over bar (-1)(1)>, < (X)over bar (-1)(2)>,...,<(X)over bar (-1)(n)>] be the corresponding L aurent polynomial ring. In a recent paper, Kirkman, Procesi, and Small considered these two rings under the assumption that S is simple and showed, for example, that the Lie ring of inner derivations of S is si mple. Furthermore, when n = 2, they determined the automorphisms of S, related its ring of inner derivations to a certain Block algebra, and proved that every derivation of R is the sum of an inner derivation a nd a derivation which sends each x(i) to a scalar multiple of itself. In this paper, we extended these results to a more general situation. Specifically, we study twisted group algebras F-t[G] where G is a comm utative group and F is a field of any characteristic. Furthermore, we consider certain subalgebras F-t[H] where H is a subsemigroup of G whi ch generates G as a group. Finally, if e: G x G --> F is a skew-symmet ric bilinear form, then we study the Lie algebra F-e[G] associated wit h e, and we consider its relationship to the Lie structure defined on various twisted group algebras F-t[G]. (C) 1995 Academic Press, Inc.