Derivations of Skew Polynomial Rings

• Published in: Journal of algebra 1995-09, Vol.176 (2), p.417-448
• Main Authors: Osborn, J.M., Passman, D.S.
• Format: Article
• Language: English
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1006/jabr.1995.1252

Let F be a field of characteristic 0 and let λi,j ∈ F. for 1 ≤ i,j ≤ n. Define R = F[x1,x2,..., xn] to be the skew polynomial ring with xixj = λi,jxjxi and let S = F[x1,x2,...,xn, x−11, x−12,...,x−1n] be the corresponding Laurent 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 simple. 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 and a derivation which sends each xi to a scalar multiple of itself. In this paper, we extended these results to a more general situation. Specifically, we study twisted group algebras Ft[G] where G is a commutative group and F is a field of any characteristic. Furthermore, we consider certain subalgebras Ft[H] where H is a subsemigroup of G which generates G as a group. Finally, if e: G × G → F is a skew-symmetric bilinear form, then we study the Lie algebra Fe[G] associated with e, and we consider its relationship to the Lie structure defined on various twisted group algebras Ft[G].