Derivations of Skew Polynomial Rings Original Research Article

Let F be a field of characteristic 0 and let λi,j set membership, variant F. for 1 ≤ i,j ≤ n. Define R = F[image1,image2,..., imagen] to be the skew polynomial ring with imageiimagej = λi,jimagejimagei and let S = F[image1,image2,...,imagen, image−11, image−12,...,image−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].