Research Projects

Let R be a commutative ring with 1. Let A be an R-algebra and N be an A − A-bimodule. Then an R-linear map γ : A → N is called a derivation if γ(ab) = γ(a)b + aγ(b) for all a, b ∈ A. A derivation is inner if there exists x ∈ N , such that γ(a) = xa − ax for all a ∈ A. Hence a derivation is a representative of an element in the degree 1 Hoschild cohomology H1(A, N ) and it is inner if it represents the zero element in H1(A, N ). Let σ, τ be two different algebra endomorphisms on A. Then a (σ, τ )-derivation is an R-linear map δ : A → σAτ satisfying δ(ab) = δ(a)τ(b)+σ(a)δ(b) for a, b ∈ A. So a (σ, τ)-derivation is the special case of a derivation with the twisted bimodule σAτ in the role of N . Here σAτ is A as R-space and on it a acts on the left as multiplication by σ(a) and on the right by multiplication by τ(a). If there exists x ∈ A such that the (σ, τ )-derivation δx : A → σAτ is of the form δx(a) = xτ (a) − σ(a)x, then δx is called a (σ, τ )-inner derivation of A induced by x. If σ = τ = id, then δ and δx are respectively the usual derivation and inner derivation of A induced by x. The aim of this project is to study twisted derivations on different algebras and their applications. We shall address the question of whether every twisted derivation in an algebra is twisted inner or not. Automorphisms of an algebra are extremely difficult objects of study and for most algebras very little is known about the automorphisms. Hence the study of twisted derivations also become quite challenging. Such twisted derivations have important use in different areas of Mathematics, like generalizing Galois theory over division rings and study of q-difference operators in number theory just to name a few. In 2006, Hartwig, Larsson and Silvestrov gave the study of (σ, τ)-derivations a new dimension when they generalized Lie algebras to hom-Lie algebras using (σ, τ)-derivations on associative algebras A over the field of Complex numbers. Just as Lie algebras were initially studied as algebras of derivations, their idea was to introduce a generalization of the structure of Lie algebras to hom-Lie algebras which can be seen as the algebra of twisted derivations with some added natural conditions. Such structures are used to study deformations and discretizations of vector fields that have widespread applications to quantum physics and complex dynamical systems. Our main aim is to study hom-Lie structures on group algebras. For that we shall study (σ, τ)-derivations of group algebras first. The study of such derivations on group algebras can be quite tough and intriguing and shall open new doors of correlating this area with the well known and important Zassenhaus Conjectures. Then we shall focus on coding theory applications of twisted derivations on finite modular group algebras. Finally we shall deal with representations of hom-Lie algebras.