ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS an‎d THEIR DUALS

Let A be an associative algebra over a commutative ring R, BiL(A) the set of R-bilinear maps from A × A to A, and arbitrarily elements x, y in A. Consider the following R-modules: Ω(A) = {(f, α) | f ∈ HomR(A, A), α ∈ BiL(A)}, TDer(A) = {(f, f0 , f00) ∈ HomR(A, A) 3 | f(xy) = f 0 (x)y + xf00(y)}. TDer(A) is called the set of triple derivations of A. We define a Lie algebra structure on Ω(A) and TDer(A) such that ϕA : TDer(A) → Ω(A) is a Lie algebra homomorphism. Dually, for a coassociative R-coalgebra C, we define the R-modules Ω(C) and TCoder(C) which correspond to Ω(A) and TDer(A), and show that the similar results to the case of algebras hold. Moreover, since C∗ = HomR(C, R) is an associative R-algebra, we give that there exist anti-Lie algebra homomorphisms θ0 : TCoder(C) → TDer(C∗) and θ1 : Ω(C) → Ω(C∗) such that the following diagram is commutative : TCoder(C) ψC −−−−−→ Ω(C)   yθ0   yθ1 TDer(C∗) ϕC∗ −−−−−→ Ω(C∗).