Abstract :
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∗).
