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, f ′, f ′′) ∈ HomR(A, A) | f(xy) = f ′(x)y + xf ′′(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∗). Mathematics Subject Classification (2010): 16W25, 16T15


Introduction
Throughout the following, R is a commutative ring with an identity 1, A an associative R-algebra and C a coassociative R-coalgebra.We do not assume that A has an identity 1 A and C has a counit ε : C → R. For any R-modules X and Y , we denote the set of R-linear maps from X to Y by Hom(X, Y ) and the symbol ⊗ means the tensor product ⊗ R over R. For an A-bimodule M , an R-linear map Dedicated to the memory of Prof. Manabu Harada.d : A → M is called a derivation if d(xy) = d(x)y+xd(y) for any x, y ∈ A, and there are several variations on this concept.For examples, f ∈ Hom(A, M ) is called a B-derivation (i.e., Bresar's derivation, cf.[1] and [8]) if there exists a derivation d ∈ Hom(A, M ) such that f (xy) = f (x)y + xd(y), and f is called an N-derivation if there exists an element m ∈ M such that f (xy) = f (x)y + xf (y) + x(my) (cf.[8] and [12]).These sets of derivations, B-derivations and N-derivations from A to M are denoted by Der(A, M ), BDer(A, M ) and NDer(A, M ), respectively.The properties of these derivations were discussed in many papers.
An R-linear map f : A → M is called a generalized derivation if there exist elements f , f ∈ Hom(A, M ) such that f (x)y + xf (y) = f (xy).This concept was introduced by G. F. Leger and E. M. Luks in a non-associative algebra in [10], and many properties of the generalized derivations for Lie algebras were given.Since f and f are not uniquely determined by f , we define that a triple (f, f , f ) ∈ Hom(A, M ) 3 is called a triple derivation if f (xy) = f (x)y + xf (y), and denote the set of triple derivations from A to M by TDer(A, M ) (cf. [8]).It is easy to see that the derivations, B-derivations and N-derivations are represented by (d, d, d), (f, f, d) and (f, f, f + m ) as triple derivations, respectively, and f + m is a derivation, where m (x) = mx.We have the following relations for these Rmodules: Der(A, M ) ⊆ NDer(A, M ) ⊆ BDer(A, M ) ⊆ TDer(A, M ) ⊂ Hom(A, M ) 3 , where Hom(A, M ) 3 = Hom(A, M ) × Hom(A, M ) × Hom(A, M ) is the direct product of R-module Hom(A, M ).
Let C be an R-coalgebra with a comultiplication ∆ : C → C⊗C.An R-module N is called a C-bicomodule if there exist C-comodule structure maps ρ + : N → N ⊗ C and ρ − : N → C ⊗ N such that the following relations hold: where the letter I always stands for the identity map (here, the identity map C → (cf. [4] and [11]).The notion of a coderivation is also extended as follows.An R-linear map f : N → C is called a B-coderivation if there exists a coderivation 105 and f is called an N-coderivation if there exists an R-linear map ξ : The middle sign ⊆ is proven by Lemma 3.1.These R-modules are also R-submodules The properties of these coderivations were discussed in [9] and [15].Note that when In Section 2, we treat an associative R-algebra A and an A-bimodule M .Writing BiL(A, M ) for the set of R-bilinear maps from A × A → M , we consider the following R-module as the direct product of R-modules Hom(A, M ) and BiL(A, M ): and define Then we see that Ω(A) = Ω(A, A) and TDer(A) = TDer(A, A) have Lie algebra structures such that ϕ A : TDer(A) → Ω(A) is a Lie algebra homomorphism.Moreover, we show that the set of generalized Lie (resp.Jordan) derivations GLDer(A) (resp.GJDer(A)) from A to A is also a Lie subalgebra of Ω(A).In the final of Section 2, we discuss the subset of BiL(A, M ) consisting of the biderivations in the sense of [2] and [17].
In Section 3, for a coassociative R-coalgebra C and a C-bicomodule N , we define the direct product of R-modules Hom(N, C) and Hom(N, C ⊗ C) Then we show that results similar to those from Section 2 hold.Moreover, since such that the following diagram is commutative: where ψ N is defined in Lemma 3.2.Especially, if N = C, then ψ N and ϕ N * are Lie algebra homomorphisms whereas θ 0 and θ 1 are anti-Lie algebra homomorphisms.

The case of algebras
In this section, A is an associative R-algebra and the letters x, y, z denote arbitrary elements in A. M is an A-bimodule, that is, M is an R-module, and a left and a right A-module such that (cf.[3, (72.13) and (72.14)] and [14]).Although f is not uniquely determined by α, we consider the set of above pairs, and denote it by As is easily seen, Λ(A, M ) is an R-submodule of Ω(A, M ), and for any (f, f , f ) ∈ TDer(A, M ), we have a bilinear map Then we have the following. (2.1) Then the map Especially, if M is a unital A-bimodule, then ϕ M is a monomorphism on BDer(A, M ).
Proof.We note that for any (f, f , f ) ∈ TDer(A, M ), α(f ) is a split factor set by shows that ϕ M is a monomorphism on BDer(A, M ).
For any triple derivations (f, f , f ), (g, g , g ) ∈ TDer(A) = TDer(A, A), gives a Lie algebra structure on TDer(A).It is easy to see that BDer(A) = BDer(A, A) is a Lie subalgebra of TDer(A) by the above operation.Now, we define a Lie algebra structure on Ω(A) = Ω(A, A), and show that the map ϕ A defined by (2.2) is a non-trivial Lie algebra homomorphism.
Let (f, α) be in Ω(A) and define an R-bilinear map α f by that is, First, we have the following.
Therefore, it is enough to show that the Jacobi identity holds.By the associativity of R-linear maps f , g, h ∈ Hom(A, A), the first And by (2.4), since ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS 109 then by Lemma 2.2, the second component of ( * ) is Therefore Ω(A) is a Lie algebra.
Finally, we show that ϕ A is a Lie algebra homomorphism.Let (f, f , f ) and (g, g , g ) be in TDer(A).Then by and are triple derivations, then by (2.1) and (2.4), we see and so Symmetrically, we see + f (x)g(y) + xf g(y).
Hence we have Therefore ϕ A is a Lie algebra homomorphism.
A bilinear map α ∈ BiL(A, M ) is called symmetric (resp.skew symmetric) if α(x, y) = α(y, x) (resp.α(x, y) = −α(y, x)) for any x, y ∈ A. We denote the set of symmetric (resp.skew symmetric) bilinear maps from A × A to M by BiL Sy (A, M ) (resp.BiL sSy (A, M )).Define These sets are R-submodules of Ω(A, M ) and we have the following R-submodules of Λ(A, M ): Moreover, for any (f, α), (g, In [12], we showed that the set of N-derivations then we see by (2.1) as a triple derivation.Thus the Lie algebra structure of NDer(A) defined by (2.5) is the same as (2.1).
It is known the following types of derivations which are not triple derivations.
Let G be a multiplicative subsemigroup of the set of R-algebra endomorphisms of for some σ, τ ∈ G.It is a generalization of (σ, τ )-derivation.The properties of (σ, τ )-derivations were discussed in many papers (cf.[5]).We denote the set of G-derivations by Der G (A, M ).A G-derivation is not a triple derivation.
Another one is as follows.g ∈ Hom(A, M ) is called a right derivation if g(xy) = g(x)y + g(y)x.The right derivations relate to the quasi-separable or differentially separable extension A over R (cf.[7] and [16]).We denote the set of right derivations by RDer(A, M ).As is easily seen, the above derivations are not triple derivations, but if we define α(f )(x, y) = f (xy)−f (x)y −xf (y) and β(g)(x, y) = g(y)x−xg(y), the equality α(f )(x, y) = f (x)(σ(y)−y)+(τ (x)−x)f (y) is then an easy consequence of this definition and by Proof.As is easily seen, GJDer(A, M ) and GLDer(A, M ) are R-submodules of Ω(A, M ).
Thus we have ) is a generalized Lie derivation.These show that GJDer(A) and GLDer(A) are Lie subalgebras of Ω(A).
In the final part of this section, we study biderivations from if B is a symmetric (resp.skew symmetric) bilinear map.We denote the sets of biderivations, symmetric biderivations and skew symmetric biderivations from A × A to M by BiDer(A, M ), SyBiDer(A, M ) and sSyBiDer(A, M ), respectively.
It is known that if A is commutative and f , g : A → A are derivations, then the map f • g defined by (f • g)(x, y) = f (x)g(y) is a biderivation, and thus f • f is a symmetric biderivation.Moreover, the map [−, −] : is a skew symmetric biderivation.The properties of biderivations and symmetric biderivations were discussed in [2], [6] and [17].Now, we consider the following R-module that is, This means that α g is not a biderivation in general.Similarly, we have and so and so which shows that α g − β f is not a biderivation, too.Therefore, Λ B (A) is not a Lie subalgebra of Λ(A).By these calculations, we have the following.
By the above calculations and using that α and β are symmetric, we have which shows that (α g − β f ) is a biderivation.
We can easily check the following.the assumption in Theorem 2.6.

The case of coalgebras
Let C be a coassociative R-coalgebra with a comultiplication ∆ : C → C ⊗ C.   Proof.Since (f, f, f + (ξ ⊗ I)ρ + ) is an N-coderivation, then by (1.3), we see and thus Assume that C has a counit ε : This means that a triple coderivation (f, f 1 , f 2 ) is an N-coderivation.Then we have which shows that ψ C is a Lie algebra homomorphism.
In [13], we showed that if f : N → C is an N-coderivation such that ).We finally give some
from A to A is a Lie algebra by the following operation[(f, a), (g, b)] = ([f, g], f (b) − g(a)) (a, b ∈ A).(2.5)Since an N-derivation (f, a) is represented by (f, f, f + a ) as a triple derivation and the Lie algebra structure of TDer(A) is given by Der G (A, M ) and RDer(A, M ) are R-submodules of Λ(A, M ).In general, Der G (A) = Der G (A, A) and RDer(A) = RDer(A, A) are not Lie subalgebras of Λ(A).If we assume that G is commutative and σd = dσ for any σ ∈ G and d ∈ Der G (A), then Der G (A) is a Lie subalgebra of Λ(A).Similarly, if [f (x), g(y)] + [f (y), g(x)] = 0 for any f , g ∈ RDer(A) and x, y ∈ A, then RDer(A) is also a Lie subalgebra of Λ(A).But we have not good examples for G-derivations and right derivations which satisfy the above relations.A pair (f, α) ∈ Ω(A, M ) is called a generalized Jordan derivation iff (x 2 ) = f (x)x + xf (x) + α(x, x) for any x ∈ A,(2.6)and similarly, (f, α) is called a generalized Lie derivation iff ([x, y]) = [f (x), y] + [x, f (y)] + α(x, y) − α(y, x) for any x, y ∈ A. (2.7)These notions were introduced in [13] (cf.[14]) and some properties of them weregiven.The sets of generalized Jordan derivations and generalized Lie derivations from A to M are denoted by GJDer(A, M ) and GLDer(A, M ), respectively.Then we have the following.Theorem 2.5.GJDer(A) = GJDer(A, A) and GLDer(A) = GLDer(A, A) are Lie subalgebras of Ω(A) by the operation (2.4).