Lie triple derivations of dihedron algebra

Let K be a 2-torsion free unital ring and D ( K ) be dihedron algebra over K . In the present article, we prove that every Lie triple derivation of D ( K ) can be written as the sum of the Lie triple derivation of K , Jordan triple derivation of K , and some inner derivation of D . We also prove that a generalized Lie triple derivation ϱ : D ( K ) → D ( K ) associated with the Lie triple derivation h : D ( K ) → D ( K ) exists if ϱ can be represented in the form ϱ ( τ ) = h ( τ ) + λτ , where λ lies in the center of D ( K ) . We ﬁ nally conclude that to obtain the complete algebra of the Lie triple derivation and generalized Lie triple of D ( K ) , we ﬁ rst need to ﬁ nd the Lie triple derivation and Jordan triple derivation of K .


Introduction
In physics, Lie groups are the symmetry groups of some physical systems, and their Lie algebras, which are the space of tangent vectors near the identity of the Lie groups, may be considered infinitesimal symmetry motions.Thus, Lie algebras and their representations are used extensively in the solution of differential equations and various branches of theoretical physics.The triple derivation of Lie algebra is apparently a generalization of derivation and is an analogy of the triple derivation of associative algebra and Jordan algebra.It was introduced independently in [1] by Muller, where it was called pre-derivation.Muller proved that if G is a Lie group endowed with a bi-invariant semi-Riemannian metric and g is its Lie algebra, then the Lie algebra of the group of isometries of G fixing the identity element is a subalgebra of Lie triple derivations, TDer(g).Thus, the study of the algebra of triple derivations is interesting not only from the viewpoint of the algebra itself but also for its applications in the studies of Lie groups and Lie algebra.
Let D be a unital ring with a center denoted by Z(D).We denote the commutator (Lie product) and Jordan product of τ 1 , τ 2 by [τ 1 , τ 2 ] = τ 1 τ 2 − τ 2 τ 1 and τ 1 ○ τ 2 = τ 1 τ 2 + τ 2 τ 1 , respectively, for all τ 1 , τ 2 ∈ D. We say that the ring D is an F-algebra (F is a field) if D is an Fvector space equipped with a bilinear product.Fields of scalars can also be replaced by any ring to give a more general notion of algebra over a ring.An F-linear map, h: D → D, is said to be an F-derivation or simple derivation (F-Jordan derivation, respectively) if h(τ 1 τ 2 ) = h(τ 1 )τ 2 + τ 1 h(τ 2 ) (h(τ 2 ) = h(τ)τ + τh(τ), respectively), for all τ ∈ D. The space of all Fderivations is denoted by Der F (D).These maps appear in diverse areas of mathematics.For example, in the algebra of real-valued differentiable function on R n , the partial derivative operator with respect to any variable is an R-derivation.Similarly, for any differentiable manifold M, the Lie derivative with respect to any vector field is an example of R derivation on the algebra of differentiable functions over M. Derivations are also useful in the study of the interaction of particles in physics [2].
Let us consider a Lie algebra D equipped with a bilinear product [,].A linear map h: For an element α ∈ D, the mapping I α : D → D given by I α (τ) = τα − ατ for all τ ∈ D is called an inner derivation of D induced by α.This is the well-known inner derivation called the adjoint and is denoted usually by (ad α ).A linear map ϱ: D → D is known as a generalized Lie triple derivation if there exists a Lie triple derivation h: In [11], the authors defined the Lie triple system from the Lie algebra by the trilinear product [x, y, z] = [x, [y, z]].A Lie triple system for a Lie algebra (D(K), [., .]) is a pair (D(K), [., ., .]),where [., ., .]: A linear map h defined on a Lie triple system D is said to be a derivation of ] for all τ 1 , τ 2 , τ 3 ∈ D. Quite similar notions for the Jordan triple system and Jordan triple derivation are discussed in [13].Lie triple systems generically arise from Lie algebras.If we have a particular Lie algebra (g, [, ]), then the triple product on g can be given as [a, b, c] = [[a, b], c].Lie triple derivations have been used in the study of symmetric spaces [10].It also has some connection with the study of the Yang-Baxter equation [12].The Lie triple derivation and Lie triple system are related to each other [11].Hom-Lie triple systems endowed with a symmetric invariant nondegenerate bilinear form are called quadratic Hom-Lie triple systems.In [13], the authors introduced the notion of double extension of Hom-Lie triple systems to give an inductive description of quadratic Hom-Lie triple systems.Baklouti et al. studied semi-simple Jordan triple systems and proved that a Jordan triple system is semi-simple if and only if its Casimir operator is nondegenerate [14].A mapping g: D → D is called a commuting map on D if [g(τ), τ] = 0 holds for all τ ∈ D. A commuting map g of an associative algebra is said to be proper if it can be written as g(τ) = λτ + ](τ), where λ lies in the center of algebra and ] is a linear map with an image in the center of algebra.It is evident that every derivation happens to be a Lie derivation and every Lie derivation is a Lie triple derivation.However, generally, the converse does not hold.For example, let D be a derivation on an algebra A and g be an additive central mapping with g([A, A]) = 0, then D + g presents an example of Lie derivation, which is not necessarily a derivation.Let ST 3 (R) be a set of three by three strictly upper triangular matrices over R.Then, the map The study of rings with derivations is a kind of subject that undergoes monumental revolutions and has become the center of discussion nowadays.A ring K is called a semi-prime ring if κKκ (0) implies that κ = 0.A natural question arises in the context of the algebra of derivations and for its subalgebras-whether a Lie derivation defined on some Lie algebra is induced by an ordinary derivation.This question is briefly examined in different manners for different rings in [15,28].The several generic extensions of derivations, which are Lie derivations, Jordan derivations, Lie triple derivations, Jordan triple derivations, and generalized Lie triple derivations, have gained significant interest from researchers.It is exhibited in [25] that the Lie triple derivation of perfect, free Lie algebras and the Lie algebras of upper triangular block matrices is a derivation.In [26], it has been shown that every Jordan triple derivation with the condition of nonlinearity on triangular algebras is a derivation.In [24], it is established that in the case of a 2-torsion free semi-prime ring, any Jordan derivation on a Lie ideal L is a derivation.It is proved in [16] that any Jordan triple derivation of a 2-torsion free semi-prime ring is a derivation.It has been proven that near-rings with derivations satisfying certain relations are commutative rings [17].Shang also proved that a prime near-ring admitting generalized derivations with some conditions is commutative.It is further established that a prime near-ring which admits a nonzero derivation satisfying certain differential identities is a commutative ring [18].For further results relating to derivations on near-prime rings, please refer to [5][6][7][8][9].The concept of derivations was first extended to Lie triple derivations by Muller in [1].The meaningful results on Lie triple derivations of some important well-known algebras, such as unital algebras, algebras of strictly upper triangular matrices over some commutative ring, and parabolic subalgebras of simple Lie algebra, are given in [29,30], respectively.The article [27] contains the decomposition of generalized Lie triple derivations on Borel subalgebra in terms of a block diagonal matrix and a Lie triple derivation.The authors in [4] characterized the Lie triple derivations of the algebra of the tensor product of some algebra T and quaternion algebra.Ghahramani et al, in [21], gave some characterizations of the generalized derivation and generalized Jordan derivation of a ring of quaternion and, in [22], discussed the characterization of the Lie derivation and its natural generic extension of the quaternion ring.[17,18] discussed the derivations of prime near-rings and the commutativity of prime near-rings.Benkoic in [23] generalized the concept of Lie derivation to Lie n-derivations for triangular algebras.
Section 2 contains some minor details about the algebra under consideration (the dihedron algebra).Quaternion and dihedron share many algebraic aspects, but dihedron algebra has not been studied in great detail.The dihedron algebra has great significance in the networking of real-world entities and their relationships.Entities can be objects, situations, concepts, or events, and they are described with formal explanations that allow both computers and people's minds to process them.Despite the significance of dihedron algebra, it is less studied among researchers, unlike quaternion algebra.As we know, derivations and their variants are sources to produce new classes and subclasses of Lie algebras.Since the algebra of Lie derivations and Lie triple derivations of quaternion algebra is recently well understood [21,22], it is natural to ask about the algebra of Lie derivations and Lie triple derivations of dihedron algebra.This paper is devoted to the Lie triple derivation and generalized Lie triple derivation of the dihedron algebra.Section 3, which is the main part of this article, contains the results on the characterization of the Lie triple derivation and its natural extension, which we call the generalized Lie triple derivation of the dihedron ring D(K), over the unital 2-torsion free ring K.

Dihedron algebra (D(K))
In this part, we discuss the main aspects of Dihedron algebra denoted as D(K).Let K be a 2-torsion free unital ring.Set D(K) where ẽ0 , ẽ1 , ẽ2 , ẽ3 are the matrices given as with the following relations, (2.2) Like a ring of quaternions, we can see that commutativity does not hold in the case of basis elements, that is, ẽ1 .ẽ 2 ẽ3 and ẽ2 .ẽ 1 −ẽ 3 ≠ ẽ1 .ẽ 2 .So, it is clear that D(K) is a noncommutative unital ring.A typical dihedron can be represented in the form . Then, "by using" the usual addition and multiplication of matrices and "taking" the commutator as a Lie bracket, we can see that D(K) forms a unital noncommutative ring over K and a Lie algebra.From the discussion in the previous section, we can gather that (D(K), [, ,]) forms a Lie triple system.As we know the term center is used to denote the set of all those elements that commute with all other elements, the element ẽ0 clearly acts as the identity, that is, ẽ0 .ẽi ẽi .ẽ0 ẽi , for i = 1, 2, 3. So, the center of D is Z(D) K.ẽ 0 K.We call it the dihedron algebra because of the great similarity between the quaternion group of order eight and the dihedral group.It is well known that up to isomorphism, there are only two noncommutative groups of order eight: one is the dihedral group, and the other is the quaternion group.As far as the Lie algebra of quaternion over any ring is concerned, it is well-established and well-studied.However, dihedron algebra is relatively less studied.Although there are similarities between these algebras, they are non-isomorphic, so it is natural to discuss the algebras of Lie triple derivations of dihedrons in a detailed way.In recent years, it was confirmed by several authors that various physical covariance groups, namely SO(3), the Lorentz group, the group of the theory of general relativity, the Clifford algebra(biquaternions) SU(2), and the conformal group, can all be related to the quaternion group and dihedrons in modern algebra [19,20]. Let where the dot and cross products are defined as 3 Lie triple and generalized Lie triple derivations of dihedron algebra D(K) This section contains the characterization of the Lie triple derivations of dihedron algebra over K.In [21], theorem 3.1 characterizes that if S is a 2-torsion free ring, R = H(S) is a quaternion ring, then the derivation of R can be decomposed in terms of derivation of S and an inner derivation of R. Here, for D(K), we have the following result.
To prove conversely, let us recall the definition of the generalized Lie triple derivation.
Let us substitute τ 1 = τ 2 in Eq. 3.5.Then, we have Remark 3.1 implies that g is a commuting map.By using the expression g(τ) = ](τ) + λτ, it follows that ϱ has the form Since λ ∈ Z(D), we have In addition, by taking into account that ](D) ⊆ Z(D), we can rewrite Eq. 3.6 as This completes the proof.It is quite evident that setting λ = 0, we obtain that the generalized Lie derivation becomes a derivation.We can write it as Substitute τ 1 = e 1 = τ 3 and τ 2 = e 2 to obtain

Conclusion
The present article focuses on the general classes of Lie triple derivations and generalized Lie triple derivations for dihedron algebra.Unlike quaternion, dihedron algebra has not been deeply studied, at least from the viewpoint of derivations and their variants.We have computed the decomposition of Lie triple derivations and generalized Lie triple derivations of D(K) in terms of Lie triple derivation and Jordan triple derivation of K and some inner derivation of D. We have also proven that a generalized Lie triple derivation ϱ: D(K) → D(K) associated with the Lie triple derivation h: D(K) → D(K) exists if ϱ can be represented in the form ϱ(τ) = h(τ) + λτ, where λ lies in the center of D(K).These results are new and exhibit complete decomposition of algebras of Lie triple derivations and generalized Lie triple derivations of dihedron algebras.As dihedron algebras are used in the geometric aspects of four-dimensional Lie groups, our results can be helpful in understanding the geometry of these manifolds.This article contains the representation of the Lie triple derivation of the dihedron ring D over a 2-torsion free unital ring K in terms of Lie triple and Jordan triple derivations of K and inner derivations of D.
be two elements of D, where v and v* are the vector parts of d and d*, respectively.The dihedron product between d and d* is defined as follows:

Example 3 .
2. A generalized Lie triple derivation ϱ: D → D associated with the Lie triple derivation h: D → D can be represented in the form ϱ τ ( ) h τ ( ) + λτ.