Structure of simple multiplicative Hom-Jordan algebras

We study the structure of simple multiplicative Hom-Jordan algebras. We discuss equivalent conditions for multiplicative Hom-Jordan algebras to be solvable, simple, and semi-simple. Moreover, we give a theorem on the classification of simple multiplicative Hom-Jordan algebras and obtain some propositions about bimodules of multiplicative Hom-Jordan algebras.


Introduction
Algebras where the identities defining the structure are twisted by a homomorphism are called Hom-algebras. These algebras have recently been investigated by many authors. The theory of Hom-algebras started from Hom-Lie algebras introduced and discussed in [6,10,11,12]. Hom-associative algebras were introduced in [15], while Hom-Jordan algebras were introduced in [14] as twisted generalization of Jordan algebras.
In recent years, vertex operator algebras are becoming more and more popular because of their importance. In [9], by using the structure of Heisenberg algebras, Lam constructed a vertex operator algebra such that the weight two space V 2 ∼ = J for a given simple Jordan algebra J of type A, B or C over C. In [2], Ashihara gave a counterexample to the following assertion: If R is a subalgebra of the Griess algebra, then the weight two space of the vertex operator subalgebra VOA(R) generated by R coincides with R by using a vertex operator algebra associated with the simple Jordan algebra of type D. H. B. Zhao constructed simple quotients V J ,r for r ∈ Z =0 using dual-pair type constructions, whereV J ,r is such that (V J ,r ) 0 = C1, (V J ,r ) 1 = {0}, and (V J ,r ) 2 is isomorphic to the type B Jordan algebra J . Moreover, in his paper [19] he reproved that V J ,r is simple if r / ∈ Z. The structure of Hom-algebras seems to be more complex because of the variety of twisted maps. But the structure of the original algebras is pretty clear. So one of the ways to study the structure of Hom-algebras is to look for relationships between Hom-algebras and their induced algebras. In [15], Makhlouf and Silvestrov Lemma 2.8. (1) Suppose that (V, µ) is a Jordan algebra and α : V → V is a homomorphism. Then (V,μ, α) is a multiplicative Hom-Jordan algebra with µ(x, y) = α(µ(x, y)), ∀x, y ∈ V . (2) Suppose that (V, µ, α) is a multiplicative Hom-Jordan algebra and α is invertible. Then (V, µ, α) is a Jordan-type Hom-Jordan algebra and its induced Jordan algebra is (V, µ ) with µ (x, y) = α −1 (µ(x, y)), ∀x, y ∈ V .

Proof.
We have thatμ is commutative, since µ is commutative.

Lemma 3.4. Suppose that an algebra A over F can be decomposed into the unique direct sum of simple ideals
contradicting the assumption that A i aren't isomorphic to each other. Hence, we have α(

Theorem 3.5. (1) Suppose that (V, µ, α) is a simple multiplicative Hom-Jordan algebra. Then its induced Jordan algebra
can be decomposed into a direct sum of isomorphic simple ideals; in addition, α acts simply transitively on simple ideals of the induced Jordan algebra.
(1) According to the proof of Lemma 2.8 (2) and Lemma 2.12, α is an automorphism both on (V, µ, α) and (V, µ ). Suppose that V 1 is the maximal solvable ideal of (V, µ ). Then there exists m ∈ Z + such that V Note that . Because there may be isomorphic Jordan algebras in V 1 , V 2 , . . . , V s , we rearrange the order as follows: According to Lemma 3.4, we have and so we have that V i1 ⊕V i2 ⊕· · ·⊕V imi are Hom-ideals of (V, µ, α). Since (V, µ, α) is simple, we have V i1 ⊕V i2 ⊕· · ·⊕V imi = 0 or V . So all but one V i1 ⊕V i2 ⊕· · ·⊕V imi must be 0. Without loss of generality, we can assume that When In addition, it is easy to show that V 11 ⊕ α(V 11 ) ⊕ · · · ⊕ α m1−1 (V 11 ) is a Hom-ideal of (V, µ, α). Therefore, That is, α acts simply transitively on simple ideals of the induced Jordan algebra.

α) is a Jordan-type Hom-Jordan algebra and its induced Jordan algebra
Hom-Jordan algebra and has a unique decomposition.

Classification of simple multiplicative Hom-Jordan algebras
In this section we present a theorem about classification of simple multiplicative Hom-Jordan algebras. First, we give a construction of n-dimensional simple Hom-Jordan algebras.  Obviously, µ(V, V ) = V . Take α ∈ End(V ) such that α(e 0 ) = pe 0 , α(e 1 ) = qe 1 , p, q ∈ C.
When n ≥ 3, let {aī | i ∈ Z n } be a basis of an n-dimensional vector space V over C. Define a bilinear symmetric binary operation µ : V × V → V as follows: all the others are zero. Then for any linear map α ∈ End(V ), (V, µ, α) is a Hom-Jordan algebra.
According to Theorem 3.5 (1) and Corollary 2.13, the dimension of a simple multiplicative Hom-Jordan algebra can only be an integer multiple of dimensions of simple Jordan algebras. Also by Theorem 3.5 (1) and Corollary 2.13, in order to classify simple multiplicative Hom-Jordan algebras, we just classify automorphisms on their induced Jordan algebras; in particular, automorphisms on semi-simple Jordan algebras which are direct sum of finite isomorphic simple ideals.

Theorem 4.2.
Let J be a semi-simple Jordan algebra such that its n simple ideals are mutually isomorphic; moreover, J can be generated by its automorphism α (or β) and any simple ideal, and α n (β n ) leaves each simple ideal of J invariant, where α n = α n and β n = β n . Then there exists an automorphism ϕ on J satisfying ϕ • α = β • ϕ if and only if there exists an automorphism ψ on the simple ideal of J satisfying ψ • α n = β n • ψ.
By Theorem 4.2, it is obvious that two simple multiplicative Hom-Jordan algebras (V 1 , µ 1 , α) and (V 2 , µ 2 , β) are isomorphic if and only if the automorphisms α n and β n on two simple ideals (as simple Jordan algebras) of the corresponding induced Jordan algebras are conjugate.
Combining Corollary 2.13, Theorem 3.5 (1) and Theorem 4.2, we get the following theorem. Theorem 4.3. All finite-dimensional simple multiplicative Hom-Jordan algebras can be denoted as (X, n, Γ α ), where X represents the type of the simple ideal (as the simple Jordan algebra) of the corresponding induced Jordan algebras, n represents numbers of simple ideals, and Γ α represents the set of conjugate classes of the automorphism α n on the simple Jordan algebra X, i.e., Then (V, µ, α) is a simple multiplicative Hom-Jordan algebra. Moreover, its induced Jordan algebra is (V, µ ), where µ : µ (e 2 , e 2 ) = e 2 , µ (e 1 , e 2 ) = µ(e 2 , e 1 ) = 0.
(V, µ ) is semi-simple and has the decomposition into simple ideals V = V 1 ⊕ V 2 , where V 1 and V 2 are simple ideals generated by e 1 and e 2 , respectively. Moreover, we get that V 1 is isomorphic to V 2 .

Bimodules of simple multiplicative Hom-Jordan algebras
In this section we mainly study bimodules of simple multiplicative Hom-Jordan algebras. We give a theorem on relationships between bimodules of Jordan-type Hom-Jordan algebras and modules of their induced Jordan algebras. Moreover, some propositions about bimoudles of simple multiplicative Hom-Jordan algebras are also obtained.
Next, we construct an example over a field of prime characteristic. (1) Let (W, α W ) be a V -bimodule of (V, µ, α) with ρ l (ρ r ) the left structure map (respectively, the right structure map). Suppose that α W is invertible and satisfies ) and the right structure map ρ r : W ⊗ V → W (ρ r (w ⊗ a) = α W (w · a)). Proof.
Proof. Assume that (W, α W ) is reducible. Then there exists W 0 = {0 W } a subspace of W such that (W 0 , α W | W 0 ) is a submodule of (W, α W ). That is, α W (W 0 ) ⊆ W 0 and a·w ∈ W 0 , for any a ∈ V, w ∈ W 0 . Hence, a· w = α −1 W (a·w) ∈ α −1 W (W 0 ) = W 0 . So W 0 is a nontrivial submodule of W for (V, µ ), a contradiction. Hence, (W, α W ) is an irreducible bimodule of (V, µ, α). Remark 5.10. In [18], the author introduced another definition of Hom-Jordan algebras, using which one could verify that all the above results are also valid.