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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 1 (2021), 1 – 24

doi:10.17398/2605-5686.36.1.1

Available online May 7, 2021

Structure and bimodules of simple
Hom-alternative algebras

S. Attan

Département de Mathématiques, Université d’Abomey-Calavi
01 BP 4521, Cotonou 01, Bénin

syltane2010@yahoo.fr

Received June 11, 2020 Presented by Rosa M. Navarro
Accepted March 03, 2021

Abstract: This paper is mainly devoted to the study of the structure of Hom-alternative algebras.

Equivalent conditions for Hom-alternative algebras being solvable, simple and semi-simple are pro-
vided. Moreover, some results about Hom-alternative bimodule are found.

Key words: Bimodules, solvable, simple, Hom-alternative algebras.

MSC (2020): 13B10, 13D20, 17A30, 17D15.

1. Introduction

Hom-algebras are new classes of algebras which have been studied exten-
sively in the literature during the last decade. They are algebras where the
identities defining the structure are twisted by a homomorphism and began
with Hom-Lie algebras [6, 9, 10, 11], motivated by quasi-deformations of Lie al-
gebras of vector fields, in particular q-deformations of Witt and Virasoro alge-
bras. Hom-associative algebras were introduced in [14] while Hom-alternative
and Hom-Jordan algebras are introduced in [13, 20] and deformations of Hom-
alternative and Hom-Malcev algebras are studied in [5].

Questions on the structure of simple algebras in this or that variety are
one of the main questions in the theory of rings. This question, for alternative
algebras, has been studied by many authors. It turns out that the only sim-
ple alternative algebras which are not associative are 8-dimensional algebras
over their centers which are generalizations of the original algebra of Cayley
numbers [15, 22, 23]. Hence, all semisimple alternative algebras are known.
Similarly, it is relevant to study simple Hom-algebras in Hom-algebras theory.
In [4], the authors gave a classification theorem about multiplicative simple
Hom-Lie algebras. Inspired by this study, a classification of multiplicative
simple Hom-Jordan algebras is obtained in [18].

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.1.1
mailto:syltane2010@yahoo.fr
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


2 s. attan

Representations (or bimodules) and deformations are important tools in
most parts of Mathematics and Physics. By means of the representation the-
ory, we would be more aware of the corresponding algebras. The study of
bimodules of Jordan algebras was initiated by N. Jacobson [7]. Subsequently
the alternative case was considered by Schafer [16]. Similarly, it is very impor-
tant to study representations of Hom-algebras. Fortunately, representations
of Hom-Lie algebras were introduced and studied in [17], see also [1, 3]. Later
the one of Hom-Jordan and Hom-alternative is presented in [2] where some
useful results are obtained. Moreover representations of simple Hom-Lie al-
gebras [4, 12] and the one of simple Hom-Jordan algebras [18] are introduced
and studied in detail. In this paper, based on [18] and on [21], we will study
structure and bimodules over simple Hom-alternative algebras.

The paper is organized as follows: In section two, we give the basics
about Hom-alternative algebras and provide some new properties. Section
three deals with the study of simple and solvable Hom-alternative algebras.
Some useful results are obtained (see Lemma 3.8 and Theorem 3.9). We
also mainly prove relevant theorems which are about solvability, simplicity
and semi-simplicity of Hom-alternative algebras (see Theorem 3.14, Theorem
3.16 and Theorem 3.20). In section four, we prove Theorem 4.5 which is
very important. It deals with the relationship between bimodules over Hom-
alternative algebras of alternative type and the ones over their induced alter-
native algebras. Moreover, some relevant propositions about bimodules over
Hom-alternative algebras are also displayed as applications of Theorem 4.5.

All vector spaces are assumed to be over a fixed ground field K of charac-
teristic 0.

2. Preliminaries

We recall some basic notions, introduced in [6, 14, 19] related to Hom-
algebras. For the map µ : A⊗2 → A, we will write sometimes µ(a ⊗ b) as
µ(a,b) or ab for a,b ∈ A.

Definition 2.1. A Hom-module is a pair (M,αM ) consisting of a
K-module M and a linear self-map αM : M → M. A morphism
f : (M,αM ) → (N,αN ) of Hom-modules is a linear map f : M → N such
that f ◦αM = αN ◦f.

Definition 2.2. ([14, 19]) A Hom-algebra is a triple (A,µ,α) in which
(A,α) is a Hom-module, µ : A⊗2 → A is a linear map. The Hom-algebra



structure and bimodules 3

(A,µ,α) is said to be multiplicative if α ◦ µ = µ ◦ α⊗2 (multiplicativity). A
morphism f : (A,µA,αA) → (B,µB,αB) of Hom-algebras is a morphism of
the underlying Hom-modules such that f ◦µA = µB ◦f⊗2.

In this paper, we will only consider multiplicative Hom-algebras.

Definition 2.3. Let (A,µ,α) be a Hom-algebra and λ ∈ K. Let R be a
linear map satisfying

µ(R(x),R(y)) = R
(
µ(R(x),y) + µ(x,R(y)) + λµ(x,y)

)
, ∀x,y ∈ A. (1)

Then, R is called a Rota-Baxter operator of weight λ and (A,µ,α,R) is called
a Rota-Baxter Hom-algebra of weight λ.

Definition 2.4. Let (A,µ,α) be a Hom-algebra.

1. The Hom-associator of A is the linear map asA : A
⊗3 → A defined as

asA = µ◦(µ⊗α−α⊗µ). A multiplicative Hom-algebra (A,µ,α) is said
to be Hom-associative algebra if asA = 0.

2. A Hom-alternative algebra [13] is a multiplicative Hom-algebra (A,µ,α)
that satisfies both

asA(x,x,y) = 0 (left Hom-alternativity), (2)

asA(x,y,y) = 0 (right Hom-alternativity) (3)

for all x,y ∈ A.
3. Let (A,µ,α) be a Hom-alternative algebra; a Hom-subalgebra of (A,µ,α)

is a linear subspace H of A, which is closed for the multiplication µ and
invariant by α, that is, µ(x,y) ∈ H and α(x) ∈ H for all x,y ∈ H. If
furthermore µ(a,b) ∈ H and µ(b,a) ∈ H for all (a,b) ∈ A×H, then H
is called a two-sided Hom-ideal of A.

Example 2.5. The octonions algebra O, also called Cayley Octaves or
Cayley algebra is 8-dimensional, with a basis (e0,e1,e2,e3,e4,e5,e6,e7), where
e0 is the identity for the multiplication. This algebra is twisted into the eight-
dimensional Hom-alternative algebra Oα = (O,µ1,α) [20] with the same basis
(e0,e1,e2,e3,e4,e5,e6,e7) where

α(e0) = e0, α(e1) = e5, α(e2) = e6, α(e3) = e7,

α(e4) = e1, α(e5) = e2, α(e6) = e3, α(e7) = e4,



4 s. attan

and the multiplication table is:

µ1 e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 e5 e6 e7 e1 e2 e3 e4
e1 e5 −e0 e1 e4 −e6 e3 −e2 −e7
e2 e6 −e1 −e0 e2 e5 −e7 e4 −e3
e3 e7 −e4 −e2 −e0 e3 e6 −e1 e5
e4 e1 e6 −e5 −e3 −e0 e4 e7 −e2
e5 e2 −e3 e7 −e6 −e4 −e0 e5 e1
e6 e3 e2 −e4 e1 −e7 −e5 −e0 e6
e7 e4 e7 e3 −e5 e2 −e1 −e6 −e0

and into the eight-dimensional Hom-alternative algebra Oβ = (O,µ2,β)
[13] with the same basis (e0,e1,e2,e3,e4,e5,e6,e7) where β(ei) = ei for
all i ∈ {0, 4, 5, 7} and β(ei) = −ei for all i ∈ {1, 2, 3, 6}. The multiplica-
tion table is:

µ2 e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 −e1 −e2 −e3 e4 e5 −e6 e7
e1 −e1 −e0 e4 e7 e2 −e6 −e5 e3
e2 −e2 −e4 −e0 e5 −e1 e3 e7 e6
e3 −e3 −e7 −e5 −e0 −e6 −e2 −e4 −e1
e4 e4 −e2 e1 e6 −e0 e7 −e3 −e5
e5 e5 e6 −e3 e2 −e7 −e0 −e1 e4
e6 −e6 e5 −e7 e4 e3 e1 −e0 −e2
e7 e7 −e3 −e6 e1 e5 −e4 e2 −e0

Nor Oα, neither Oβ, are alternative algebras. Moreover, both α and β are
automorphisms of O.

Similarly as in [13], it easy to prove the following:

Proposition 2.6. Let (A,µ,α) be a Hom-alternative algebra and β :
A → A be a morphism of (A,µ,α). Then (A,β◦µ,β◦α) is a Hom-alternative
algebra. In particular, if (A,µ) is an alternative algebra and β is a morphism
of (A,µ), then (A,β ◦µ,β) is a Hom-alternative algebra [13].

Definition 2.7. Let (A,µ,α) be a Hom-alternative algebra. If there is
an alternative algebra (A,µ′) such that µ = α ◦ µ′, we say that (A,µ,α) is
alternative-type and (A,µ′) is its compatible alternative algebra or the untwist
of (A,µ,α).



structure and bimodules 5

It is noticed in [13] that a Hom-alternative algebra with an invertible
twisting map has the compatible alternative algebra. More precisely, we get:

Corollary 2.8. ([13]) Let (A,µ,α) be a Hom-alternative algebra where
α is invertible then (A,µ′ = α−1 ◦ µ) is an alternative algebra and α is an
automorphism with respect to µ′. Hence (A,µ,α) is alternative-type and
(A,µ′ = α−1 ◦µ) is its compatible alternative algebra.

Proposition 2.9. Let (A1,µ1,α1) and (A2,µ2,α2) be Hom-alternative
algebras and ϕ : (A1,µ1,α1) → (A2,µ2,α2) be an invertible morphism of
Hom-algebras. If (A1,µ1,α1) is alternative-type and (A1,µ

′
1) is its compatible

alternative algebra then (A2,µ2,α2) is alternative-type with compatible alter-
native algebra (A2,µ

′
2 = ϕ◦µ

′
1◦(ϕ

−1⊗ϕ−1)) such that ϕ : (A1,µ′1) → (A2,µ
′
2)

is an algebra morphism.

Proof. First, let us prove that (A2,µ
′
2) is an alternative algebra such that

µ′2 = ϕ◦µ
′
1 ◦ (ϕ

−1 ⊗ϕ−1). Denote by as1 and as2 the associators of (A1,µ1)
and (A2,µ

′
2) respectively. Then

as2(u,u,v) = µ
′
2(µ
′
2(u,u),v) −µ

′
2(u,µ

′
2(u,v))

= ϕ◦µ′1
(
ϕ−1 ◦ϕ◦µ′1(ϕ

−1(u),ϕ−1(u)),ϕ−1(v)
)

−ϕ◦µ′1
(
ϕ−1(u),ϕ−1 ◦ϕ◦µ′1(ϕ

−1(u),ϕ−1(v))
)

= ϕ◦µ′1
(
µ′1(ϕ

−1(u),ϕ−1(u)),ϕ−1(v)
)

−ϕ◦µ′1
(
ϕ−1(u),µ′1(ϕ

−1(u),ϕ−1(v))
)

= ϕ◦ as1(ϕ−1(u),ϕ−1(u),ϕ−1(v)) = ϕ(0) = 0.

Similarly, we prove that as2(u,v,v) = 0. Hence (A,µ
′
2) is an alternative

algebra.

Next, we have α2 ◦ϕ = ϕ◦α1 and ϕ defines µ2 by µ2 ◦ϕ⊗2 = ϕ◦µ1 since
ϕ is a morphism from (A1,µ1,α1) to (A2,µ2,α2), i.e.,

µ2 = ϕ◦µ1 ◦ (ϕ−1 ⊗ϕ−1)

= ϕ◦α1 ◦µ′1 ◦ (ϕ
−1 ⊗ϕ−1) = α2 ◦ϕ◦µ′1 ◦ (ϕ

−1 ⊗ϕ−1).

Hence, let take µ′2 = ϕ◦µ
′
1 ◦ (ϕ

−1 ⊗ϕ−1).
Finally, the fact that ϕ : (A1,µ

′
1) → (A2,µ

′
2) is an algebra morphism

follows from the definition of µ′2.



6 s. attan

The following characterization was given for Hom-Lie algebras in [17] and
Hom-associative algebras in [21].

Proposition 2.10. Given two Hom-alternative algebras (A,µA,α) and
(B,µB,β), there is a Hom-alternative algebra (A ⊕ B,µA⊕B,α + β), where
the bilinear map µA⊕B : (A⊕B)×2 → (A⊕B) is given by

µA⊕B(a1 + b1,a2 + b2) = µA(a1,a2) + µB(b1,b2) ∀a1,a2 ∈ A, ∀b1,b2 ∈ B,

and the linear map (α + β) : (A⊕B) → (A⊕B) is given by

(α + β)(a + b) = (α(a) + β(b)) ∀ (a,b) ∈ A×B. (4)

Proof. First, (α + β) is multiplicative with respect to µA⊕B. Indeed,

(α + β)◦(µA⊕B)(a1 + b1,a2 + b2)
= (α + β)(µA(a1,a2) + µB(b1,b2))

= α◦µA(a1,a2) + β ◦µB(b1,b2)
= µA(α(a1),α(a2)) + µB(β(b1),β(b2)

= µA⊕B(α(a1) + β(b1),α(a2) + β(b2))

= µA⊕B((α + β)(a1 + b1), (α + β)(a2 + b2)).

Secondly we prove the left Hom-alternativity (2) for A⊕B as follows

asA⊕B(a1 + b1,a1 + b1,a2 + b2)

= µA⊕B
(
µA⊕B(a1 + b1,a1 + b1), (α + β)(a2 + b2)

)
−µA⊕B

(
(α + β)(a1 + b1),µA⊕B(a1 + b1,a2 + b2)

)
= µA⊕B(µA(a1,a1) + µB(b1,b1),α(a2) + β(b2)

)
−µA⊕B

(
α(a1) + β(b1),µA(a1,a2) + µ(b1,b2)

)
= µA(µA(a1,a1),α(a2)) + µB(µB(b1,b1),β(b2))

−µA(α(a1),µA(a1,a2)) −µB(β(b1),µB(b1,b2))
= asA(a1,a1,a2) + asB(b1,b1,b2) = 0.

Similarly, we prove the right Hom-alternativity (3) for A ⊕ B. Hence (A ⊕
B,µA⊕B,α + β) is a Hom-alternative algebra.



structure and bimodules 7

Proposition 2.11. Let (A,µA,α) and (B,µB,β) be two Hom-alternative
algebras and ϕ : A → B be a linear map. Denote by Γϕ ⊂ A⊕B the graph
of ϕ. Then ϕ is a morphism from the Hom-alternative algebra (A,µA,α) to
the Hom-alternative algebra (B,µB,β) if and only if its graph Γϕ is a Hom-
subalgebra of (A⊕B,µA⊕B,α + β).

Proof. Let φ : (A,µA,α) → (B,µB,β) be a morphism of Hom-alternative
algebras. Then we have for all u,v ∈ A,

µA⊕B
(
(u,ϕ(u)), (v,ϕ(v))

)
=

(
µA(u,v),µB(ϕ(u),ϕ(v))

)
=

(
µA(u,v),ϕ(µA(u,v))

)
.

Thus the graph Γϕ is closed under the multiplication µA⊕B. Furthermore since
ϕ◦α = β ◦ϕ, we have (α⊕β)(u,ϕ(u)) = (α(u),β ◦ϕ(u)) = (α(u),ϕ◦α(u)),
which implies that Γϕ is closed under α⊕β. Thus Γϕ is a Hom-subalgebra of
(A⊕B,µA⊕B,α⊕β).

Conversely, if the graph Γϕ ⊂ A⊕B is a Hom-subalgebra of (A⊕B,µA⊕B,
α⊕β), then we have

µA⊕B
(
(u,ϕ(u)), (v,ϕ(v))

)
=

(
µA(u,v),µB(ϕ(u),ϕ(v))

)
∈ Γϕ,

which implies that
µB(ϕ(u),ϕ(v)) = ϕ(µA(u,v)).

Furthermore, (α⊕β)(Γϕ) ⊂ Γϕ implies

(α⊕β)(u,ϕ(u)) = (α(u),β ◦ϕ(u)) ∈ Γϕ,

which is equivalent to the condition β ◦φ(u) = φ◦α(u), i.e., β ◦ϕ = ϕ◦α.
Therefore, ϕ is a morphism of Hom-alternative algebras.

Proposition 2.12. Let (A,µ,α,R) be a Rota-Baxter Hom-alternative al-
gebra of weight 0 such that R commutes with α. Define a new multiplication
on A by

µR(x,y) = µ(R(x),y) + µ(x,R(y)) for any x,y ∈ A.

Then AR = (A,µR,α) is a Hom-alternative algebra.

Proof. The multiplicativity of α with respect to µR follows from the one
of α with respect to µ and the hypothesis R ◦ α = α ◦ R. To prove the left
Hom-alternative identity, let pick x,y ∈ A. Then,



8 s. attan

asR(x,x,y) = µR(µR(x,x),α(y)) −µR(α(x),µR(x,y))
= µR(µ(R(x),x) + µ(x,R(x)),α(y))

−µR
(
α(x),µ(R(x),y) + µ(x,R(y))

)
= µ(R(µ(R(x),x) + µ(x,R(x))),α(y))

+ µ(µ(R(x),x) + µ(x,R(x)),R◦α(y))
−µ

(
R◦α(x),µ(R(x),y) + µ(x,R(y))

)
−µ

(
α(x),R

(
µ(R(x),y) + µ(x,R(y))

))
= µ(µ(R(x),R(x)),α(y)) + µ(µ(R(x),x),α◦R(y))

+ µ(µ(x,R(x)),α◦R(y)) −µ(α◦R(x),µ(R(x),y))
−µ

(
α◦R(x),µ(x,R(y))

)
−µ

(
α(x),µ(R(x),R(y))

)
(using (1) with λ = 0 and α◦R = R◦α)

= asA(R(x),R(x),y) + asA(R(x),x,R(y)) + asA(x,R(x),R(y))

= 0 + 0 = 0 (by (2) in A).

Hence, we get (2) for AR. Similarly, we prove (3) for AR and therefore, the
conclusion follows.

3. Structures of Hom-alternative algebras

In this section, we study simple and solvable Hom-alternative algebras.
This study is inspired by the study given in [18] and [21]. We discuss necessary
and sufficient conditions for Hom-alternative algebras to be solvable, simple
and semi-simple. In the classical case, we already know that every simple
alternative algebra is either an associative algebra or a Cayley-Dickson algebra
over its center [15, 22]. As it turns out, there is one non-associative simple
alternative algebra. Recall that the Cayley-Dickson algebras are a sequence
A0, A1, . . . of non-associative R-algebras with involution. The first few are
familiar: A0 = R, A1 = C, A2 = H (the quaternions) and A3 = O (the
octonions). Each algebra An is constructed from the previous one An−1 by a
doubling procedure. The first three Cayley-Dickson algebras are associative
and O is the only one non-associative alternative Cayley-Dickson algebra.
Alternativity fails in the higher Cayley-Dickson algebras. Basing on this fact,
we will give in this section an example of non Hom-associative simple Hom-
alternative algebra.



structure and bimodules 9

Definition 3.1. Let (A,µ,α) be a Hom-alternative algebra. Define its
derived sequences as follows:

A(0) = A, A(1) = µ(A,A), A(2) = µ
(
A(1),A(1)

)
, . . . ,

A(k) = µ
(
A(k−1),A(k−1)

)
, . . . .

The following elementary result will be very useful.

Lemma 3.2. Let (A,µ,α) be a Hom-alternative algebra and k ∈ N. Then

· · · ⊆ A(k+1) ⊆ A(k) ⊆ A(k−1) ⊆ ··· ⊆ A(2) ⊆ A(1) ⊆ A

and A(k) is a two-sided Hom-ideal of (A,µ,α).

Proof. First, it is clear that A(1) ⊆ A(0) = A. Next let k ∈ N and assume
that A(k) ⊆ A(k−1). Then, we have

A(k+1) = µ
(
A(k),A(k)

)
⊆ µ

(
A(k−1),A(k−1)

)
= A(k).

To prove that A(k) is a two-sided Hom-ideal of (A,µ,α) for every k ∈ N, by the
inclusion condition, it suffices to prove the case n = 1. Thanks to A(1) ⊆ A,
we have first

α(A(1)) = α
(
µ(A,A)

)
= µ

(
α(A),α(A)

)
⊆ µ(A,A) = A(1),

and next

µ
(
A(1),A

)
⊆ µ(A,A) = A(1) and µ

(
A,A(1)

)
⊆ µ(A,A) = A(1).

Thus A(1) is a two-sided Hom-ideal of (A,µ,α).

Definition 3.3. Let (A,µ,α) be a Hom-alternative algebra; (A,µ,α) is
said to be solvable if there exists n ∈ N∗ such that A(n) = {0}.

We get the following example of solvable and non solvable Hom-alternative
algebras respectively.

Example 3.4. (a) Consider the 3-dimensional Hom-alternative algebra
(A,µ,α) with basis (e1,e2,e3) where µ(e2,e2) = e1,µ(e2,e3) = e1,µ(e3,e2) =
e1,µ(e3,e3) = e1,µ(e3,e2) = e1 and α(e1) = e1, α(e2) = e1 + e2, α(e3) =
e3. Actually, (A,µ,α) is a Hom-associative algebra (see [21], Theorem 3.6,



10 s. attan

Hom-algebra A37). Then A
(1) is a one-dimensional Hom-alternative algebra

generated by (e1) defined as follows: µ(e1,e1) = 0 and α(e1) = e1. It follows
that A(2) = {0} and therefore (A,µ,α) is solvable.

(b) Consider in the Example 2.5, the Hom-alternative algebras Oα and
Oβ. From their multiplication tables, we get (Oα)(1) = Oα and (Oβ)(1) = Oβ.
Hence for every k ∈ N we have (Oα)(k) = Oα and (Oβ)(k) = Oβ. It follows
that neither Oα and nor Oβ are solvable.

Definition 3.5. Let (A,µ,α) (α 6= 0) be a non trivial Hom-alternative
algebra.

1. (A,µ,α) is said to be a simple Hom-alternative algebra if A(1) 6= {0}
and it has no proper two-sided Hom-ideal.

2. (A,µ,α) is said to be a semi-simple Hom-alternative algebra if

A = A1 ⊕A2 ⊕A3 ⊕···⊕Ap

where Ai (1 ≤ i ≤ p) are simple two-sided Hom-ideals of (A,µ,α).

Let give the following example of non-simple Hom-alternative algebra.

Example 3.6. Consider the three-dimensional Hom-alternative algebra
(A,µ,α) over K with basis (e1,e2,e3) defined by µ(e1,e1) = e1, µ(e2,e2) = e2,
µ(e3,e3) = e1, µ(e1,e3) = µ(e3,e1) = −e3 and α(e1) = e1, α(e3) = −e3.
Actually, (A,µ,α) is a Hom-associative algebra (see [21], Theorem 3.12, Hom-
algebra A′

3
3). Consider the subspace I = span(e1,e3) of A. Then one can

observe that (I,µ,α) is a proper two-sided Hom-ideal of (A,µ,α). Hence the
Hom-alternative (A,µ,α) is not simple.

We have the following elementary result which will be used in next sections.

Proposition 3.7. Let (A,µ,α) be a simple Hom-alternative algebra.
Then, for each k ∈ N, A(k) = A.

Proof. Thanks to the definition of A(k), it suffices to prove that A(1) = A.
By the simplicity of (A,µ,α), we have A(1) 6= {0}. Moreover by Lemma
3.2, A(1) is a two-sided Hom-ideal of (A,µ,α) which has no proper two-sided
Hom-ideal, then A(1) = A.

The following lemma is useful for next results.



structure and bimodules 11

Lemma 3.8. Let (A,µ,α) be a Hom-alternative algebra. Then, (Ker(α),
µ,α) is a two-sided Hom-ideal of (A,µ,α).

Proof. Obvious, α(x) = 0 ∈ Ker(α) for all x ∈ Ker(α). Next, let x,z ∈ A
and y ∈ Ker(α). Then α(µ(x,y)) = µ(α(x),α(y)) = µ(α(x), 0) = 0 and
α(µ(y,z)) = µ(α(y),α(z)) = µ(0,α(z)) = 0. Thus µ(x,y) ∈ Ker(α) and
µ(y,z) ∈ Ker(α) and it follows that (Ker(α),µ,α) is a two-sided
Hom-ideal.

Proposition 3.9. Let (A,µ,α) be a finite dimensional simple Hom-
alternative algebra. Then the Hom-alternative algebra is alternative-type and
α is an automorphism of both (A,µ,α) and its induced algebra.

Proof. By Lemma 3.8, Ker(α) is a two-sided Hom-ideal of the simple Hom-
alternative algebra (A,µ,α). Therefore Ker(α) = {0} or Ker(α) = A. Since
the Hom-alternative algebra is non trivial, it follows that Ker(α) = {0} and
α is an automorphism. Thus, A is alternative-type (see Corollary 2.8).

Let (A,µ′ = α−1 ◦µ) be the induced (the compatible) alternative algebra
of the simple Hom-alternative algebra (A,µ,α). We have

α◦µ′ = α◦α−1 ◦µ = α−1 ◦µ◦α⊗2 = µ′ ◦α⊗2,

i.e., α is both automorphism of (A,µ′) and (A,µ,α).

As in Hom-associative algebras case [21], by the above proposition, there
exists an induced alternative algebra of any simple Hom-alternative algebra
(A,µ,α) and α is an automorphism of the induced alternative algebra. More-
over, their products are mutually determined.

Theorem 3.10. Two finite dimensional simple Hom-alternative algebras
(A1,µ1,α) and (A2,µ2,β) are isomorphic if and only if there exists an alter-
native algebra isomorphism ϕ : A1 → A2 (between their induced alternative
algebras) which renders conjugate the two alternative algebra automorphisms
α and β that is ϕ◦α = β ◦ϕ.

Proof. Since (A1,µ1,α) and (A2,µ2,β) are finite dimensional simple Hom-
alternative algebras, they are alternative-type. Let (A1,µ

′
1) and (A2,µ

′
2) be

their induced alternative algebras respectively. If ϕ : (A1,µ1,α) → (A2,µ2,β)



12 s. attan

is an isomorphism of Hom-alternative algebras, then ϕ ◦ α = β ◦ ϕ, thus
β−1 ◦ϕ = ϕ◦α−1. Moreover,

ϕ◦µ′1 = ϕ◦α
−1 ◦α◦µ′1 = ϕ◦α

−1 ◦µ1
= β−1 ◦ϕ◦µ1 = β−1 ◦µ2 ◦ϕ⊗2 = µ′2 ◦ϕ

⊗2.

So, ϕ is an isomorphism between the induced Hom-alternative algebras.
On the other hand, if there exists an isomorphism between the induced

alternative algebras satisfying ϕ◦α = β ◦ϕ, then

ϕ◦µ1 = ϕ◦α◦µ′1 = β ◦ϕ◦µ
′
1 = β ◦µ

′
2 ◦ϕ

⊗2 = µ2 ◦ϕ⊗2.

Let recall the following:

Proposition 3.11. ([2]) Let (A,µ,α) be a Hom-alternative algebra and
I be a two-sided Hom-ideal of (A,µ,α). Then (A/I,µ̄, ᾱ) is a Hom-alternative
algebra where µ̄(x̄, ȳ) = µ(x,y) and ᾱ(x̄) = ¯α(x) for all x̄, ȳ ∈ A/I.

Proof. First, note that the multiplicativity of µ̄ with respect to ᾱ follows
from the one of µ with respect to α. Next, pick x̄, ȳ ∈ A/I. Then the left
Hom-alternativity (2) in (A/I,µ̄, ᾱ) is proved as follows

asA/I(x̄, x̄, ȳ) = ū
(
ū(x̄, x̄), ᾱ(ȳ)

)
− µ̄

(
ᾱ(x̄), µ̄(x̄, ȳ)

)
= µ

(
µ(x,x)α(y)

)
−µ

(
α(x),µ(x,y)

)
= asA(x,x,y)) = 0̄.

Hence we get (2) for (A/I,µ̄, ᾱ). Similarly, we get (3) and therefore (A/I,µ̄, ᾱ)
is a Hom-alternative algebra.

Corollary 3.12. Let (A,µ,α) be a finite dimensional Hom-alternative
algebra such that α2 = α. Then, (A/ Ker(α), µ̄, ᾱ) is a Hom-alternative alge-
bra of alternative-type.

Proof. It is clear that (A/ Ker(α), µ̄, ᾱ) is a Hom-alternative algebra by
Proposition 3.11 since by Lemma 3.8, Ker(α) is a two-sided Hom-ideal of A.
• If α is invertible, i.e., Ker(α) = {0}, then (A/ Ker(α), µ̄, ᾱ) = (A,µ,α)

and (A/ Ker(α), µ̄, ᾱ) is a Hom-alternative algebra of alternative-type (see
Corollary 2.8).
• If α is not invertible, then Ker(α) 6= {0}. Therefore we have to show

that ᾱ is invertible on the Hom-alternative algebra (A/ Ker(α), µ̄, ᾱ). Assume



structure and bimodules 13

that x̄ ∈ Ker(ᾱ). Then ¯α(x) = ᾱ(x̄) = 0̄, i.e., α(x) ∈ Ker(α). Since α2 = α,
we have

α(x) = α2(x) = α(α(x)) = 0,

which means that x ∈ Ker(α), i.e., x̄ = 0̄. If follows that ᾱ is invertible and
thanks to the Corollary 2.8, the Hom-alternative algebra (A/ Ker(α), µ̄, ᾱ) is
alternative-type.

Corollary 3.13. Let (A,µ,α) be a Hom-alternative algebra such that
α is invertible. Then, (A/ Ker(α), µ̄, ᾱ) is a Hom-alternative algebra of
alternative-type.

Theorem 3.14. Let (A,µ,α) be a Hom-alternative algebra such that α
is invertible. Then (A,µ,α) is solvable if and only if its induced alternative
algebra (A,µ′) is solvable.

Proof. Let (A,µ,α) be a Hom-alternative algebra such that α is invertible.

Denote the derived sequences of (A,µ′) and (A,µ,α) by A(i) and A
(i)
α (i =

1, 2, · · ·) respectively.
Suppose that (A,µ′) is solvable. Then there exists p ∈ N∗ such that

A(p) = {0}. Note that

A(1)α = µ(A,A) = α◦µ
′(A,A) = α

(
A(1)

)
,

A(2)α = µ
(
A(1)α ,A

(1)
α

)
= µ

(
α
(
A(1)

)
,α

(
A(1)

))
= α2 ◦µ′

(
A(1),A(1)

)
= α2

(
A(2)

)
,

so by induction A
(p)
α = α

p
(
A(p)

)
. It follows that A

(p)
α = {0}, which means

that (A,µ,α) is solvable.

On the other hand, assume that (A,µ,α) is solvable. Then there

exists q ∈ N∗ such that A(q)α = {0}. By the above proof, we get αq
(
A(q)

)
=

A
(q)
α . Note that α

q is invertible since α is, then A(q) = {0} that is (A,µ′) is
solvable.

Lemma 3.15. ([18]) Let A be an algebra over a field K that has the
unique decomposition of direct sum of simple ideals A = ⊕si=1Ai where the
Ai are not isomorphic to each other and α ∈ Aut(A). Then α(Ai) = Ai
(i = 1, 2, · · · ,s).



14 s. attan

Theorem 3.16. (i) Let (A,µ,α) be a finite dimensional simple Hom-
alternative algebra. Then its induced alternative algebra (A,µ′) is semi-
simple. Moreover, (A,µ′) can be decomposed into direct sum of iso-
morphic simple ideals. In addition, α acts simply transitively on simple
ideals of the induced alternative algebra.

(ii) Let (A,µ′) be a simple alternative algebra and α ∈ Aut(A). Then
(A,µ = α◦µ′,α) is a simple Hom-alternative algebra.

Proof. (i) Thanks to Corollary 2.8, α is both automorphism with respect
to µ′ and µ.

Assume that A1 is the maximal solvable two-sided ideal of (A,µ
′). Then

there exists p ∈ N∗ such that A(p)1 = {0}. Since

µ′
(
A,α(A1)

)
= µ′

(
α(A),α(A1)

)
= α

(
µ′(A,A1)

)
⊆ α(A1),

µ′
(
α(A1),A

)
= µ′

(
α(A1),α(A)

)
= α

(
µ′(A1,A)

)
⊆ α(A1),(

α(A1)
)(p)

= α
(
A

(p)
1

)
= {0},

we obtain that α(A1) is also a solvable two-sided ideal of (A,µ
′). Then

α(A1) ⊆ A1. Moreover

µ(A,A1) = α
(
µ′(A,A1)

)
⊆ α(A1) ⊆ A1,

µ(A1,A) = α
(
µ′(A1,A)

)
⊆ α(A1) ⊆ A1.

It follows that A1 is a two-sided Hom-ideal of (A,µ,α) and we get that A1 =
{0} or A1 = A since (A,µ,α) is simple.

If A1 = A, thanks to the proof of Theorem 3.14, we obtain

A(p)α = α
p
(
A(p)

)
= αp

(
A

(p)
1

)
= {0}.

On the other hand, by the simplicity of (A,µ,α), we have by Proposition 3.7,

A
(p)
α = A, which is a contradiction. It follows that A1 = {0} and thus, (A,µ′)

is semi-simple.
Now, by the semi-simplicity of (A,µ′), we have A = ⊕si=1Ai where for

all i ∈ {1, . . . ,s}, Ai is a simple two-sided ideal of (A,µ′). Since there may
be isomorphic alternative algebras among A1, . . . ,As, we can rewrite A as
follows:

A = A11 ⊕A12 ⊕···⊕A1m1 ⊕A21 ⊕A22 ⊕···
⊕A2m2 ⊕···⊕At1 ⊕At2 ⊕···⊕Atmt,



structure and bimodules 15

where (Aij,µ
′) ∼= (Aik,µ′), 1 ≤ j,k ≤ mi, i = 1, 2, . . . , t. Thanks to Lemma

3.15, we have

α
(
Ai1 ⊕Ai2 ⊕···⊕Aimi

)
= Ai1 ⊕Ai2 ⊕···⊕Aimi.

Then

µ
(
Ai1 ⊕Ai2 ⊕···⊕Aimi,A

)
= α

(
µ′
(
Ai1 ⊕Ai2 ⊕···⊕Aimi,A

))
⊆ α

(
Ai1 ⊕Ai2 ⊕···⊕Aimi

)
= Ai1 ⊕Ai2 ⊕···⊕Aimi

and

µ
(
A,Ai1 ⊕Ai2 ⊕···⊕Aimi

)
= α

(
µ′
(
A,Ai1 ⊕Ai2 ⊕···⊕Aimi

))
⊆ α

(
Ai1 ⊕Ai2 ⊕···⊕Aimi

)
= Ai1 ⊕Ai2 ⊕···⊕Aimi.

It follows that Ai1⊕Ai2⊕···⊕Aimi are two-sided Hom-ideals of (A,µ,α).
Then, the simplicity of (A,µ,α) implies Ai1 ⊕ Ai2 ⊕ ··· ⊕ Aimi = {0} or
Ai1 ⊕Ai2 ⊕···⊕Aimi = A. Therefore, all but one of Ai1 ⊕Ai2 ⊕···⊕Aimi
must be equal to A. Without lost of generality, assume that

A = A11 ⊕A12 ⊕···⊕A1m1.

If m1 = 1 then (A,µ
′) is simple. Else,

α(A1p) = A1l (1 ≤ l 6= p ≤ m1)

since, if

α(A1p) = A1p (1 ≤ p ≤ m1),

then, A1p would be a non trivial two-sided Hom-ideal of (A,µ,α) which con-
tradicts the simplicity of (A,µ,α).

In addition, it is clear that A11 ⊕α(A11) ⊕α2(A11) ⊕···αm1−1(A11) is a
two-sided Hom-ideal of (A,µ,α). Therefore,

A = A11 ⊕α(A11) ⊕α2(A11) ⊕···⊕αm1−1(A11).

In other words, α acts simply transitively on simple ideals of the induced
alternative algebra.



16 s. attan

(ii) By Proposition 2.6, it is clear that (A,µ,α) is a Hom-alternative
algebra.

Assume that A1 is a non-trivial two-sided Hom-ideal of (A,µ,α, then we
get

µ′(A,A1) = α
−1(µ(A,A1)) ⊆ α−1(A1) ⊆ A1,

µ′(A1,A) = α
−1(µ(A1,A)) ⊆ α−1(A1) ⊆ A1.

It follows that A1 is a non trivial two-sided ideal of (A,µ
′), contradiction. It

follows that (A,µ,α) has no trivial ideals.
If µ(A,A) = {0}, then,

µ′(A,A) = α−1
(
µ(A,A)

)
= {0},

which is in contradicts the fact that (A,µ′) is simple. It follows that (A,µ,α)
is simple.

Next, we will give an example of non Hom-associative simple Hom-alter-
native algebra using some results about Cayley-Dickson algebras.

Example 3.17. Consider Hom-alternative algebras Oα = (O,µ1,α) and
Oβ = (O,µ2,β) (see Example 2.5) obtained from the octonions algebra O
which is an eight dimensional simple non-associative alternative algebra. Since
α ∈ Aut(O) and β ∈ Aut(O), thanks to Theorem 3.16 (ii), both Oα and Oβ
are eight dimensional simple Hom-alternative algebras.

Proposition 3.18. The eight dimensional simple Hom-alternative alge-
bras Oα and Oβ are not isomorphic.

Proof. Suppose that the simple Hom-alternative algebras Oα and Oβ are
isomorphic. Then by Theorem 3.10, there exists an alternative algebra iso-
morphism ϕ : O → O (between their induced alternative algebras) such that
α ◦ ϕ = ϕ ◦ β. This means by the definition of β that for all i ∈ {1, 2, 3, 6},
α(ϕ(ei)) = −ϕ(ei), i.e., λ = −1 is an eigenvalue of α (contradiction since the
characteristic polynomial of α is X8 −X7 −X + 1).

Remark 3.19. We know in the classical case that there exists only one
example of a nonassociative simple alternative algebra that is, the Cayley-
Dickson algebra over its center [15, 22]. In the Hom-algebra setting, Propo-
sition 3.18, clearly proves that there is more than one simple non Hom-
associative Hom-alternative algebras.



structure and bimodules 17

Theorem 3.20. (i) Let (A,µ,α) be a finite dimensional semi-simple
Hom-alternative algebra. Then (A,µ,α) is alternative-type and its
induced alternative algebra (A,µ′) is also semi-simple.

(ii) Let (A,µ′) be a semi-simple alternative algebra such that A has a de-
composition A = ⊕siAi where Ai (1 ≤ i ≤ s) are simple two-sided ideal
of (A,µ′). Moreover let α ∈ Aut(A) satisfying α(Ai) = Ai (1 ≤ i ≤ s).
Then (A,µ = α ◦ µ′,α) is a semi-simple Hom-alternative algebra and
has the unique decomposition.

Proof. (i) Suppose that (A,µ,α) is a finite dimensional semi-simple Hom-
alternative algebra.Then A has the decomposition A = ⊕siAi where Ai (1 ≤
i ≤ s) are simple two-sided Hom-ideal of (A,µ,α). Then (Ai,µ,α|Ai ) (1 ≤ i ≤
s) are simple finite dimensional Hom-alternative algebras. According to the
proof of Proposition 3.9, α|Ai is invertible and therefore α is invertible. Thus
thanks to Corollary 2.8, the Hom-alternative algebra (A,µ,α) is alternative-
type and its induced alternative algebra is (A,µ′) with µ′ = α−1 ◦µ.

On the other hand, by the proof of Theorem 3.16 (ii), Ai (1 ≤ i ≤ s) are
two-sided ideal of (A,µ′). Moreover (Ai,µ

′|Ai ) (1 ≤ i ≤ s) are induced alterna-
tive algebra of finite dimensional simple Hom-alternative algebras (Ai,µ,α|Ai )
(1 ≤ i ≤ s) respectively. Thanks to Theorem 3.16 (i), (Ai,µ′) are semi-simple
alternative algebras and can be decomposed into direct sum of isomorphic
simple two-sided ideals Ai = Ai1 ⊕ Ai2 ⊕ ···⊕ Aimi . It follows that (A,µ

′)
is semi-simple and has the decomposition of direct sum of simple two-sided
ideals

A = A11 ⊕A12 ⊕···⊕A1m1 ⊕A21 ⊕A22 ⊕···
⊕A2m2 ⊕···⊕As1 ⊕As2 ⊕···⊕Asms.

(ii) We know by Proposition 2.6 that (A,µ,α) is a Hom-alternative alge-
bra. Next, for all 1 ≤ i ≤ s, the condition α(Ai) = Ai implies

µ(Ai,A) = α
(
µ′(Ai,A)

)
⊆ α(Ai) = Ai,

µ(A,Ai) = α
(
µ′(A,Ai)

)
⊆ α(Ai) = Ai.

It follows that Ai are two-sided Hom-ideals of (A,µ,α).
If there exits non trivial two-sided Hom-ideal Ai0 of (Ai,µ,α), then we

have

µ(Ai0,A) = µ
(
Ai0,A1 ⊕A2 ⊕···⊕As

)
= µ(Ai0,Ai) ⊆ Ai0,

µ(A,Ai0 ) = µ
(
A1 ⊕A2 ⊕···⊕As,Ai0

)
= µ(Ai0,Ai) ⊆ Ai0.



18 s. attan

It follows that Ai0 is a non trivial two-sided Hom-ideal of (A,µ,α). Thanks
to the proof of Theorem 3.16 (ii), Ai0 is also a non trivial two-sided ideal of
(A,µ′). Hence, Ai0 is also a non trivial two-sided ideal of (Ai,µ

′) which is
a contradiction. It follows that Ai (i = 1, . . . ,s) are simple two-sided Hom-
ideals of (A,µ,α) and therefore (A,µ,α) is semi-simple and has the unique
decomposition.

Proposition 3.21. Let (A,µ,α) be a Hom-alternative algebra such that
α2 = α. Then (A,µ,α) is isomorphic to the decomposition of direct sum of
Hom-alternative algebras, i.e.,

A ∼= (A/ Ker(α)) ⊕ Ker(α).

Proof. It is clear that (Ker(α),µ,α) is a Hom-alternative algebra and
thanks to Proposition 3.11 since Ker(α) is a two-sided Hom-ideal of (A,µ,α),
the quotient Hom-algebra (A/ Ker(α), µ̄, ᾱ) is a Hom-alternative algebra. Now
set A1 = (A/ Ker(α)) ⊕ Ker(α) and define µ1 : A×21 → A1 and α1 : A1 → A1
by

µ1
(
(x̄,h), (ȳ,k)

)
:=

(
µ(x,y),µ(h,k)

)
, α1

(
(x̄,h)

)
:=

(
α(x), 0

)
.

Then one can show that (A1,µ1,α1) is a Hom-alternative algebra which is
a decomposition of direct sum of Hom-alternative algebras. Next, let
y ∈ Ker(α) ∩ Im(α). Then, there exists x ∈ A such that y = α(x). Moreover,
we have

0 = α(y) = α2(x) = α(x) = y.

It follows that Ker(α) ∩ Im(α) = {0}, and then A = Ker(α) ⊕ Im(α) since
for any x ∈ A we have x = (x − α(x)) + α(x) with x − α(x) ∈ Ker(α) and
α(x) ∈ Im(α).

Now, let show that (Im(α),µ,α) ∼=
(
V/ Ker(α), µ̄, ᾱ

)
. Note that, it is clear

that
(

Im(α),µ,α
)

is a Hom-alternative algebra. Define ϕ : V/ Ker(α) →
Im(α) by ϕ(x̄) = α(x) for all x̄ ∈ A/ Ker(α). Clearly, ϕ is bijective and for all
x̄, ȳ ∈ A/ Ker(α), we have

ϕ
(
µ̄(x̄, ȳ)

)
= ϕ

(
µ(x,u)

)
= α

(
µ(x,y)

)
= µ

(
α(x),α(y)

)
= µ

(
ϕ(x̄),ϕ(ȳ)

)
,

ϕ
(
ᾱ(x̄)

)
= ϕ

(
α(x)

)
= α2(x) = α

(
ϕ(x̄)

)
,

i.e., ϕ ◦ µ̄ = µ̄ ◦ ϕ⊗2 and ϕ ◦ ᾱ = α ◦ ϕ. It follows that (Im(α),µ,α) ∼=(
V/ Ker(α), µ̄, ᾱ

)
and therefore

V = Ker(α) ⊕ Im(α) ∼= (A/ Ker(α)) ⊕ Ker(α).



structure and bimodules 19

4. Bimodules over simple Hom-alternative algebras

In this section, we mainly study bimodules over Hom-alternative algebras
of alternative-type. We give a theorem about the relationship between bimod-
ules over Hom-alternative algebras of alternative-type and the ones over their
induced alternative algebras. Moreover, some relevant propositions about
bimodules over Hom-alternative algebras are also displayed. As a conse-
quence, interesting results about bimodules over finite dimensional simple
Hom-alternative algebras are given.

Definition 4.1. Let A′ = (A,µ) be any algebra and V be a K-module.

1. A left (resp. right) structure map on V is a morphism δl : A⊗V → V ,
a⊗v 7−→ a ·v (resp. δr : V ⊗A → V , v ⊗a 7−→ v ·a) of Hom-modules.

2. Let δl and δr be structure maps on V . Then the module associator of
V is a trilinear map ( , , )A,V defined as:

( , , )A,V ◦ IdV⊗A⊗A = δr ◦ (δr ⊗ IdA) −δl ◦ (IdV ⊗µ),

( , , )A,V ◦ IdA⊗V⊗A = δr ◦ (δl ⊗ IdA) −δl ◦ (IdA ⊗δr),

( , , )A,V ◦ IdA⊗A⊗V = δl ◦ (µ⊗ IdV ) −δl ◦ (IdA ⊗δl).

A bimodule over alternative algebras is given in [8, 16].

Definition 4.2. ([8, 16]) Let A′ = (A,µ) be an alternative algebra. An
alternative A′-bimodule is a K-module V that comes equipped with a (left)
structure map δl : A⊗V → V (δl(a⊗v) = a ·v) and a (right) structure map
δr : V ⊗A → V (δr(v ⊗a) = v ·a) such that the following equalities holds:

(a,v,b)A,V = −(v,a,b)A,V = (b,a,v)A,V = −(a,b,v)A,V . (5)

The notion of alternative bimodules has been extended to the Hom-
alternative bimodules. More precisely, we get

Definition 4.3. ([2]) Let A = (A,µ,αA) be a Hom-alternative algebra.
A Hom-alternative A-bimodule is a Hom-module (V,αV ) that comes equipped
with a (left) structure map ρl : A⊗V → V (ρl(a⊗v) = a ·v) and a (right)
structure map ρr : V ⊗ A → V (ρr(v ⊗ a) = v · a) such that the following
equalities

asA,V (a,v,b) = −asA,V (v,a,b) = asA,V (b,a,v) = −asA,V (a,b,v) (6)



20 s. attan

hold for all a,b,c ∈ A and v ∈ V , where asA,V is the module Hom-associator
of the Hom-module (V,αV ) defined by

asA,V ◦IdV⊗A⊗A = ρr ◦ (ρr ⊗αA) −ρl ◦ (αV ⊗µ),

asA,V ◦IdA⊗V⊗A = ρr ◦ (ρl ⊗αA) −ρl ◦ (αA ⊗ρr),

asA,V ◦IdA⊗A⊗V = ρl ◦ (µ⊗αV ) −ρl ◦ (αA ⊗ρl).

Remark 4.4. If αA = IdA and αV = IdV , the notion of Hom-alternative
bimodule is reduced to the one of alternative bimodule.

Theorem 4.5. (i) Let A = (A,µ,αA) be a Hom-alternative algebra of
alternative-type with A′ = (A,µ′) its induced alternative algebra and
(V,αV ) be a Hom-alternative A-bimodule with the structure maps ρl
and ρr such that αV is invertible. Then V is an alternative A′-bimodule
with the structures maps δl = α

−1
V ◦ρl and δr = α

−1
V ◦ρr.

(ii) Let A′ = (A,µ′) be an alternative algebra and A = (A,µ = αA ◦µ′,αA)
a corresponding Hom-alternative algebra. Let V be an alternative A′-
bimodule with the structure maps δl and δr and αV ∈ End(V ) such that
αV ◦δl = δl◦(αA⊗αV ) and αV ◦δr = δr ◦(αV ⊗αA). Then (V,αV ) is a
Hom-alternative A-bimodule with the structures maps ρl = αV ◦δl and
ρr = αV ◦δr where αA is a morphism of (A,µ′).

Proof. (i) Let (V,αV ) be an alternative A-bimodule with the structure
maps ρl and ρr such that αV is invertible. Note that the structure maps
δl = α

−1
V ◦ρl and δr = α

−1
V ◦ρr satisfy conditions

αV ◦ δl = δl ◦ (αA ⊗αV ) and αV ◦ δr = δr ◦ (αV ⊗αA). (7)

Therefore,

asA,V ◦IdA⊗V⊗A = ρr ◦ (ρl ⊗αA) −ρl ◦ (αA ⊗ρr)
= αV ◦ δr ◦ (αV ◦δl ⊗αA) −αV ◦ δl ◦ (αA ⊗αV ◦ δr)

= α2V ◦ δr ◦ (δl ⊗ IdA) −α
2
V ◦ δl ◦ (IdA ⊗δr) (by (7))

= α2V ◦ ( , , )A,V ◦ IdA⊗V⊗A.

Thus, ( , , )A,V ◦ IdA⊗V⊗A = (α2V )
−1 ◦ asA,V ◦IdA⊗V⊗A. Similarly, we get

( , , )A,V ◦ IdA⊗A⊗V = (α2V )
−1 ◦ asA,V ◦IdA⊗A⊗V ,

( , , )A,V ◦ IdV⊗A⊗A = (α2V )
−1 ◦ asA,V ◦IdV⊗A⊗A.



structure and bimodules 21

Finally, Condition (5) follows from Condition (6).
(ii) Similar to (i).

Corollary 4.6. Let A = (A,µ,αA) be a finite dimensional simple Hom-
alternative algebra and (V,αV ) be a Hom-alternative A-bimodule with the
structure maps ρl and ρr such that αV is invertible. Then V is an alternative
A′-bimodule with the structures maps δl = α−1V ◦ρl and δr = α

−1
V ◦ρr where

A′ = (A,µ′) is the induced alternative algebra.

Definition 4.7. Let A = (A,µ,αA) be a Hom-alternative algebra and
(V,αV ) be a Hom-alternative A-bimodule with the structure ρl and ρr.

1. A subspace V0 of V is called an A-subbimodule of (V,αV ) if αV (V0) ⊆ V0,
ρl(A⊗V0) ⊆ V0 and ρr(V0 ⊗A) ⊆ V0.

2. The Hom-alternative A-module (V,αV ) is said to be irreducible if it
has no non trivial A-subbimodules and completely reducible if V =
V1 ⊕V2 ⊕···⊕Vs where Vi (1 ≤ i ≤ s) are irreducible A-subbimodules
of (V,αV ).

Proposition 4.8. Let A = (A,µ,αA) be a Hom-alternative algebra and
(V,αV ) a Hom-alternative A-bimodule with the structure maps ρl and ρr.
Then Ker(αV ) is an A-subbimodule of (V,αV ). Moreover if αA is surjective
then Im(αV ) is an A-subbimodule of (V,αV ) and we have the isomorphism
of A-bimodules ᾱV : V/ Ker(αV ) → Im(αV ).

Proof. Obvious, we have αV (Ker(αV )) ⊆ Ker(αV ).
Next, let (v,a) ∈ Ker(αV ) ×A. Then we get

αV (ρl(a⊗v)) = ρl(αA(a) ⊗αV (v)) = 0,
αV (ρr(v ⊗a)) = ρr(αV (v) ⊗αA(a)) = 0

since αV (v) = 0. Therefore Ker(αV ) is an A-subbimodule of (V,αV ).
Similarly, it is obvious that αV (Im(αV)) ⊆ Im(αV). Let (v,a) ∈ Im(αV)×A.

Then there exits v′ ∈ V and if αA is surjective a′ ∈ A such that v = αV (v′)
and a = αA(a

′). Therefore

ρl(a⊗v) = ρl
(
αA(a

′) ⊗αV (v′)
)

= αV
(
ρl(a

′ ⊗v′)
)
∈ Im(αV ),

ρr(v ⊗a) = ρr
(
αV (v

′) ⊗αA(a′)
)

= αV
(
ρr(v

′ ⊗a′)
)
∈ Im(αV ).

Thus Im(αV ) is an A-subbimodule of (V,αV ).



22 s. attan

Finally, if define the map ᾱV : V/ Ker(αV ) → Im(αV ) by ᾱV (v̄) = αV (v),
then it is easy to prove that ᾱV is an isomorphism.

Corollary 4.9. Let A = (A,µ,αA) be a Hom-alternative algebra and
(V,αV ) be a finite dimensional irreducible Hom-alternative A-bimodule. Then
αV is invertible.

Proposition 4.10. Let A = (A,µ,αA) be a Hom-alternative algebra of
alternative-type and (V,αV ) a Hom-alternative A-bimodule with the struc-
ture maps ρl and ρr such that αV is invertible. If the alternative A′-bimodule
V over the induced alternative algebra A′ = (A,µ′) with the structures maps
δl = α

−1
V ◦ ρl and δr = α

−1
V ◦ ρr is irreducible, then the Hom-alternative

A-bimodule (V,αV ) is also irreducible.

Proof. Assume that the Hom-alternative A-bimodule (V,αV ) is reducible.
Then there exists a non trivial subspace V0 such that (V0,αV |V0 ) is an
A-subbimodule of (V,αV ). Therefore αV (V0) ⊆ V0, ρl(a ⊗ v)) ∈ V0 and
ρr(v ⊗ a) ∈ V0 for all (a,v) ∈ A × V0. Hence δl(a ⊗ v)) ∈ α−1V (V0) = V0
and δr(v ⊗ a) ∈ α−1V (V0) = V0. Thus V0 is a non trivial A

′-subbimodule of
V , contradiction. It follows that (V,αV ) is an irreducible Hom-alternative
A-bimodule.

Corollary 4.11. Let A = (A,µ,αA) be a finite dimensional simple Hom-
alternative algebra and (V,αV ) a Hom-alternative A-bimodule with the struc-
ture maps ρl and ρr such that αV is invertible. If the alternative A′-bimodule
V over the induced alternative algebra A′ = (A,µ′) with the structures maps
δl = α

−1
V ◦ ρl and δr = α

−1
V ◦ ρr is irreducible, then the Hom-alternative

A-bimodule (V,αV ) is also irreducible.

Let recall the following result from [16] which is very useful for the next result.

Theorem 4.12. ([16]) Let A′ = (A,µ′) be a semi-simple alternative al-
gebra. Then any representation (S,T) of A′ is completely reducible.

Since the notion of completely reducible representation of A′, is equivalent
to the notion of a completely reducible alternative A′-bimodule, thanks to
Theorem 3.20 (i) and Theorem 4.5 (i), we get the following important result.

Corollary 4.13. Let A = (A,µ,α) be a finite dimensional semi-simple
Hom-alternative algebra. Then any Hom-alternative A-bimodule (V,αV ) such
that αV is invertible, is completely reducible.



structure and bimodules 23

Example 4.14. Consider the following Hom-algebra A = (A,µ,α) where
the non-zero products are given by µ(e1,e1) = e1, µ(e2,e2) = e2, µ(e3,e3) =
e1, µ(e1,e3) = µ(e3,e1) = −e3 and α(e1) = e1, α(e3) = −e3. Then A is a
semi-simple Hom-alternative algebra with the decomposition A1 ⊕A2 where
A1 and A2 are simple two sided-Hom-ideals generated by (e2) and (e1,e3)
respectively. Therefore, any Hom-alternative A-bimodule (V,αV ) such that
αV is invertible, is completely reducible.

Acknowledgements

The author would like to thank the anonymous reviewers for their
valuable comments which improved the paper.

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https://arxiv.org/abs/1906.04561v1
https://arxiv.org/abs/0712.3515v1

	Introduction
	Preliminaries
	Structures of Hom-alternative algebras
	Bimodules over simple Hom-alternative algebras