A non-abelian Hom-Leibniz tensor product andapplications

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Linear and Multilinear Algebra

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A non-abelian Hom-Leibniz tensor product and applications

J. M. Casas, E. Khmaladze & N. Pacheco Rego

To cite this article: J. M. Casas, E. Khmaladze & N. Pacheco Rego (2017): A non- abelian Hom-Leibniz tensor product and applications, Linear and Multilinear Algebra, DOI:

10.1080/03081087.2017.1338651

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https://doi.org/10.1080/03081087.2017.1338651

A non-abelian Hom-Leibniz tensor product and applications

J. M. Casas

^a

, E. Khmaladze

^b

and N. Pacheco Rego

^c

aDepartamento Matemática Aplicada I, Universidad de Vigo, Pontevedra, Spain;^bA. Razmadze Math. Inst. of I.

Javakhishvili Tbilisi State University, Georgia & University of Georgia, Tbilisi, Georgia;^cIPCA, Departamento de Ciências, Campus do IPCA, Barcelos, Portugal

ABSTRACT

The notion of non-abelian Hom–Leibniz tensor product is introduced and some properties are established. This tensor product is used in the description of the universal (α-)central extensions of Hom–Leibniz algebras. We also give its application to the Hochschild homology of Hom-associative algebras.

ARTICLE HISTORY Received 28 January 2016 Accepted 1 June 2017 COMMUNICATED BY L.-H. Lim

KEYWORDS Hom–Leibniz algebra;

non-abelian tensor product;

universal (α)-central extension; Hom-associative algebra; Hochschild homology AMS SUBJECT CLASSIFICATIONS 17A30; 17B55; 17B60;

18G35; 18G60

1. Introduction

Since the invention of Hom–Lie algebras as the algebraic model of deformed Lie algebras coming from twisted discretizations of vectors fields by Hartwig, Larsson and Silvestrov in [1], many papers appeared dealing with Hom-type generalizations of various algebraic structures (see for instance [2–12] and related references given therein). In particular, Makhlouf and Silvestrov introduced the notion of Hom–Leibniz algebras in [7], which simultaneously is a non-commutative generalization of Hom–Lie algebras and Hom-type generalization of Leibniz algebras. In this generalized framework, it is natural to seek for possible extensions of results in the categories of Lie or Leibniz algebras to the categories of Hom–Lie or Hom–Leibniz algebras.

Recently, in [13], we developed the non-abelian Hom–Lie tensor product, extending the non-abelian Lie tensor product by Ellis [14] from Lie to Hom–Lie algebras. It has applications in universal ( α -)central extensions of Hom–Lie algebras and cyclic homology of Hom-associative algebras.

In this paper, we have chosen to work with Hom–Leibniz algebras. This work is a continuation and non-commutative generalization of the work already begun in [13].

Thus, we introduce a non-abelian Hom–Leibniz tensor product, extending the non- abelian Leibniz tensor product by Gnedbaye [15], which itself is the Leibniz algebra

CONTACT J. M. Casas jmcasas@uvigo.es

© 2017 Informa UK Limited, trading as Taylor & Francis Group

(non-commutative) version of the non-abelian Lie tensor product [14,16]. Then, we investigate properties of the non-abelian Hom–Leibniz tensor product (Section 3) and give its applications to universal ( α -)central extensions of Hom–Leibniz algebras (Section 4) and Hochschild homology of Hom-associative algebras (Section 5).

One observes that not all results can be transferred from Leibniz to Hom–Leibniz algebras. For example, results on universal central extensions of Leibniz algebras cannot be extended directly to Hom–Leibniz algebras because the category of Hom–Leibniz algebras doesn’t satisfy the so-called universal central extension condition [17], which means that the composition of central extensions is not central in general [4]. By this reason, the notion of α -central extension of Hom–Leibniz algebras is introduced in [4], and classical results are divided between universal central and universal α -central extensions of Hom–Leibniz algebras (see Theorem 4.1). Further, Hom-type version of Gnedbaye’s result relating Hochschild and Milnor type Hochschild homology of associative algebras [15], doesn’t hold for all Hom-associative algebras and requires an additional condition (see Theorem 5.2).

Notations

Throughout this paper we fix K as a ground field. Vector spaces are considered over K and linear maps are K-linear maps. We write ⊗ for the tensor product ⊗

_K

over K. For any vector space (resp. Hom–Leibniz algebra) L, a subspace (resp. a two-sided ideal) L

and x ∈ L we write x to denote the coset x + L

. We denote by Lie and Lb the categories of Lie and Leibniz algebras, respectively.

2. Preliminaries on Hom–Leibniz algebras 2.1. Basic definitions

In this section, we review some terminology on Hom–Leibniz algebras and recall notions used in the paper. We also introduce notions of actions and semi-direct product of Hom–

Leibniz algebras.

Definition 1 ([7]): A Hom–Leibniz algebra is a triple ( L, [− , −] , α

L

) consisting of a vector space L, a bilinear map [−, −] : L × L → L, called bracket operation, and a linear map α

L

: L → L satisfying:

[α

L

(x), [y, z]] = [[x, y], α

L

(z)] − [[x, z], α

L

(y)] (Hom–Leibniz identity) (2.1) for all x, y, z ∈ L.

In the whole paper we only deal with (the so-called multiplicative [12]) Hom–Leibniz algebras (L, [−, −], α

L

) such that α

L

preserves the bracket operation, that is, α

L

[x, y] = [α

L

(x), α

L

(y)], for all x, y ∈ L.

Example 1:

(a) Taking α = id in Definition 1, we obtain the definition of a Leibniz algebra [18].

Hence any Leibniz algebra L can be considered as a Hom–Leibniz algebra with

α

L

= id.

(b) Any Leibniz algebra L can be considered as a Hom–Leibniz algebra with α

L

= 0.

In fact, any vector space L endowed with any bracket operation is a Hom–Leibniz algebra with α

L

= 0.

(c) Any Hom–Lie algebra [1] is a Hom–Leibniz algebra whose bracket operation satisfies the skew-symmetry condition.

(d) Any Hom-dialgebra [11] ( D, , , α

D

) becomes a Hom–Leibniz algebra ( D, [− , −] , α

D

) with the bracket given by [x, y] = x y − y x, for all x, y ∈ D .

(e) Let ( L, [− , −]) be a Leibniz algebra and α

L

: L → L a Leibniz algebra endomor- phism. Define [−, −]

_α

: L ⊗ L → L by [x, y]

_α

= [α(x), α(y)], for all x, y ∈ L. Then ( L, [− , −]

α

, α

L

) is a Hom–Leibniz algebra.

(f) Any Hom-vector space ( V, α

V

) , (i.e. V is a vector space and α

V

: V → V is a linear map) together with the trivial bracket [−, −] (i.e. [x, y] = 0 for all x, y ∈ V) is a Hom–Leibniz algebra, called abelian Hom–Leibniz algebra.

(g) The two-dimensional C-vector space L with basis {a

1

, a

₂

}, endowed with the bracket operation [ a

₂

, a

₂

] = a

₁

and zero elsewhere, and the endomorphism α

L

given by the matrix

1 1 0 1

is a non-Hom–Lie Hom–Leibniz algebra.

In the sequel we shall use the shortened notation (L, α

L

) for (L, [−, −], α

L

).

Definition 2: A homomorphism of Hom–Leibniz algebras f : (L, α

L

) → (L

, α

L

) is a linear map f : L → L

such that

f ([ x, y ]) = [ f ( x ) , f ( y )] , f ◦ α

L

= α

L

◦ f , for all x, y ∈ L.

Hom–Leibniz algebras and their homomorphisms form a category, which we denote by HomLb. Example 1 a), b) and c) say that there are full embedding functors

I

₀

, I

₁

: Lb −→ HomLb, I

₀

(L) = (L, 0), I

₁

(L) = (L, id) and

inc : HomLie −→ HomLb, ( L, α

L

) → ( L, α

L

) ,

where HomLie denotes the category of Hom–Lie algebras. Note that the restrictions of I

₀

and I

₁

to the category of Lie algebras are full embeddings

I

₀

, I

₁

: Lie −→ HomLie, I

₀

( L ) = ( L, 0 ) , I

₁

( L ) = ( L, id ).

Definition 3: A Hom–Leibniz subalgebra ( H, α

H

) of a Hom–Leibniz algebra (L, α

L

) consists of a vector subspace H of L, which is closed under the bracket and invariant under the map α

L

(i.e. α

L

( H ) ⊆ H), together with the linear map α

H

: H → H being the restriction of α

L

on H. In such a case we may write α

_L|

for α

H

.

A Hom–Leibniz subalgebra ( H, α

H

) of ( L, α

L

) is said to be a two-sided Hom-ideal if [x, y], [y, x] ∈ H, for all x ∈ H, y ∈ L.

If ( H, α

H

) is a two-sided Hom-ideal of ( L, α

L

) , then the quotient vector space L / H

together with the endomorphism α

L

: L / H → L / H induced by α

L

, naturally inherits a

structure of Hom–Leibniz algebra, and it is called the quotient Hom–Leibniz algebra.

The commutator of two-sided Hom-ideals ( H, α

L|

) and ( K , α

L|

) of a Hom–Leibniz algebra (L, α

L

), denoted by ([H, K], α

_L|

), is the Hom–Leibniz subalgebra of (L, α

L

) spanned by the brackets [ h, k ] and [ k, h ] for all h ∈ H, k ∈ K.

The following lemma can be readily checked.

Lemma 2.1: Let ( H, α

H

) and ( K , α

K

) be two-sided Hom-ideals of a Hom–Leibniz algebra (L, α

L

). The following statements hold:

(a) (H ∩ K , α

_L|

) and (H + K , α

_L|

) are two-sided Hom-ideals of (L, α

L

);

(b) [ H, K ] ⊆ H ∩ K;

(c) ([H, K ], α

_L|

) is a two-sided Hom-ideal of (H, α

H

) and (K , α

K

). In particular, ([L, L], α

_L|

) is a two-sided Hom-ideal of (L, α

L

);

(d) (α

L

( L ) , α

_L|

) is a Hom–Leibniz subalgebra of ( L, α

L

) ;

(e) If H, K ⊆ α

L

(L), then ([H, K ], α

_L|

) is a two-sided Hom-ideal of (α

L

(L), α

_L|

);

(f) If α

L

is surjective, then ([ H, K ] , α

L|

) is a two-sided Hom-ideal of ( L, α

L

) .

Definition 4: Let ( L, α

L

) be a Hom–Leibniz algebra. The subspace Z ( L ) = { x ∈ L | [x, y] = 0 = [y, x], for all y ∈ L} of L is said to be the center of (L, α

L

).

Note that if α

L

: L → L is a surjective homomorphism, then (Z(L), α

_L|

) is a two-sided Hom-ideal of L.

Remark 1: Any Hom–Leibniz algebra ( L, α

L

) gives rise to a Hom–Lie algebra ( L

_Lie

, α

L

) , which is obtained as the quotient of L by the relation [x, x] = 0, x ∈ L. Here α

L

is induced by α

L

. This defines a functor ( − )

Lie

: HomLb −→ HomLie: Moreover, the canonical epimorphism π : (L, α

L

) (L

Lie

, α

L

) is universal among all homomorphisms from ( L, α

L

) to a Hom–Lie algebra, implying that the functor ( − )

Lie

is left adjoint to the inclusion functor inc : HomLie −→ HomLb. This adjunction is a natural extension of the well-known adjunction Lie

⊥

inc

// Lb

(−)Lie

oo between the categories of Lie and Leibniz algebras, in the sense that the following inner and outer diagrams

Lie

⊥

inc

//

I_i

Lb

(−)Lie

oo

Ii

HomLie

inc

//

HomLb

(−)Lie

oo

are commutative for i = 0, 1.

Proof: Sice α

L

preserves the bracket, then α

L

: L

Lie

→ L

Lie

is a well-defined linear

map preserving the (induced) bracket. Given a homomorphism of Hom-Leibiz algebras

f : (L, α

L

) → (L

, α

L

), where (L

, α

L

) is any Hom–Lie algebra, then there is a unique

homomorphism of Hom–Lie algebras g : ( L

_Lie

, α

L

) → ( L

, α

L

) , g ( x ) = f ( x ) , such that

gπ = f . This means that the functor ( − )

Lie

is left adjoint to the inclusion functor. Finally,

it is readily checked that the declared diagrams are commutative.

2.2. Hom–Leibniz actions and semi-direct product Definition 5: Let

L, α

L

and M, α

M

be Hom–Leibniz algebras. A Hom–Leibniz action of

L, α

L

on M, α

M

is a couple of linear maps, L ⊗ M → M, x ⊗ m →

^x

m and M ⊗ L → M, m ⊗ x → m

^x

, satisfying the following identities:

(a) α

M

m

_[x,y]

=

m

^x

_αLy

−

m

^y

_αLx

, (b)

^[x,y]

α

M

m

=

_x

m

_αLy

−

^αLx

m

^y

, (c)

^αLx

_y

m

= −

^αLx

m

^y

, (d)

^αLx

m, m

=

_x

m, α

M

m

−

_x

m

, α

M

m , (e)

m, m

_α

L x

= m

^x

, α

M

m

+

α

M

m , m

^x

, (f)

α

M

m ,

^x

m

= − α

M

m , m

^x

, (g) α

M

_x

m

=

^αLx

α

M

m , (h) α

M

m

^x

= α

M

m

_αLx

, for all x, y ∈ L and m, m

∈ M.

The action is called trivial if

^x

m = 0 = m

^x

, for all x ∈ L, m ∈ M.

Example 2:

(a) A Hom-representation of a Hom–Leibniz algebra ( L, α

L

) is a Hom-vector space (M, α

M

) equipped with two linear operations, L ⊗ M → M, x ⊗ m →

^x

m and M ⊗ L → M, m ⊗ x → m

^x

, satisfying the axioms a), b), c), g), h) of Definition 5.

Therefore, representations over a Hom–Leibniz algebra ( L, α

L

) are abelian Leibniz algebras enriched with Hom–Leibniz actions of (L, α

L

).

(b) Any action of a Leibniz algebra L on another Leibniz algebra M (see e.g. [19]) gives a Hom–Leibniz action of (L, id

_L

) on (M, id

_M

).

(c) Let K, α

K

be a Hom–Leibniz subalgebra of a Hom–Leibniz algebra L, α

L

and H, α

H

be a two-sided Hom-ideal of L, α

L

. There exists a Hom–Leibniz action of K , α

K

on H, α

H

given by the bracket in L, α

L

.

(d) Let 0 → (M, α

M

) →

ⁱ

(K , α

K

) →

^π

(L, α

L

) → 0 be a split short exact sequence of Hom–Leibniz algebras, i. e. there exists a homomorphism of Hom–Leibniz algebras s : ( L, α

L

) → ( K, α

K

) such that π ◦ s = id

_L

. Then there is a Hom–Leibniz action of (L, α

L

) on (M, α

M

) defined in the standard way:

^x

m = i

⁻¹

[s(x), i(m)] and m

^x

= i

⁻¹

[ i ( m ) , s ( x )] for all x ∈ L, m ∈ M.

Definition 6: Let M, α

M

and L, α

L

be Hom–Leibniz algebras together with a Hom–

Leibniz action of L, α

L

on M, α

M

. Their semi-direct product

M L, α

is the Hom–

Leibniz algebra with the underlying vector space M ⊕ L, endomorphism α : M L → M L given by α

m, l

= α

M

m , α

L

l

and bracket m

₁

, l

₁

, m

₂

, l

₂

=

[m

1

, m

₂

] +

^αLl1

m

₂

+ m

^αL

l2

1

,

l

₁

, l

₂

. Let

M, α

M

and L, α

L

be Hom–Leibniz algebras with a Hom–Leibniz action of L, α

L

on

M, α

M

. Then we have the following short exact sequence of Hom–Leibniz algebras

0 → M, α

M

i

→

M L, α

_π

→ L, α

L

→ 0 , (2.2)

where i : M → M L, i ( m ) = m, 0

, and π : M L → L, π m, x

= x. Moreover, this sequence splits by the Hom–Leibniz homomorphism L → M L, x →

0, x .

2.3. Homology of Hom–Leibniz algebras

In this subsection, we recall from [4] the construction of homology vector spaces of a Hom–Leibniz algebra with coefficients in a Hom-co-representation.

Definition 7: A Hom-co-representation of a Hom–Leibniz algebra ( L, α

L

) is a Hom- vector space ( M, α

M

) together with two linear maps L ⊗ M → M, x ⊗ m →

^x

m and M ⊗ L → M, m ⊗ x = m

^x

, satisfying the following identities:

(a)

^[x,y]

α

M

(m) =

^αL(x)

(

^y

m) −

^αL(y)

(

^x

m), (b) α

M

( m )

^[x,y]

= (

^y

m )

^αL(x)

−

^αL(y)

( m

^x

) , (c) (m

^x

)

^αL(y)

= −

^αL(y)

(m

^x

),

(d) α

M

(

^x

m ) =

^αL(x)

α

M

( m ) , (e) α

M

(m

^x

) = α

M

(m)

^αL(x)

, for any x, y ∈ L and m ∈ M Example 3:

(a) Let M be a co-representation of a Leibniz algebra L [19], then ( M, id

_M

) is a Hom- co-representation of the Hom–Leibniz algebra (L, id

_L

).

(b) The underlying Hom-vector space of a Hom–Leibniz algebra ( L, α

L

) has a Hom- co-representation structure given by

^x

y = −[y, x] and y

^x

= [y, x], x, y ∈ L.

Let (L, α

L

) be a Hom–Leibniz algebra and (M, α

M

) be a Hom-co-representation of ( L, α

L

) . The homology HL

^α

( L, M ) of ( L, α

L

) with coefficients in ( M, α

M

) is defined to be the homology of the chain complex (CL

^α

(L, M), d

), where

CL

_n^α

( L, M ) := M ⊗ L

^⊗n

, n ≥ 0

and d

_n

: CL

^α_n

(L, M) → CL

^α_n−1

(L, M), n ≥ 1, is the linear map given by d

_n

( m ⊗ x

₁

⊗ · · · ⊗ x

_n

) = m

^x1

⊗ α

L

( x

₂

) ⊗ · · · ⊗ α

L

( x

_n

)

+

n

i=2

( − 1)

^{i xi}

m ⊗ α

L

(x

1

) ⊗ · · · ⊗ α

L

(x

i

) ⊗ · · · ⊗ α

L

(x

n

)

+

1≤i<j≤n

( − 1 )

^j+1

α

M

( m ) ⊗ α

L

( x

₁

) ⊗ · · · ⊗ α

L

( x

_i−1

) ⊗ [ x

_i

, x

_j

] ⊗ α

L

( x

_i+1

)

⊗ · · · ⊗ α

L

( x

_j

) ⊗ · · · ⊗ α

L

( x

_n

) ,

where the notation α

L

(x

i

) indicates that the variable α

L

(x

i

) is omitted. Hence HL

^α_n

( L, M ) := H

_n

( CL

^α

( L, M ) , d

) , n ≥ 0 .

Direct calculations show that HL

^α₀

( L, M ) = M / M

^L

, where M

^L

= { m

^x

| m ∈ M, x ∈ L } , and if ( M, α

M

) is a trivial Hom-co-representation of ( L, α

L

) , that is m

^x

=

^x

m = 0, then HL

^α₁

(L, M) =

M ⊗ L /

α

M

(M) ⊗ [L, L]

. In particular, if M = K then HL

^α₁

(L, K) =

L /[ L, L ] . Later on we write HL

^α_n

( L ) for HL

^α_n

( L, K) and it is called homology with trivial coefficients.

3. Non-abelian Hom–Leibniz tensor product

Let (M, α

M

) and (N , α

N

) be Hom–Leibniz algebras acting on each other. We denote by M ∗ N the vector space spanned by all symbols m ∗ n, n ∗ m and subject to the following relations:

λ(m ∗ n) = (λm) ∗ n = m ∗ (λn), λ(n ∗ m) = (λn) ∗ m = n ∗ (λm), ( m + m

) ∗ n = m ∗ n + m

∗ n,

m ∗ (n + n

) = m ∗ n + m ∗ n

, ( n + n

) ∗ m = n ∗ m + n

∗ m, n ∗ (m + m

) = n ∗ m + n ∗ m

, α

M

(m) ∗

n, n

= m

ⁿ

∗ α

N

(n

) − m

ⁿ

∗ α

N

(n), α

N

( n ) ∗

m, m

= n

^m

∗ α

M

( m

) − n

^m

∗ α

M

( m ) , m, m

∗ α

N

( n ) =

^m

n ∗ α

M

( m

) − α

M

( m ) ∗ n

^m

, n, n

∗ α

M

( m ) =

ⁿ

m ∗ α

N

( n

) − α

N

( n ) ∗ m

ⁿ

, α

M

( m ) ∗

^m

n = −α

M

( m ) ∗ n

^m

,

α

N

(n) ∗

ⁿ

m = −α

N

(n) ∗ m

ⁿ

, m

ⁿ

∗

^m

n

=

^m

n ∗ m

ⁿ

,

m

ⁿ

∗ n

^m

=

^m

n ∗

ⁿ

m

,

n

m ∗

^m

n

= n

^m

∗ m

ⁿ

,

n

m ∗ n

^m

= n

^m

∗

ⁿ

m

,

(3.1)

for all λ ∈ K, m, m

∈ M, n, n

∈ N.

We claim that ( M ∗ N , α

M∗N

) is a Hom-vector space, where α

M∗N

is the linear map induced by α

M

and α

N

, i.e.

α

_M∗N

m ∗ n

= α

M

m

∗ α

N

n

, α

_M∗N

n ∗ m

= α

N

n

∗ α

M

m . Indeed, it can be checked readily that α

M∗N

preserves all the relations in (3.1).

To be able to introduce the non-abelian Hom–Leibniz tensor product, we need to assume that the actions are compatible in the following sense.

Definition 8: Let ( M, α

M

) and ( N , α

N

) be Hom–Leibniz algebras with Hom–Leibniz actions on each other. The actions are said to be compatible if

(^mn)

m

= [ m

ⁿ

, m

] ,

^(nm)

n

= [ n

^m

, n

] ,

(n^m)

m

= [

ⁿ

m, m

] ,

^(mn)

n

= [

^m

n, n

] , m

^(mn)

= [m, m

ⁿ

], n

^(nm)

= [n, n

^m

], m

^(nm)

= [m,

ⁿ

m

], n

^(mn)

= [n,

^m

n

],

(3.2)

for all m, m

∈ M and n, n

∈ N .

Example 4: If ( H, α

H

) and ( H

, α

H

) both are Hom-ideals of a Hom–Leibniz algebra (L, α

L

), then the Hom–Leibniz actions of (H, α

H

) and (H

, α

H

) on each other, considered in Example 2 c), are compatible.

Now we have the following property:

Proposition 3.1: Let ( M, α

M

) and ( N, α

N

) be Hom–Leibniz algebras acting compatibly on each other, then the Hom-vector space (M ∗N , α

_M∗N

) endowed with the following bracket operation

m ∗ n, m

∗ n

= m

ⁿ

∗

^m

n

, m ∗ n, n

∗ m

= m

ⁿ

∗ n

^m

, n ∗ m, m

∗ n

=

ⁿ

m ∗

^m

n

, n ∗ m, n

∗ m

=

ⁿ

m ∗ n

^m

,

(3.3)

is a Hom–Leibniz algebra.

Proof: Routine calculations show that the bracket given by (3.3) is compatible with the defining relations in (3.1) of M ∗ N and can be extended from generators to any elements.

The verification of the Hom–Leibniz identity (2.1) is straightforward using compatibility conditions in (3.2). Finally, it follows directly by definition of α

_M∗N

that it preserves the bracket given by (3.3).

Definition 9: The above Hom–Leibniz algebra structure on (M ∗ N, α

_M∗N

) is called the non-abelian Hom–Leibniz tensor product of the Hom–Leibniz algebras

M, α

M

and N, α

N

.

Remark 2: If α

M

= id

_M

and α

N

= id

_N

, then M ∗ N coincides with the non-abelian tensor product of Leibniz algebras introduced in [15].

Remark 3: Let ( M, α

M

) and ( N , α

N

) be Hom–Lie algebras. One can readily check that the following assertions hold:

(a) Any Hom–Lie action of (M, α

M

) on (N, α

N

), M ⊗ N → N, m ⊗ n →

^m

n (see [13, Definition 1.7] for the definition), gives a Hom–Leibniz action of ( M, α

M

) on (N, α

N

) by letting n

^m

= −

^m

n for all m ∈ M and n ∈ N.

(b) For compatible, Hom–Lie actions (see [13, Definition 2.1]) of (M, α

M

) and (N, α

N

) on each other, the induced Hom–Leibniz actions are also compatible.

(c) If (M, α

M

) and (N, α

N

) act compatibly on each other, then there is an epimorphism of Hom–Leibniz algebras ( M ∗ N , α

M∗N

) ( M N, α

MN

) defined on generators by m ∗ n → mn and n ∗ m → −mn, where denotes the non-abelian Hom–Lie tensor product (see [13, Definition 2.4]).

Sometimes the non-abelian Hom–Leibniz tensor product can be described as the tensor product of vector spaces. In particular, we have the following:

Proposition 3.2: If the Hom–Leibniz algebras (M, α

M

) and (N, α

N

) act trivially on each other and both α

M

, α

N

are epimorphisms, then there is an isomorphism of abelian Hom–

Leibniz algebras

(M ∗ N, α

M∗N

) ∼ =

(M

^ab

⊗ N

^ab

) ⊕ (N

^ab

⊗ M

^ab

), α

_⊕

,

where M

^ab

= M /[ M, M ] , N

^ab

= N /[ N , N ] and α

_⊕

denotes the linear self-map of (M

^ab

⊗ N

^ab

) ⊕ (N

^ab

⊗ M

^ab

) induced by α

M

and α

N

.

Proof: Since the actions are trivial, then relations (3.3) enables us to see that (M ∗N , α

_M∗N

) is an abelian Hom–Leibniz algebra.

Since α

M

and α

N

are epimorphisms, the defining relations (3.1) of the non-abelian tensor product say precisely that the vector space M ∗ N is the quotient of (M ⊗ N)⊕

( N ⊗ M ) by the relations

m ∗ [n, n

] = [m, m

] ∗ n = [n, n

] ∗ m = n ∗ [m, m

] = 0

for all m, m

∈ M, n, n

∈ N . The later is isomorphic to ( M

^ab

⊗ N

^ab

) ⊕ ( N

^ab

⊗ M

^ab

) and this isomorphism commutes with the endomorphisms α

_⊕

and α

_M∗N

.

The non-abelian Hom–Leibniz tensor product is functorial in the following sense: if f : (M, α

M

) → (M

, α

M

) and g : (N, α

N

) → (N

, α

N

) are homomorphisms of Hom–

Leibniz algebras together with compatible actions of ( M, α

M

) (resp. ( M

, α

M

)) and ( N, α

N

) (resp. (N

, α

N

)) on each other such that f , g preserve these actions, that is

f (

ⁿ

m) =

^g(n)

f (m), f (m

ⁿ

) = f (m)

^g(n)

, g(

^m

n) =

^f(m)

g(n), g(n

^m

) = g(n)

^f(m)

,

for all m ∈ M, n ∈ N , then we have a homomorphism of Hom–Leibniz algebras f ∗ g : ( M ∗ N , α

M∗N

) → ( M

∗ N

, α

M∗N

)

defined by (f ∗ g)(m ∗ n) = f (m) ∗ g(n), (f ∗ g)(n ∗ m) = g(n) ∗ f (m).

Proposition 3.3: Let 0 → (M

1

, α

M1

) →

^f

(M

2

, α

M2

) →

^g

(M

3

, α

M3

) → 0 be a short exact sequence of Hom–Leibniz algebras. Let ( N , α

N

) be a Hom–Leibniz algebra together with compatible Hom–Leibniz actions of (N , α

N

) and (M

i

, α

Mi

) (i = 1, 2, 3) on each other and f , g preserve these actions. Then there is an exact sequence of Hom–Leibniz algebras

(M

1

∗ N, α

M1∗N

)

^f

−→

^∗idN

(M

2

∗ N, α

M2∗N

)

^g

−→

^∗idN

(M

3

∗ N , α

M3∗N

) −→ 0.

Proof: Clearly g ∗ id

_N

is an epimorphism and Im

f ∗ id

_N

⊆ Ker

g ∗ id

_N

. Since Im

f ∗ id

_N

is generated by all elements of the form f (m

1

) ∗ n, n ∗ f (m

1

), with m

₁

∈ M

₁

, n ∈ N , it is a two-sided Hom-ideal in ( M

₂

∗ N, α

M2∗N

) because of the following equalities:

f ( m

₁

) ∗ n, m

₂

∗ n

= f ( m

₁

)

ⁿ

∗

^m2

n

= f m

₁ⁿ

∗

^m2

n

∈ Im

f ∗ id

_N

,

f (m

1

) ∗ n, n

∗ m

₂

= f (m

1

)

ⁿ

∗ n

^m2

= f m

₁ⁿ

∗ n

^m2

∈ Im

f ∗ id

_N

,

n ∗ f (m

1

), m

₂

∗ n

=

ⁿ

f (m

1

) ∗

^m2

n

= f

_n

m

₁

∗

^m2

n

∈ Im

f ∗ id

_N

,

n ∗ f ( m

₁

) , n

∗ m

₂

=

ⁿ

f ( m

₁

) ∗ n

^m2

= f

_n

m

₁

∗ n

^m2

∈ Im

f ∗ id

_N

,

m

₂

∗ n

, f ( m

₁

) ∗ n

= m

₂ⁿ

∗

^f(m1)

n = m

₂ⁿ

∗ f

_m

1

n

∈ Im

f ∗ id

_N

,

m

₂

∗ n

, n ∗ f ( m

₁

)

= m

₂ⁿ

∗ n

^f(m1)

= m

ⁿ₂

∗ f n

^m1

∈ Im

f ∗ id

_N

,

n

∗ m

₂

, f ( m

₁

) ∗ n

=

ⁿ

m

₂

∗

^f(m1)

n =

ⁿ

m

₂

∗ f

_m

1

n

∈ Im

f ∗ id

_N

,

n

∗ m

₂

, n ∗ f ( m

₁

)

=

ⁿ

m

₂

∗ n

^f(m1)

=

ⁿ

m

₂

∗ f n

^m1

∈ Im

f ∗ id

_N

,

α

M2∗N

f (m

1

) ∗ n

= f (α

M2

m

₁

) ∗ α

N

n

∈ Im

f ∗ id

_N

, α

M2∗N

n ∗ f ( m

₁

)

= f (α

N

n

∗ α

M2

m

₁

) ∈ Im

f ∗ id

_N

,

for any m

₁

∈ M

₁

, m

₂

∈ M

₂

, n, n

∈ N . Thus, g ∗ id

_N

induces a homomorphism of Hom–Leibniz algebras

M

₂

∗ N / Im

f ∗ id

_N

, α

M2∗N

−→( M

₃

∗ N , α

M3∗N

) ,

given on generators by m

₂

∗ n → g(m

2

) ∗ n and n ∗ m

₂

→ n ∗ g(m

2

), which is an isomorphism with the inverse map

M

₃

∗ N , α

M3∗N

−→

M

₂

∗ N /Im

f ∗ id

_N

, α

M2∗N

defined by m

₃

∗ n → m

₂

∗ n and n ∗ m

₃

= n ∗ m

₂

, where m

₂

∈ M

₂

such that g(m

2

) = m

₃

. Then the required exactness follows.

Proposition 3.4: If ( M, α

M

) is a two-sided Hom-ideal of a Hom–Leibniz algebra ( L, α

L

) , then there is an exact sequence of Hom–Leibniz algebras

( M ∗ L ) ( L ∗ M ) , α

_σ

−→ ( L ∗ L, α

L∗L

) −→

^τ

( L / M ∗ L / M, α

L/M∗L/M

) −→ 0 . where the Hom–Leibniz action of L ∗ M on M ∗ L is naturally given by the bracket in L ∗ M.

Proof: First we note that τ is the functorial homomorphism induced by projection L, α

L

L/M, α

_L/M

and clearly it is an epimorphism.

Let σ

: ( M ∗ L, α

M∗L

) → ( L ∗ L, α

L∗L

) , σ

: ( L ∗ M, α

L∗M

) → ( L ∗ L, α

L∗L

) be the functorial homomorphisms induced by the inclusion (M, α

M

) → (L, α

L

) and the identity map ( L, α

L

) → ( L, α

L

). Then σ :

M ∗ L

L ∗ M , α

→ ( L ∗ L, α

L∗L

) , is defined by σ

x, y

= σ

(x) + α

_L∗L

◦ σ

(y) for all x ∈ M ∗ L and y ∈ L ∗ M, i. e.

σ ((m

1

∗ l

₁

), (l

2

∗ m

₂

)) = m

₁

∗ l

₁

+ α

L

(l

2

) ∗ α

M

(m

2

), σ ((l

1

∗ m

₁

), (l

2

∗ m

₂

)) = l

₁

∗ m

₁

+ α

L

(l

2

) ∗ α

M

(m

2

), σ ((m

1

∗ l

₁

), (m

2

∗ l

₂

)) = m

₁

∗ l

₁

+ α

M

(m

2

) ∗ α

L

(l

2

), σ ((l

1

∗ m

₁

), (m

2

∗ l

₂

)) = l

₁

∗ m

₁

+ α

M

(m

2

) ∗ α

L

(l

2

), for all m

₁

, m

₂

∈ M and l

₁

, l

₂

∈ L.

The exactness can be checked in the same way as in the proof of Proposition 3.3 and we omit it.

Proposition 3.5: Let (M, α

M

) and (N , α

N

) be Hom–Leibniz algebras with compatible actions on each other.

(a) There are homomorphisms of Hom–Leibniz algebras

ψ

1

: (M ∗ N, α

M∗N

) → (M, α

M

), ψ

1

(m ∗ n) = m

ⁿ

, ψ

1

(n ∗ m) =

ⁿ

m,

ψ

2

: ( M ∗ N, α

M∗N

) → ( N, α

N

) , ψ

2

( m ∗ n ) =

^m

n, ψ

2

( n ∗ m ) = n

^m

.

(b) There is a Hom–Leibniz action of ( M, α

M

) (resp. ( N , α

N

) ) on the non-abelian tensor product (M ∗ N, α

_M∗N

) given, for all m, m

∈ M, n, n

∈ N, by

m

(m ∗ n) = [m

, m] ∗ α

N

(n) −

^m

n ∗ α

M

(m),

m

(n ∗ m) =

^m

n ∗ α

M

(m) − [m

, m] ∗ α

N

(n), (m ∗ n)

^m

= [m, m

] ∗ α

N

(n) + α

M

(m) ∗ n

^m

, (n ∗ m)

^m

= n

^m

∗ α

M

(m) + α

N

(n) ∗ [m, m

], resp.

ⁿ

(m ∗ n) =

ⁿ

m ∗ α

N

(n) −

n

, n

∗ α

M

(m),

n

(n ∗ m) = n

, n

∗ α

M

(m) −

ⁿ

m ∗ α

N

(n

), (m ∗ n)

ⁿ

= m

ⁿ

∗ α

N

(n) + α

M

(m) ∗

n, n

, ( n ∗ m )

ⁿ

=

n, n

∗ α

M

( m ) + α

N

( n ) ∗ m

ⁿ

. (c) Ker (ψ

1

) and Ker (ψ

2

) both are contained in the center of ( M ∗ N, α

M∗N

) .

(d) The induced Hom–Leibniz action of Im(ψ

1

) (resp. Im(ψ

2

)) on Ker(ψ

1

) (resp. Ker(ψ

2

)) is trivial.

(e) ψ

1

and ψ

2

satisfy the following properties for all m, m

∈ M, n, n

∈ N:

(i) ψ

1

(

^m

( m ∗ n )) = [α

M

( m

) , ψ

1

( m ∗ n )] , (ii) ψ

1

((m ∗ n)

^m

) = [ψ

1

(m ∗ n), α

M

(m

)], (iii) ψ

1

(

^m

(n ∗ m)) = [α

M

(m

), ψ

1

(n ∗ m)], (iv) ψ

1

(( n ∗ m )

^m

) = [ψ

1

( n ∗ m ) , α

M

( m

)] , (v) ψ

2

(

ⁿ

(m ∗ n)) = [α

N

(n

), ψ

2

(m ∗ n)], (vi) ψ

2

(( m ∗ n )

ⁿ

) = [ψ

2

( m ∗ n ) , α

N

( n

)] , (vii) ψ

2

(

ⁿ

(n ∗ m)) = [α

N

(n

), ψ

2

(n ∗ m)], (viii) ψ

2

(( n ∗ m )

ⁿ

) = [ψ

2

( n ∗ m ) , α

N

( n

)] ,

(ix)

^ψ1(m∗n)

(m

∗ n

) = [α

M∗N

(m ∗ n), m

∗ n

] =

^ψ2(m∗n)

(m

∗ n

), (x)

^ψ1(m∗n)

(n

∗ m

) = [α

M∗N

(m ∗ n), n

∗ m

] =

^ψ2(m∗n)

(n

∗ m

), (xi)

^ψ1(n∗m)

( m

∗ n

) = [α

M∗N

( n ∗ m ) , m

∗ n

] =

^ψ2(n∗m)

( m

∗ n

) , (xii)

^ψ1(n∗m)

(n

∗ m

) = [α

M∗N

(n ∗ m), n

∗ m

] =

^ψ2(n∗m)

(n

∗ m

), (xiii) ( m

∗ n

)

^ψ1(m∗n)

= [ m

∗ n

, α

M∗N

( m ∗ n )] = ( m

∗ n

)

^ψ2(m∗n)

, (xiv) (n

∗ m

)

^ψ1(m∗n)

= [n

∗ m

, α

M∗N

(m ∗ n)] = (n

∗ m

)

^ψ2(m∗n)

, (xv) ( m

∗ n

)

^ψ1(n∗m)

= [ m

∗ n

, α

M∗N

( n ∗ m )] = ( m

∗ n

)

^ψ2(n∗m)

, (xvi) (n

∗ m

)

^ψ1(n∗m)

= [n

∗ m

, α

M∗N

(n ∗ m)] = (n

∗ m

)

^ψ2(n∗m)

.

Proof: The proof requires routine combinations of equations in Definitions 5 and 8 with the defining relations (3.1) and (3.3) of the non-abelian Hom–Leibniz tensor product.

Remark 4:

(a) If α

M

= id

_M

and α

N

= id

_N

then the statement e) of Lemma 3.5 says that both ψ

1

and ψ

2

are crossed modules of Leibniz algebras (see [15]).

(b) If M = N and the action of M on itself is given by the bracket in M, then ψ

1

= ψ

2

and they are defined on generators by m ∗ m

→ [m, m

]. In such a case we write

ψ

M

for ψ

1

.