Extensions of hom-Lie color algebras

1 Introduction

The investigations of various quantum deformations (or q-deformations) of Lie algebras began a period of rapid expansion in the 1980s stimulated by the introduction of quantum groups motivated by applications to the quantum Yang–Baxter equation, quantum inverse scattering methods and constructions of the quantum deformations of universal enveloping algebras of semi-simple Lie algebras. Since then several other versions of q-deformed Lie algebras have appeared, especially in physical contexts such as string theory, vertex models in conformal field theory, quantum mechanics and quantum field theory in the context of deformations of infinite-dimensional algebras, primarily the Heisenberg algebras, oscillator algebras and Witt and Virasoro algebras [3, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 38, 39, 40]. In these pioneering works it was in particular discovered that in these q-deformations of Witt and Virasoro algebras and some related algebras, some interesting q-deformations of the Jacobi identity, extending the Jacobi identity for Lie algebras, are satisfied. This was one of the initial motivations for the development of general quasi-deformations and discretizations of Lie algebras of vector fields using more general σ-derivations (twisted derivations) in [24], and the introduction of abstract quasi-Lie algebras and subclasses of quasi-hom-Lie algebras and hom-Lie algebras as well as their general colored (graded) counterparts in [24, 34, 33, 51, 35]. These generalized Lie algebra structures with (graded) twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within the same algebraic framework such quasi-deformations and discretizations of Lie algebras of vector fields using σ-derivations, describing general discretizations and deformations of derivations with twisted Leibniz rule, and the well-known generalizations of Lie algebras such as color Lie algebras which are the natural generalizations of Lie algebras and Lie superalgebras. Quasi-Lie algebras are non-associative algebras for which the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps and also the Jacobi identity in quasi-Lie and quasi-hom-Lie algebras in general contains six twisted triple bracket terms. Hom-Lie algebras are a special class of quasi-Lie algebras with the bilinear product satisfying the non-twisted skew-symmetry property as in Lie algebras, whereas the Jacobi identity contains three terms twisted by a single linear map, reducing to the Jacobi identity for ordinary Lie algebras when the linear twisting map is the identity map. Subsequently, hom-Lie admissible algebras have been considered in [43] where also the hom-associative algebras have been introduced and shown to be hom-Lie admissible natural generalizations of associative algebras corresponding to hom-Lie algebras. In [43], moreover, several other interesting classes of hom-Lie admissible algebras generalizing some non-associative algebras, as well as examples of finite-dimensional hom-Lie algebras have been described.

Since these pioneering works [24, 34, 35, 33, 36, 43], hom-algebra structures have become a popular area with increasing number of publications in various directions. Hom-Lie algebras, hom-Lie superalgebras and hom-Lie color algebras are important special classes of color (graded) quasi-Lie algebras introduced first by Larsson and Silvestrov [35, 33]. Hom-Lie algebras and hom-Lie superalgebras have been studied in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yau and other authors [4, 13, 32, 42, 43, 44, 45, 46, 49, 50, 52, 53, 58, 59, 60, 62, 63, 64, 65, 66], and hom-Lie color algebras have been considered for example in [1, 14, 15, 65]. We wish to mention especially [6], where the constructions of hom-Lie and quasi-hom Lie algebras based on twisted discretizations of vector fields [24] and hom-Lie admissible algebras have been extended to hom-Lie superalgebras, a subclass of graded quasi-Lie algebras [35, 33]. We also wish to mention that ℤ3-graded generalizations of supersymmetry, ℤ3-graded algebras, ternary structures and related algebraic models for classifications of elementary particles and unification problems for interactions, quantum gravity and non-commutative gauge theories [2, 28, 29, 30, 61] also provide interesting examples related to hom-associative algebras, graded hom-Lie algebras, twisted differential calculi and n-ary hom-algebra structures. It would be a project of great interest to extend and apply all the constructions and results in the present paper in the relevant contexts of the articles [2, 5, 6, 10, 11, 9, 28, 29, 30, 35, 33, 43].

In Section 2 of this paper, hom-Lie algebras, hom-Lie superalgebras, hom-Lie color algebras and some useful related definitions are presented. In Section 3, we introduce hom-Lie color algebra extensions with an emphasize on geometric aspects. Finally, in Section 4, introducing the Chevalley cohomology for hom-Lie color algebras, we find a cohomological obstruction to the existence of extensions.

2 Definitions and notations for hom-Lie algebras, hom-Lie superalgebras, hom-Lie color algebras

A hom-Lie algebra is called a multiplicative hom-Lie algebra if α is an algebraic morphism with

for any x,y∈𝔤.

We call a hom-Lie algebra regular if α is an automorphism.

A linear subspace 𝔥⊆𝔤 is a hom-Lie sub-algebra of (𝔤,[⋅,⋅],α) if

and 𝔥 is closed under the bracket operation, i.e.,

for all x1,x2∈𝔥.

Let (𝔤,[⋅,⋅],α) be a multiplicative hom-Lie algebra. Denote by αk the k-times composition of α by itself for any nonnegative integer k, i.e.,

where we define α0=Id and α1=α. For a regular hom-Lie algebra 𝔤, let

The elements of Xγ are called homogenous of degree γ for all γ∈Γ. Let 𝔤=⊕γ∈Γ𝔤γ and 𝔥=⊕γ∈Γ𝔥γ be two Γ-graded linear spaces. A linear mapping f:𝔤→𝔥 is said to be graded or homogenous of degree μ∈Γ if

for all γ∈Γ. A graded linear mapping f is said to be homogenous of degree zero if

holds for any γ∈Γ. Sometimes such f are said to be even.

A special class of color quasi-Lie algebras [35, 33] are hom-Lie color algebras.

In particular, if α is a morphism of color Lie algebras, i.e.,

then we call (𝔤,[⋅,⋅],ε,α) a multiplicative hom-Lie color algebra.

Denote by Derαkγ⁡(𝔤) the set of all homogenous αk-derivations of the multiplicative hom-Lie color algebra (𝔤,[⋅,⋅],α). The space

provided with the color-commutator and the linear map

is a color Lie algebra.

For any x∈𝔤 satisfying α⁢(x)=x, define

by

for all y∈𝔤.

It is shown in [1] that adk⁡(x) is a αk+1-derivation, which we call an inner αk-derivation. So,

3 Extensions of hom-Lie color algebras

In this section, we clarify what we mean by an extension of a hom-Lie color algebra. Although one can see that extensions of a given Abelian hom-Lie color algebra are characterized by elements of their second cohomology group, we concentrate on some geometric aspects in this research. The cohomology has been studied for Lie superalgebras and color Lie algebras [20, 48, 56, 57] and hom-Lie color algebras [1].

Two extensions

are equivalent if there is an isomorphism f:𝔢1→𝔢2 such that f∘i1=i2 and p2∘f=p1.

We want to study possible extensions, so suppose there exists an extension

and let s:𝔤→𝔢 be a graded linear map of even degree such that p∘s=I⁢d𝔤. We define

and

The following lemma shows some properties of the above maps which we will use later in this research.

Therefore, by using φ and ρ which satisfy (3.3) and (3.4), the hom-Lie color algebra structure on 𝔢=𝔥⊕s⁢(𝔤) will be in the following form:

We can see φ as a connection in the sense of [31] and ρ as its curvature. Moreover, in order to give (3.4) a more geometric image, we have to use a graded version of the Chevalley coboundary operator which makes it a Γ-graded exterior covariant derivative. Throughout this research, we will use the notation which is a modification of the one introduced in [55]:

for all a1,…,ak∈Γ and σ∈𝒮k, where 𝒮k is the set of permutations of k symbols (the symmetric group). If we write

for a=(a1,…,ak)∈(Γ)k, then we have

Moreover, we simply show the degree of any homogenous element xi∈𝔤 by xi¯ through the rest.

Now, for a linear space V, the space of graded p-linear skew-symmetric maps 𝔤p→V of degree w is denoted by Agskewp,w⁢(𝔤,V), i.e., the space of maps of the form

for all homogenous elements xi, i=1,…,k. We see that when ψ∈Agskewp,w⁢(𝔤,𝔥), we have

We see that for ψ∈Agskewp,w⁢(𝔤,𝔥) and ζ∈Agskewq,z⁢(𝔤,ℝ), which is a form of degree q and weight z, the operator δφ satisfies the condition for a graded covariant exterior derivative, i.e.,

where

is the color version of the Chevalley coboundary operator and the module structure is given by

where ηi⁢(σ,x)=xσ⁢1¯+⋯+xσ⁢i¯.

Also, for ψ∈Agskewp,w⁢(𝔤,𝔥) and ζ∈Agskewq,z⁢(𝔤,𝔥) we have

Thus, we can see s as a connection in the sense of the horizontal lift of vector fields on the base of a bundle. Moreover, φ is an induced connection; see [31] for more background information.

Thus formula (3.4) becomes the Bianchi identity

Moreover, we deduce that

Therefore, if φ:𝔤→Derαk⁡(𝔥) is a homomorphism of hom-Lie color algebras or

is a representation in a graded vector space, then (3.5) shows the hom-color version application of the Chevalley cohomology.

We can now complete the hom-Lie color algebra structure on 𝔢=𝔥⊕s⁢(𝔤) as follows:

One can check that (3.6) gives 𝔥⊕s⁢(𝔤) a hom-Lie color algebra structure if φ and ρ satisfy (3.3) and (3.4). If we put s′=s+b instead of s, where b:𝔤→𝔥 is an even linear map, we will have

and

i.e.,

Thus, we have proved the following theorem.

It may be a good idea to illustrate the motivation of the above theorem by an example below.

In the special case of Theorem 3.4, we have the following corollary.

4 Cohomological obstruction to existence of extensions

In this section, we present a proposition which shows that if there exists a hom-Lie color algebra extension, there should be a trivial member of the third cohomology. We have to give some remarks first.