Hom-algebra structures

A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli's isotopies of associative and Lie algebras.


Introduction
In [2,4,5], the class of quasi-Lie algebras and subclasses of quasi-hom-Lie algebras and hom-Lie algebras has been introduced. These classes of algebras are tailored in a way suitable for simultaneous treatment of the Lie algebras, Lie superalgebras, the color Lie algebras and the deformations arising in connection with twisted, discretized or deformed derivatives and corresponding generalizations, discretizations and deformations of vector fields and differential calculus. It has been shown in [2,4,5,6] that the class of quasi-hom-Lie algebras contains as a subclass on one hand the color Lie algebras and in particular Lie superalgebras and Lie algebras, and on another hand various known and new single and multi-parameter families of algebras obtained using twisted derivations and constituting deformations and quasi-deformations of universal enveloping algebras of Lie and color Lie algebras and of algebras of vector-fields. The main feature of quasi-Lie algebras, quasi-hom-Lie algebras and hom-Lie algebras is that the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps and also in quasi-Lie and quasi-hom-Lie algebras the Jacobi identity in general contains 6 twisted triple bracket terms.
In this paper, we provide a different way for constructing hom-Lie algebras by extending the fundamental construction of Lie algebras from associative algebras via commutator bracket multiplication. To this end we define the notion of homassociative algebras generalizing associative algebras to a situation where associativity law is twisted, and show that the commutator product defined using the multiplication in a hom-associative algebra leads naturally to hom-Lie algebras. We introduce also hom-Lie-admissible algebras and more general G-hom-associative algebras with subclasses of hom-Vinberg and pre-hom-Lie algebras, generalizing to the twisted situation Lie-admissible algebras, G-associative algebras, Vinberg and pre-Lie algebras respectively, and show that for these classes of algebras the operation of taking commutator leads to hom-Lie algebras as well. We construct all the twistings such that the bracket [X 1 , determine a three dimensional hom-Lie algebras. Finally, we provide for a subclass of twistings, a the list of all three-dimensional hom-Lie algebras. This list contains all three-dimensional Lie algebras for some values of structure constants. The families of hom-Lie algebras in these list can be viewed as deformations of Lie algebras into a class of hom-Lie algebras.

Definitions
Let K be an algebraically closed field of characteristic 0 and V be a linear space over K. Let α be a homomorphism in Hom(V, V ).
1.1. Hom-associative algebras. Definition 1.1. A Hom-associative algebra is a triple (V, µ, α) consisting of a linear space V , a bilinear map µ : V × V → V and a homomorphism α : V → V with respect to µ satisfying µ(α(x), µ(y, z)) = µ(µ(x, y), α(z)) 1.2. Hom-Leibniz algebras. A class of quasi Leibniz algebras was introduced in [5] in connection to general quasi-Lie algebras following the standard Loday's conventions for Leibniz algebras (i.e. right Loday algebras). In the context of the subclass of Hom-Lie algebras one gets a class of Hom-Leibniz algebras. In terms of the (right) adjoint homomorphisms Ad Y : V → V defined by Ad Y (X) = [X, Y ], the identity (1.1) can be written as or in pure operator form One may introduce the notion of α-derivation for a Hom-binary algebra.
The condition of Hom-Leibniz algebras may be rephrased as the adjoint homomorphisms Ad α (z) being α-derivations with respect to the bracket multiplication [·, ·] for all z ∈ V .  x,y,z [α(x), [y, z]] = 0 (Hom-Jacobi condition) for all x, y, z from V , where x,y,z denotes summation over the cyclic permutation on x, y, z.
Using the skew-symmetry, one may write the previous equation in the form 1.2. Note that if a Hom-Leibniz algebra is skewsymmetric then it is a Hom-Lie algebra. Proof. The Hom-Jacobi condition is satisfied for any triple (x, x, y).
The Hom-Lie algebras are special case of a Quasi-Hom-Lie algebra and the more general Quasi-Lie algebras [4,5]. In the following, we recall the definition of Quasi-Lie algebras.
Let L K (V ) be the set of linear maps of the linear space L over the field K.
such that the following conditions hold: • (ω-symmetry) The product satisfies a generalized skew-symmetry condition • (quasi-Jacobi identity) The bracket satisfies a generalized Jacobi identity The class of Quasi-Lie algebras incorporates as special cases hom-Lie algebras and more general quasi-hom-Lie algebras (qhl-algebras) which appear naturally in the algebraic study of σ-derivations (see [2]) and related deformations of infinitedimensional and finite-dimensional Lie algebras. To get the class of qhl-algebras one specifies θ = ω and restricts attention to maps α and β satisfying the twisting condition [α(x), α(y)] = β • α[x, y]. Specifying this further by taking D ω = V × V , β = id and ω = −id, one gets the class of hom-Lie algebras including Lie algebras when α = id. The class of quasi-Lie algebras contains also color Lie algebras and in particular Lie superalgebras.

Functor Hom-Lie.
Proposition 1.7. To any Hom-associative algebra defined by the multiplication µ and a homomorphism α over a K-linear space V , one may associate a Hom-Lie algebra defined for all x, y ∈ V by the bracket Proof. The bracket is obviously skewsymmetric and with a direct computation we have The Lie-admissible algebras was introduced by A. A. Albert in 1948. Physicists attempted to introduce this structure instead of Lie algebras. For instance, the validity of Lie-Admissible algebras for free particles is well known. These algebras arise also in classical quantum mechanics as a generalization of conventional mechanics (see [8], [9]). In this section, we extend to Hom-algebras the classical concept of Lie-Admissible algebras.
Definition 2.1. Let A be a Hom-algebra structure on V defined by the multiplication µ and a homomorphism α. Then A is said to be Hom-Lie-Admissible algebra over V if the bracket defined for all x, y ∈ V by satisfies the Hom-Jacobi identity, that is Remark 2.2. Since the bracket is also skewsymmetric then it defines a Hom-Lie algebra.
In the following, we explore some other Hom-Lie-Admissible algebras. We introduce the following notation.
Definition 2.4. Let a α,µ be a trilinear map over V associated to a product µ and a homomorphism α defined by We call a α,µ the α-associator of µ.
Definition 2.5. Let G be a subgroup of the permutations group S 3 , a binary Hom-algebra on V defined by the multiplication µ and a homomorphism α is said G-Hom-associative if where x i are in V and (−1) ε(σ) is the signature of the permutation σ.
The condition 2.1 may be written Remark 2.6. If µ is the multiplication of a Hom-Lie-Admissible Lie algebra then the condition (2.1) is equivalent to the property that the bracket defined by satisfies the Hom-Jacobi condition or equivalently to (3) )) = 0 which may be written as Proposition 2.7. Let G be a subgroup of the permutations group S 3 . Then any G-Hom-associative algebra is a Hom-Lie-Admissible algebra.
Proof. The skewsymmetry follows straightaway from the definition.
We have a subgroup G in S 3 . Take the set of conjugacy class {gG} g∈I where I ⊆ G, and for any σ 1 , σ 2 ∈ I, The G-associative algebra in classical case was studied in [3]. The result may be extended to Hom-structures in the following way.
The subgroups of S 3 are where A 3 is the alternating group and τ ij is the transposition between i and j. We obtain the following type of Hom-Lie-admissible algebras.

Flexible Hom-Lie admissible algebras
The flexible Lie-admissible algebras was initiated by Albert [1] and investigated by number of authors Myung, Okubo, Laufer, Tomber and Santilli, see [7] and [10]. The aim of this section is to extend the results on Lie-admissible structure to Hom-structures. (2) For any x, y in V , a µ,α (x, y, x) = 0.
Let A = (V, µ, α) be a Hom-algebra where µ is the multiplication and α a homomorphism. We denote by A + the Hom-algebra over V with a multiplication x • y = 1 2 (µ(x, y) + µ(y, x)). We denote by A − the Hom-algebra over V where the multiplication is given by the commutator [x, y] = µ(x, y) − µ(y, x).
Recall that an endomorphism f is an α-derivation for the Hom-algebra (

Algebraic varieties of Hom-structures
Let V be a n-dimensional K-linear space and {e 1 , · · · , e n } be a basis of V . A Hom-algebra structure on V with product µ is determined by n 3 structure constants C k ij , where µ(e i , e j ) = n k=0 C k ij e k and homomorphism α which is given by n 2 structure constants a ij , where α(e i ) = n j=0 a ij e j . If we require that this algebra structure to be Hom-associative, then this limits the set of structure constants (C k ij , a ij ) to a subvariety of K n 3 +n 2 defined by the following system of polynomial equations : For i, j, k, s = 1, · · · , n n l,m=1 The algebraic variety of n-dimensional Hom-associative algebras is denoted by HomAss n . Note that the equations are given by cubic polynomials.
If we consider the n-dimensional unitary Hom-associative algebras, where e 1 is the unit, then we obtain a subvariety which we denote by HomAlg n and determined by the following polynomial equations : For i, j, k, s = 1, · · · , n n l,m=1 a il C m jk C s lm − a km C l ij C s lm = 0 If we require that this algebra structure to be Hom-Lie, then the structure constants {(C k ij ) i<j , (a ij )} determine a subvariety of K n 2 (n+1)/2 defined by the following system of polynomial equations n l,m=1 a il C m jk C s lm + a jl C m ki C s lm + a kl C m ij C s lm = 0 where C k ij = −C k ji . The variety of n-dimensional Hom-Lie algebras is denoted by HomLie n . One can do the same for every Hom-Lie-admissible structure. If the algebras are parameterized by their structure constants and the structure constants of the homomorphism, the set of n-dimensional algebras of given structure carries a structure of algebraic varieties imbedded in K n 3 +n 2 defined by a system of cubic polynomials.
A point in such an algebraic variety (e.g. , HomAss n , HomLie n ) represents an n-dimensional algebra (e.g. Hom-associative, or Hom-Lie), along with a particular choice of basis. A change of basis may give rise to a (possibly) different point of the algebraic variety. The group GL(n, K) acts on the algebraic varieties of Hom-structures by the so-called "transport of structure" action defined as follows : Let A be an algebra (e.g. Hom-associative, or Hom-Lie) defined by multiplication µ and homomorphism α. Given f ∈ GL(n, K), the action f · A transports the structures by The orbit of an algebra (e.g. Hom-associative, or Hom-Lie) A is given by The orbits are in 1-1-correspondence with the isomorphism classes of n-dimensional algebras. The stabilizer subgroup of A (stab (A) = {f ∈ GL n (K) : A = f · A}) is exactly Aut (A) , the automorphism group of A.
The orbit ϑ (A) is identified with the homogeneous space GL n (K) /Aut (A). Then dim ϑ (A) = n 2 − dim Aut (A)

Three-dimensional Hom-Lie algebras
By a direct calculation, we obtain the following class of Hom-Lie algebras Proposition 5.1. Let V be a three dimensional K-linear space and (X 1 , X 2 , X 3 ) being its basis. Any Hom-Lie algebra with the following brackets This may be viewed as a quasi-deformation of sl(2) in the class of Hom-Lie algebras. In case where the matrix is the identity then one gets the classical Lie algebras sl(2).
Remark 5.2. It would be interesting to compare this list with the quasi-hom-Lie and Hom-Lie algebras of quasi-deformations of sl(2) constructed in [6] using a quasi-hom-Lie structure on twisted vector fields.
We provide in the following a list of all Hom-Lie algebras associated to the homomorphism given with respect to the basis e 1 , e 2 , e 3 of the 3-dimensional V by the following matrix