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^{a}Universit´e de Haute Alsace, Laboratoire de Math´ematiques, Informatique et Applications,
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^{b}Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden

**E-mail:** [email protected]

**Received date:** June 25, 2007

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Hom-algebra structures are given on linear spaces by multiplications twisted by linear maps. Hom-Lie algebras and general quasi-Hom-Lie and quasi-Lie algebras were introduced by Hartwig, Larsson and Silvestrov as algebras embracing Lie algebras, super and color Lie algebras and their quasi-deformations by twisted derivations. In this paper we introduce and study Hom-associative, Hom-Leibniz and Hom-Lie admissible algebraic structures generalizing associative, Leibniz and Lie admissible algebras. Also, we characterize flexible Hom-algebras and explain some connections and differences between Hom-Lie algebras and Santilli’s isotopies of associative and Lie algebras.

Hom-Lie algebra, Hom-Associative algebra, Hom-Leibniz algebra, Hom-Lie admissible algebra, flexible algebra, classification.

In [4, 5, 6], the class of quasi-Lie algebras and subclasses of quasi-Hom-Lie algebras and Hom- Lie algebras had been introduced. These classes of algebras are tailored in a way suitable for simultaneous treatment of the Lie algebras, Lie superalgebras, color Lie algebras and deformations arising in connection with twisted, discretized or deformed derivatives and corresponding generalizations, discretizations and deformations of vector fields and differential calculus. It had been shown in [4, 5, 6, 7] 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 Hom-associative 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 so that the brackets

determine a three-dimensional Hom-Lie algebra, generalizing sl(2). Finally, we provide for a subclass of twistings, a list of all three-dimensional Hom-Lie algebras. For some values of structure constants, this list contains all three-dimensional Lie algebras. The families of Hom- Lie algebras in these list can be viewed as deformations of Lie algebras into a class of Hom-Lie algebras.

The paper is organized as follows. In Section 1, we summarize the definitions of Homassociative,
Hom-Leibniz algebra and Hom-Lie algebra. In Section 2, we extend the classical
concept of Lie admissible algebra to Hom-Lie admissible algebra. In Section 3 we review and
show the connections to Santilli’s isotopies of associative and Lie algebras. In Section 4, we
explore some general classes of Hom-Lie admissible algebras. For any subgroup *G* of the permutation
group S_{3}, we introduce the *G*-Hom-associative algebras, show that they are Hom-Lie
admissible and describe all these classes. Section 5 is devoted to some properties of flexible
Hom-algebras. We extend the notion of flexibility to Hom-algebras and characterize flexible
Hom-algebras using Jordan and Lie parts of the multiplication. In Section 6, we consider the
algebraic varieties of finite-dimensional Hom-associative and Hom-Lie algebras. We also provide
a characterization of all Hom-Lie algebras of sl(2) type and construct all the 3-dimensional
Hom-Lie algebras for a class of twisting homomorphisms.

Let be an algebraically closed field of characteristic 0 and V be a linear space over . Let α
be a homomorphism of *V* . In the sequel we mean by Hom-algebra a triple (*V, μ, α*) consisting of
a linear space *V* , a bilinear map and a linear space homomorphism

**Definition 1.1.** A Hom-associative algebra is a triple (*V, μ, α*) consisting of a linear space *V* , a
bilinear map and a linear space homomorphism satisfying

A class of quasi-Leibniz algebras was introduced in [6] 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.

**Definition 1.2.** A Hom-Leibniz algebra is a triple (*V*, [·, ·], *α*) consisting of a linear space *V* ,
bilinear map and a linear space homomorphism satisfying

In terms of the (right) adjoint homomorphisms defined by *Ad _{Y}* (

or in pure operator form

(1.3)The Hom-Lie algebras were initially introduced by Hartwig, Larson and Silvestrov in [4] motivated initially by examples of deformed Lie algebras coming from twisted discretizations of vector fields.

**Definition 1.3** (Hartwig, Larsson, Silvestrov [4]). A Hom-Lie algebra is a triple (*V*, [·, ·], *α*)
consisting of a linear space *V* , bilinear map and a linear space homomorphism satisfying

for all *x, y, z* from *V* , where denotes summation over the cyclic permutation on *x, y, z*.

Using the skew-symmetry, one may write the Hom-Jacobi identity in the form (1.2). Thus, if a Hom-Leibniz algebra is skewsymmetric, then it is a Hom-Lie algebra.

**Proposition 1.4**. *Every skew-symmetric bilinear map on a 2-dimensional linear space defines
a Hom-Lie algebra*.

**Proof**. The Hom-Jacobi identity (1.5) is satisfied for any triple (*x, x, y*).

If we add the condition that the linear map *α* is an algebra homomorphism with respect to
the bracket then we get restricted Hom-Lie algebras which are a special case of Quasi-Hom-Lie
algebras. All these classes are a special case of the more general Quasi-Lie algebras [5, 6]. In
the following, we recall the definition of Quasi-Lie algebras.

We denote by the set of linear maps of the linear space *V* over the field .

**Definition 1.5** (Larsson, Silvestrov [6]). A *quasi-Lie algebra* is a tuple consisting
of

• *V* is a linear space over ,

• is a bilinear map that is called a product or bracket in *V* ;

• are linear maps,

• and are maps with definition domains , such that the following conditions hold:

• (*ω*-symmetry) The product satisfies a generalized skew-symmetry condition

[*x, y*] = *ω*(*x, y*)[*y, x*], for all

• (quasi-Jacobi identity) The bracket satisfies a generalized Jacobi identity

for all .

Note that [*x, y*] = *ω*(*x, y*)[*y, x*] = *ω*(*x, y*)*ω*(*y, x*)[*x, y*] and hence (*ω*(*x, y*)*ω*(*y, x*)−*id*)[*x, y*] = 0,
if .

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 [4]) and related deformations of infinite-dimensional 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 . Specifying this further by
taking *Dω = 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.

**Proposition 1.6.** *To any Hom-associative algebra (V, μ, α), one may associate a Hom-Lie
algebra defined for all* *by the bracket* [*x, y*] = *μ*(*x, y*) − *μ*(*y, x*).

**Proof. **The bracket is obviously skewsymmetric and with a direct computation we have

The Lie admissible algebras were 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 [9], [10]). In this section, we extend to Homalgebras 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 by

satisfies the Hom-Jacobi identity for all.

*Remark* 2.2. Since the commutator bracket (2.1) is always skew-symmetric, it makes any Hom-
Lie admissible algebra into a Hom-Lie algebra.

*Remark* 2.3. According to Proposition 1.6, any Hom-associative algebra is Hom-Lie admissible
with the same twisting map.

It can be also checked that any Hom-Lie algebra with a twisting *α* is Hom-Lie admissible
with the same twisting *α*.

**Proposition 2.4.** *Any Hom-Lie algebra (V, [·, ·], α) is Hom-Lie admissible with the same twisting
map α*.

**Proof.** For the commutator product one has

For our further considerations it is convenient to introduce the following notation.

Definition 2.5. The *α*-associator, associated to a product *μ* and a linear map *α*, is a trilinear map defined by .

In the following, we give a new characterization of Hom-Lie admissible algebras.

Let be a Hom-algebra and let [*x, y*] = *μ*(*x, y*) − *μ*(*y, x*) be its commutator.
We introduce the ternary map defined by *S*(*x, y, z*) := *a _{μ,α}(x, y, z)+a_{μ,α}(y, z, x)+a_{μ,α}(z, x, y)*.
Then we have the following properties.

**Lemma 2.6.** *For all* , *the following equality holds*

*S(x, y, z) = [μ(x, y), α(z)] + [μ(y, z), α(x)] + [μ(z, x), α(y)]*

**Proof.** The assertion follows by expanding commutators on the right hand side:

**Proposition 2.7.** *A Hom-algebra A is Hom-Lie admissible if and only if S(x, y, z) = S(x, z, y)
for all* .

**Proof.** The assertion follows from

Already as early as 1967, Santilli considered in [11] the two-parametric deformations (mutations)
of the Lie commutator bracket in an associative algebra, *(A,B) = pAB − qBA*, where p and *q* are scalar parameters and *A* and *B* are elements in the associative algebra (typically algebra
of matrices or linear operators); and in 1978, Santilli [12, 13, 14] extended it to “operatordeformations”
of the Lie product as follows *(A,B) = APB − BQA*, where *P* and *Q* are fixed
elements in the underlying associative algebra. The motivation for introducing non-associative
generalizations of the bracket multiplication came from attempts to resolve certain limitations of
conventional formalism of classical and quantum mechanics. Subsequently, in numerous works
including articles and books by Santilli and other authors, the evolution equations based on
such deformed brackets, physical applications and consequences of introducing such generalized
models have been investigated. The deformations of the commutator bracket multiplication
introduced by Santilli in investigations of foundations of classical and quantum mechanics and
hadronic physics, have reappeared in many incarnations both in Mathematics and Physics. We
refer the reader to [2, 12, 13, 14, 15, 16, 17, 18, 19, 20] for further discussion and references on the
models based on introduction of such non-associative deformed commutator bracket multiplications
instead of commutator (Lie) bracket multiplication, their bi-module type generalizations
(genoalgebras) as well as for a review of relation with other appearances of so deformed commutator
brackets in physics and mathematics, for example in contexts related to quantum algebras,
quantum groups, Lie algebras and superalgebras, Jordan and other classes of algebras.

The relation between Hom-Lie admissible algebras and non-associative algebras with Santilli’s deformed bracket products is certainly an interesting direction for further investigation. Here we would like to highlight some interrelations and differences between considered algebraic objects.

The Santilli’s products with arbitrary *P* and *Q* are not anti-symmetric in general except when *P* and *Q* are specially interrelated within the underlying algebra, for example when *P* = *Q* that is *(A,B) = AQB − BQA*. It can also be checked that the Lie-Santilli bracket product *(A,B) = AQB −BQA* satisfies the Lie algebras Jacobi identity, which is not the case in general
for . Thus the associative algebras with the modified product *A × _{Q} B = AQB* are Lie
admissible algebras. Since

*Remark* 3.1. The Hom-Lie admissible algebras are Lie admissible algebras typically when the
twisting mapping *α* built into their definition is the scalar multiple of the identity mapping. It
could thus be of interest to describe, for a given twisting *α*, as sharp as possible conditions on *P* and *Q* in order for the general Santilli’s bracket product *(A,B) = APB − BQA* to define a
Hom-Lie algebra or more general Hom-Leibniz algebra with twisting *α*.

*Remark* 3.2. It is possible also that Hom-Lie admissible algebras can be studied using Santilli’s
formalism of genoalgebras, the abstract bi-module type extension of the deformed commutator
brackets *(A,B) = APB − BQA*, but after this formalism is appropriately modified to suit the
Hom-Lie algebras context.

Santilli has considered also so called isotopies of associative and Lie algebras. Algebraic
problem can be formulated as follows. Any associative algebra is Lie admissible since the commutator
bracket on any associative algebra satisfies axioms of a Lie algebra. How can associative
products in associative algebras be modified to yield as general as possible Lie admissible algebras?
Can any Lie admissible algebra be obtained by such modifications from some associative
algebra? Such modifications of associative and corresponding Lie algebras where called isotopies
of associative and Lie algebras. In [12, 13, 14, 15, 17, 18, 19], several general isotopies
of associative products and associated Lie products have been identified. The most general of
all presented there isotopies of a product for elements in an associative algebra *A* over a field is given by *x * y = awxwTwyw*, where and *T* is some extra
element. The product * is associative if *w ^{2} = w* and . Santilli allows

in particular, the equalities *u(xTy) = (ux)T(y), (xTy)v = (x)T(yv), u(xTy)v = (ux)T(yv)* hold for Then the expression *x * y = a(w)(x)(w)T(w)(y)(w)* yields again an
element from *A _{T}* for , and we get the product * on the algebra

using *w ^{2} = w*, one gets

One may show that this algebra carries a structure of Hom-associative algebra in the following
way. Let *α(s) = wsw* for all . Then

The map is a linear map on *A _{T}* , and (

If , then the above reduction of Hom-associativity with the twisting map to associativity is not working. Moreover, on many associative algebras *A*, there are linear maps *α*, which can not be represented on the form at all. This explains why the classes
of Hom-associative and Hom-Lie (admissible) algebras are different from Lie-Santilli algebras
and isotopies of associative and Lie algebras. Whereas the relation between Hom-associative
and Hom-Lie (admissible) algebras resembles that between associative and Lie algebras, the
Hom-associative algebras are mostly not associative, and Hom-Lie (admissible) algebras are
mostly not Lie (admissible) algebras. Only with special choices of *α* like above one recovers the
associative and Lie algebra properties holding also in the case of Santilli’s isotopies of associative
and Lie algebras.

*Remark* 3.3. We assumed that for simplicity of exposition. All conclusions above hold
however even if w as *T* is an extra element possibly outside A, but of cause with *A _{T}* replaced by
a properly defined algebra

The other fundamental algebraic issue tackled by Santilli is imbedding of the scalar field into
the algebras over this filed. If an algebra *A* over the field with a unit has a unit 1_{A}, then
there is a canonical imbedding of the field into the algebra given by for . Also
one has 1_{Ax} = *x*1_{A} = *x*. If the multiplication in *A _{T}* is defined as

In the following, we explore some other general classes of Hom-Lie admissible algebras, *G*-Homassociative
algebras, extending the class of Hom-associative algebras.

**Definition 4.1.** Let *G* be a subgroup of the permutations group S_{3}, a Hom-algebra on *V* defined
by the multiplication *μ* and a homomorphism *α* is said to be *G*-Hom-associative if

where *x _{i}* are in

The condition (4.1) may be written

(4.2)where .

*Remark* 4.2. If *μ* is the multiplication of a Hom-Lie admissible Lie algebra, then the condition
(4.1) is equivalent to the property that the bracket defined by *[x, y] = μ(x, y) − μ(y, x)* satisfies
the Hom-Jacobi identity, or equivalently

which may be written as

(4.4)**Proposition 4.3.** *Let G be a subgroup of the permutations group S _{3}. Then any G-Homassociative
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 where , and for
any . Then

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 where is the transposition between

• The *G*_{1}-Hom-associative algebras are the Hom-associative algebras defined above.

• The *G*_{2}-Hom-associative algebras satisfy the condition

• The *G*_{3}-Hom-associative algebras satisfy the condition

When *α* is the identity the algebra is called pre-Lie algebra or right symmetric algebra.

• The *G*_{4}-Hom-associative algebras satisfy the condition

• The *G*_{5}-Hom-associative algebras satisfy the condition

If the product *μ* is skewsymmetric then the previous condition is exactly the Hom-Jacobi
identity.

• The *G*_{6}-Hom-associative algebras are the Hom-Lie admissible algebras.

Then, one may set the following generalization of pre-Lie (and Vinberg) algebras.

**Definition 4.4.** A Hom-pre-Lie algebra is a triple (*V, μ, α*) consisting of a linear space *V* , a
bilinear map and a homomorphism *α* satisfying

*Remark* 4.5. A Hom-Vinberg algebra is the opposite algebra of a Hom-pre-Lie algebra.

The study of flexible Lie admissible algebras was initiated by Albert [1] and investigated by number of authors, Myung, Okubo, Laufer, Tomber and Santilli, see [8] and [15]. The aim of this section is to extend the notions and results about flexible algebras to Hom-structures.

**Definition 5.1.** A Hom-algebra is called flexible if for any *x, y* in *V*

Remark 5.2. Using the *α*-associator the condition
(5.1) may be written as

The *α*-associator *a _{μ,α}* is a useful tool in the study of Hom-Lie and Hom-Lie admissible
algebras.

**Lemma 5.3.** *Let* *be a Hom-algebra. The following assertions are equivalent*

*1. A is flexible.*

*2. For any x, y in V* ,.

*3. For any x, y, z in V *, *a _{μ,α}(x, y, z) = −a_{μ,α}(z, y, x)*.

**Proof.** The equivalence of the first two assertions follows from the definition, and of last two
assertions since *a _{μ,α}(x − z, y, x − z)* = 0 holds by definition and is equivalent to

**Corollary 5.4.** *Any Hom-associative algebra is flexible*.

Let be a Hom-algebra, where *μ* is the multiplication and *α* a homomorphism.
We denote by the Hom-algebra over *V* with a multiplication . We
also denote by the Hom-algebra over *V* where the multiplication is given by the commutator .

**Proposition 5.5.** *A Hom-algebra* *is flexible if and only if*

**Proof.** Let *A* be a flexible Hom-algebra. By Lemma 5.3, this is equivalent to *a _{μ,α}(x, y, z) + a_{μ,α}(z, y, x)* = 0 for any

By expansion, the previous relation is equivalent to . Conversely, assume we have the condition (5.3) in Proposition. By setting *x = z* in (5.4), one
gets *a _{μ,α}(x, y, x)* = 0. Therefore

Let *V* be a *n*-dimensional -linear space and {*e _{1}, · · · , e_{n}*} be a basis of

If we require this algebra structure to be Hom-associative, then this limits the set of structure constants to a subvariety of defined by the following polynomial equations:

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 unital Hom-associative algebras, then we obtain a subvariety
(which we denote by *HomAlg _{n}*) determined by the following polynomial equations:

If we require this algebra structure to be Hom-Lie, then the structure constants determine a subvariety of defined by the following system of polynomial equations

where .

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 in defined by a system of cubic polynomials.

A point in such an algebraic variety (e.g. *HomAss _{n}*,

Let *A* be an algebra (e.g. Hom-associative, or Hom-Lie) defined by multiplication *μ* and a
linear map *α*. Given , 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.

In this section, we provide a characterization of all Hom-Lie algebras of sl(2) type and construct all the 3-dimensional Hom-Lie algebras for a class of homomorphisms.

By a direct calculation, we obtain the following class of Hom-Lie algebras of sl(2) type.

**Proposition 7.1.** *Let V be a three-dimensional* *-linear space and let (X _{1},X_{2},X_{3}) be its basis.
Any Hom-Lie algebra with the following brackets*

*is given by homomorphism α defined, with respect to the previous basis, by a matrix of the form where a, c, d, e, f are any parameters in* .

This may be viewed as a quasi-deformation of sl(2) in the class of Hom-Lie algebras. In the case where the matrix is the identity matrix one gets the classical Lie algebra sl(2).

*Remark* 7.2. 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 [7] 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 . The list is obtained by solving the system of equations for the structure constants equivalent
to axioms of Hom-Lie algebras. This list may be considered as a first partial step towards
establishing a classification of 3-dimensional hom-Lie algebras. In the list, for

This work was partially supported by the Crafoord foundation, The Swedish Foundation of International Cooperation in Research and Higher Education (STINT), The Royal Swedish Academy of Sciences, The Royal Physiographic Society in Lund, and the Liegrits network.

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