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Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark, OH 43055, USA **E-mail:** [email protected]

**Received Date:** May 11, 2007; **Revised Date:** September 06, 2007

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

Enveloping algebras of Hom-Lie and Hom-Leibniz algebras are constructed. 2000 MSC: 05C05, 17A30, 17A32, 17A50, 17B01, 17B35, 17D25 1

A Hom-Lie algebra is a triple (L, [−,−], α), where α is a linear self-map, in which the skewsymmetric bracket satisfies an α-twisted variant of the Jacobi identity, called the Hom-Jacobi identity. When α is the identity map, the Hom-Jacobi identity reduces to the usual Jacobi identity, and *L* is a Lie algebra. Hom-Lie algebras and related algebras were introduced in [1] to construct deformations of the Witt algebra, which is the Lie algebra of derivations on the Laurent polynomial algebra C[z^{±1}].

An elementary but important property of Lie algebras is that each associative algebra *A* gives rise to a Lie algebra *Lie*(*A*) via the commutator bracket. In [8], Makhlouf and Silvestrov introduced the notion of a Hom-associative algebra (*A*, *μ*, α), in which the binary operation *μ* satisfies an α-twisted version of associativity. Hom-associative algebras play the role of associative algebras in the Hom-Lie setting. In other words, a Hom-associative algebra *A* gives rise to a Hom-Lie algebra *H Lie*(*A*) via the commutator bracket.

The first main purpose of this paper is to construct the enveloping Hom-associative algebra *U _{H Lie}*(

The second main purpose of this paper is to construct the counterparts of the functors *H Lie* and *U _{H Lie}* for Hom-Leibniz algebras. Leibniz algebras (also known as

Each Hom-Lie algebra can be thought of as a Hom-Leibniz algebra. Likewise, to every Homassociative algebra is associated a Hom-dialgebra in which both binary operations are equal to the original one. The functors described above give rise to the following diagram of categories and (adjoint) functors.

Moreover, the commutativity, , holds.

This paper is the first part of a bigger project to study (co)homology theories of the various Hom-algebras. In papers under preparation, the author aims to construct:

• Hochschild-type (co)homology and its corresponding cyclic (co)homology for Homassociative algebras;

• an analogue of the Chevalley-Eilenberg Lie algebra (co)homology for Hom-Lie algebras;

• an analogue of Loday’s Leibniz algebra (co)homology for Hom-Leibniz algebras.

From the point-of-view of homological algebra, it is often desirable to interpret homology and cohomology in terms of resolutions and the derived functors *Tor* and *Ext*, respectively. In the classical case of Lie algebra (co)homology, this requires the enveloping algebra functor *U*, since for a Lie algebra *L*, where is the ground field. It is reasonable to expect that our enveloping algebra functors *U _{H Lie}* and

A major reason to study Hom-Lie algebra cohomology is to provide a proper context for the *Hom-Lie algebra extensions* constructed in [1, Section 2.4]. After constructing a certain q-deformation of the Witt algebra, Hartwig, Larsson, and Silvestrov used the machinery of Hom-Lie algebra extensions to construct a corresponding deformation of the Virasoro algebra [1, Section 4]. In the classical case of Lie algebras, equivalence classes of extensions are classified by the cohomology module . The author expects that Hom-Lie algebra extensions will admit a similar interpretation in terms of the second Hom-Lie algebra cohomology module.

A important result in the theory of Lie algebra homology is the Loday-Quillen Theorem [7] relating Lie algebra and cyclic homology. In another paper, the author hopes to extend the Loday-Quillen Theorem to the Hom-algebra setting. Recall that the Loday-Quillen Theorem states that, for an associative algebra *A*, is isomorphic to the graded symmetric algebra of *HC*_{*−1}(*A*). It is not hard to check that, if *A* is a Hom-associative algebra, then so is gl(A). Therefore, using the commutator bracket of Makhlouf and Silvestrov [8], *gl*(*A*) can also be regarded as a Hom-Lie algebra. There is an analogue of the Loday-Quillen Theorem for Leibniz algebra and Hochschild homology due to Loday [2, Theorem 10.6.5], which the author also hopes to extend to the Hom-algebra case.

**Organization**

The rest of this paper is organized as follows.

The next section contains preliminary materials on binary trees. In Section 3, free Homnonassociative algebras of Hom-modules are constructed (Theorem 1). This leads to the construction of the enveloping Hom-associative algebra functor *U _{H Lie}* in Section 4 (Theorem 2).

In Section 5, Hom-dialgebras are introduced together with several classes of examples. It is then observed that Hom-dialgebras give rise to Hom-Leibniz algebras via a version of the commutator bracket (Proposition 1). The enveloping Hom-dialgebra functor *U _{H Leib}* for Hom- Leibniz algebras is constructed in Section 6 (Theorem 3).

The purpose of this section is to collect some basic facts about binary trees that are needed for the construction of the enveloping algebra functors in later sections. Sections 2.1 and 2.2 below follow the discussion in [4, Appendix A1] but with slightly different notation.

**Planar binary trees**

For n ≥ 1, let *T _{n}* denote the set of planar binary trees with

**Grafting of trees**

Let and be two trees. Define an (*n*+*m*)-tree , called the *grafting* of and , by joining the roots of and together, which forms the new lowest internal vertex that is connected to the new root. Pictorially, we have

Note that grafting is a nonassociative operation.

Conversely, by cutting the two upward branches from the lowest internal vertex, each *n*-tree can be uniquely represented as the grafting of two trees, say, , where *p* + *q* = *n*. By iterating the grafting operation, one can show by a simple induction argument that every *n*-tree (*n* ≥ 2) can be obtained as an iterated grafting of *n* copies of the 1-tree.

**Weighted trees**

By a *weighted n-tree*, we mean a pair , in which:

1. is an *n*-tree and

2. *w* is a function from the set of internal vertices of to the set ≥0 of non-negative integers.

If *v* is an internal vertex of , then we call *w*(*v*) the weight of *v*. The n-tree is called the *underlying n-tree* of , and *w* is called the *weight function* of . Let denote the set of all weighted *n*-trees. Since the 1-tree has no internal vertex, we have that .

We can picture a weighted *n*-tree by drawing the underlying *n*-tree and putting the weight *w*(*v*) next to each internal vertex *v*. For example, here is a weighted 4-tree,

(2.1)

**Grafting of weighted trees**

Let and be two weighted trees. Define their *grafting* to be the weighted (*p*+*q*)-tree with underlying tree The weight function is given by

The grafting can be pictured as

**The** +*m* **operation**

Let be a weighted *p*-tree, and let *m* be a non-negative integer. Define a new weighted *p*-tree with the same underlying *p*-tree . The weight function is given by

Pictorially, if

then

By cutting the two upward branches from the lowest internal vertex, every weighted *n*-tree can be written uniquely as

(2.2)

where with *p* + *q* = *n*, and *r* is the weight of the lowest internal vertex of . The same process can be applied to and and so on. In particular, every weighted *n*-tree for *n* ≥ 2 can be obtained from *n* copies of the 1-tree by iterating the operations and [*r*] (*r* ≥ 0). For example, denoting the 1-tree by *i*, the weighted 4-tree in (2.1) can be written as

The purpose of this section is to construct the free Hom-nonassociative algebra functor on which the enveloping algebra functors are based.

Throughout the rest of this paper, let denote a field of characteristic 0. Unless otherwise specified, modules, , Hom, and End (linear endomorphisms) are all meant over .

**Hom-modules**

By a *Hom-module*, we mean a pair (*V*, α) consisting of:

1. a module *V* and

2. a linear endomorphism .

A morphism of Hom-modules is a linear map such that . The category of Hom-modules is denoted by **HomMod**.

**Hom-nonassociative algebras**

By a *Hom-nonassociative algebra*, we mean a triple (*A*, *μ*, *α*) in which:

1. *A* is a module,

2. *μ*: is a bilinear map,

3. .

A morphism is a linear map such that and . The category of Hom-nonassociative algebras is denoted by **HomNonAs**.

**Parenthesized monomials**

In a Hom-nonassociative algebra (*A*, *μ*, *α*), we will often abbreviate *μ*(*x*, *y*) to *xy* for *x*, *y* *A.* In general, given elements *x*_{1}, . . . , *x*_{n} *A*, there are #*T _{n}* =

**Ideals**

Let (*A*, *μ*, *α*) be a Hom-nonassociative algebra, and let be a non-empty subset of elements of *A*. Then the *two-sided ideal* generated by *S* is the smallest sub-- module of *A* containing *S* such that and , but is not necessarily closed under α. The two-sided ideal generated by *S* always exists and can be constructed as the sub--module of *A* spanned by all the parenthesized monomials *x*_{1} · · · *x _{n}* in

It should be noted that a *two-sided ideal* as defined in the previous paragraph is *not* an ideal in the category **HomMod**, since it is not necessarily closed under α.

**Products using weighted trees**

Each weighted *n*-tree provides a way to multiply *n* elements in a Hom-nonassociative algebra (*A*, *μ*, *α*). More precisely, we define maps

(3.1)

inductively via the rules:

1. (*x*)_{i} = *x* for *x * *A*, where *i* denotes the 1-tree.

2. If as in (2.2), then

where (*r* times). This is a generalization of the parenthesized monomials discussed above. For example, if is the weighted 4-tree in (2.1), then

for . Note that

**Free Hom-nonassociative algebras**

Let *E*:** HomNonAs → HomMod** be the forgetful functor that forgets about the binary operations.

**Theorem 1**. *The functor E admits a left adjoint F _{H N As}* :

Proof. First we equip *F _{H N As}* (

The linear map is defined by the rules:

Let denote the obvious inclusion map.

To show that *F _{H N As}* is the left adjoint of

(3.2)

where the right-hand side is defined as in (3.1). It is clear that . Next we show that *g* is a morphism of Hom-nonassociative algebras.

To show that *g* commutes with *α*, first note that coincides with when restricted to *V* , since *f* commutes with *α*. For *n* ≥ 2, we compute as follows:

To show that *g* is compatible with *μ*, we compute as follows:

This shows that *g* is compatible with *μ* as well, so *g* is a morphism of Hom-nonassociative algebras.

Note that, by a simple induction argument, every generator can be obtained from by repeatedly taking products *μF* and applying *αF* . Indeed, if as in (2.2), then . The same argument then applies to , and so on. This process has to stop after a finite number of steps, since in each step both *p* and *q* are strictly less than *n*. Since *g* is required to be a morphism of Hom-nonassociative algebras, this remark implies that *g* is determined by its restriction to *V* , which must be equal to *f*. This shows that *g* is unique.

We call the *free Hom-nonassociative algebra* of **HomMod**. This object is the analogue in the Hom-nonassociative setting of the non-unital tensor algebra Other free Hom-algebras can be obtained from the free Hom-nonassociative algebra. One such example is given in Section 4.3.

The purpose of this section is to construct the enveloping Hom-associative algebra functor that is left adjoint to the functor *H Lie*, which we first recall.

**The functor H Lie**

A *Hom-associative algebra* [8, Definition 1.1] is a Hom-nonassociative algebra (*A*, *μ*, *α*) such that the following *α*-twisted associativity holds for

*α*(*x*) (*yz*) = (*xy*) *α*(*z*) (4.1)

As before, we abbreviate *μ*(*x*, *y*) to *xy*. The full subcategory of **HomNonAs** whose objects are the Hom-associative algebras is denoted by **HomAs**.

A *Hom-Lie algebra* [8, Definition 1.4] (first introduced in [1]) is a Hom-nonassociative algebra (*L*, [−,−], *α*), satisfying the following two conditions:

1. [*x*, *y*] = −[*y*, *x*] (skew-symmetry),

2. 0 = [*α*(*x*), [*y*, *z*]] + [*α*(*z*), [*x*, *y*]] + [*α*(*y*), [*z*, *x*]] (Hom-Jacobi identity)

for . The full subcategory of **HomNonAs** whose objects are the Hom-Lie algebras is denoted by **HomLie**.

Given a Hom-associative algebra (*A*, *μ*, *α*), one can associate to it a Hom-Lie algebra

(*H Lie*(*A*), [−,−], *α*)

in which *H Lie*(*A*) = *A* as a - module and [8, Proposition 1.7]. The bracket defined is clearly skew-symmetric. The Hom-Jacobi identity can be verified by writing out all 12 terms and observing that their sum is 0.

This construction gives a functor *H Lie*: **HomAs → HomLie** that is the Hom-algebra analogue of the functor *Lie* that associates a Lie algebra to an associative algebra via the commutator bracket. The functor *Lie* has as its left adjoint the enveloping algebra functor *U*. We now construct the Hom-algebra analogue of the functor *U*, which is denoted by *U _{HLie}*.

**Enveloping algebras**

Let (L, [−,−], α) be a Hom-Lie algebra. Consider the free Hom-nonassociative algebra (*F _{HN As}*(

Here the linear space *L* is identified with its image under the inclusion , and xy denotes . Inductively, we set

**Lemma 1**. *The submodule * *is a two-sided ideal and is closed under αF . The quotient*, *together with the induced maps of μF and αF , is a Hom-associative algebra*.

**Proof.** Given elements , we know that for some *n* < *∞*. Therefore, both *xy* and *yx* lie in . The two-sided ideal *I*^{∞} is closed under *αF* because, again, every element in *I*^{∞} must lie in some *I ^{n}*, and .

To show that the quotient *F _{HN As}*(

The first projection map sends the element to 0, since it is in the image of the map and, therefore, in *I*^{1}. It follows that the image of the element (*αF* (*x*)(*yz*) − (*xy*)*αF* (*z*)) in the quotient *F _{H N As}*(

From now on, we will denote the Hom-associative algebra (*F _{H N As}*(

**Theorem 2.** *The functor* *U _{H Lie}* :

**Proof.** Let (*L*, [−,−], *αL*) be a Hom-Lie algebra, and let *j* : *L* **→** *U _{H Lie}*(

*f*([*x*, *y*]) = *f*(*x*)*f*(*y*) − *f*(*y*)*f*(*x*)

for . We must show that there exists a unique morphism *h*: *U _{H Lie}*(

By Theorem 1, there exists a unique morphism *g* : *F _{H N As}*(

It follows that *g*(*I*^{1}) = 0, again because *g* commutes with *μ*. By induction, if *g*(*I*^{n}) = 0, then *g*(*αF* (*I ^{n}*)) =

The previous paragraph shows that *g* factors through *F _{H N As}*(

The uniqueness of *h* follows exactly as in the last paragraph of the proof of Theorem 1. This finishes the proof of the Theorem.

**Free Hom-associative algebras**

The construction of the functor *U _{H Lie}* can be slightly modified to obtain the free Hom-associative algebra functor. Indeed, all we need to do is to redefine the ideals

Essentially the same argument as above shows that *J*^{∞}is a two-sided ideal that is closed under *α*. Moreover, the quotient module

(4.2)

equipped with the induced maps of *μF* and *αF* , is the free Hom-associative algebra of (*V*, *α*). In other words, *F _{H As}* :

Conversely, if (*A*, *μ*, *α*) is a Hom-associative algebra, then the adjoint of the identity map on *A* is a surjective morphism of Hom-associative algebras. The kernel of *g* is a two-sided ideal in *F _{H As}*(

(4.3)

where the isomorphism is induced by *g*.

Other free Hom-algebras, including free Hom-dialgebras, free Hom-Lie algebras, and free Hom-Leibniz algebras, can be constructed similarly from the free Hom-nonassociative algebra.

The purposes of this section are (i) to introduce Hom-dialgebras and give some examples and (ii) to show how Hom-dialgebras give rise to Hom-Leibniz algebras.

**Dialgebras**

First we recall the definition of a dialgebra from [4]. A *dialgebra* *D* is a -module equipped with two bilinear maps , satisfying the following five axioms:

(5.1)

for . Many examples of dialgebras can be found in [4, pp. 16-18].

**Hom-dialgebras**

We extend this notion to the Hom-algebra setting. A *Hom-dialgebra* is a tuple , where *D* is a -module, are bilinear maps, and , such that the following five axioms are satisfied for :

(5.2)

We will often denote such a Hom-dialgebra by *D. A morphism* of Hom-dialgebras is a linear map that is compatible with *α* and the products and . The category of Hom-dialgebras is denoted by **HomDi**.

Note that if *D* is a Hom-dialgebra, then, by axioms (1) and (5), respectively, both (*D*, , *α*) and (*D*, , *α*) are Hom-associative algebras.

**Examples of Hom-dialgebras**

1.If (*A*, *μ*, *α*) is a Hom-associative algebra, then (*A*, , , *α*) is a Hom-dialgebra in which = *μ* =.

2. If (*D*, , ) is a dialgebra, then (*D*, , , *α* = Id_{D}) is a Hom-dialgebra.

3. This example is an extension of [4, Example 2.2(d)]. First we need some definitions. Let (*A*, *μA*, *αA*) be a Hom-associative algebra, and let (*M*, *αM*) be a Hom-module. A *Hom-A-bimodule* structure on (*M*, *αM*) consists of:

such that the following three conditions hold for

A *morphism* *f* : *M* → *N* of Hom-*A*-bimodules is a morphism *f* : (*M*, *αM*) → (*N*, *αN*) of Hom-modules such that *f*(*am*) = *af*(*m*) and *f*(*ma*) = *f*(*m*)*a* for

For example, if *g* : *A* → *B* is a morphism of Hom-associative algebras, then *B* becomes a Hom-*A*-bimodule via the actions, *ab* = *g*(*a*)*b* and *ba* = *bg*(*a*), for . In particular, the identity map Id_{A} makes *A* into a Hom-*A*-bimodule, and *g* : *A* → *B* becomes a morphism of Hom-*A*-bimodules.

Now let (*M*, *αM*) be a Hom-*A*-bimodule, and let *f* : *M* → *A* be a morphism of Hom-*A*-bimodules. Then the tuple (*M*, , , *αM*) is a Hom-dialgebra in which

for . The five Hom-dialgebra axioms (5.2) are easy to check. For example, given elements , we have that

This shows (1) in (5.2). The other four axioms are checked similarly.

**From Hom-dialgebras to Hom-Leibniz algebras**

Recall from [8, Definition 1.2] that a *Hom-Leibniz algebra* is a triple (L, [−,−], α), in which L is a -module,, and [−,−] : is a bilinear map, that satisfies the Hom-Leibniz identity,

(5.3)

for The full subcategory of **HomNonAs** whose objects are Hom-Leibniz algebras is denoted by **HomLeib**.

Note that Hom-Lie algebras are examples of Hom-Leibniz algebras. Also, if *α* = Id* _{L}* in a Hom-Leibniz algebra (

**Proposition 1**. *Let* (*D*, , , *α*) *be a Hom-dialgebra. Define a bilinear map* [−,−] : *by setting*

(5.4)

*Then* (*D*, , , *α*) *is a Hom-Leibniz algebra*.

**Proof**. We write down all twelve terms involved in the Hom-Leibniz identity (5.3):

Using the five Hom-dialgebra axioms (5.2), it is immediate to see that (5.3) holds.

We write (*H Leib*(*D*), [−,−], *α*) for the Hom-Leibniz algebra (*D*, [−,−], *α*) in Proposition 1. This gives a functor

*H Leib*: **HomDi** → **HomLeib** (5.5)

which is the Hom-Leibniz analogue of the functor *H Lie* [8, Proposition 1.7].

The purpose of this section is to construct the left adjoint *U*_{H Leib} of the functor *H Leib*. On the one hand, this is the Leibniz analogue of the functor *U _{H Lie}* (Theorem 2). On the other hand, this is the Hom-algebra analogue of the enveloping algebra functor of Leibniz algebras [5].

As in the case of *U _{H Lie}*, the construction of

**Diweighted trees**

By a diweighted n-tree, we mean a pair in which:

1. is an *n*-tree, called the *underlying n-tree* of , and

2. *w* is a function from the set of internal vertices of to the set ≥0 × {, }. We call *w* the *weight function* of .

The set of diweighted n-trees is denoted by . As in the case of weighted trees, we have . Every diweighted *n*-tree can be pictured by drawing the underlying *n*-tree and putting the weight *w*(*v*) next to each internal vertex *v* of .

Let be two diweighted trees. Define the *left grafting* to be the diweighted (*n* + *m*)-tree, , where the weight function is given by

The *right grafting* is defined in exactly the same way, except that *ω*(*v*) = (0,) if *v* is the lowest internal vertex of .

Let *m* be a non-negative integer. Suppose that is a diweighted n-tree in which

where *v* is the lowest internal vertex of . Define a new diweighted *n*-tree, , in which the weight function is given by

In other words, adds *m* to the integer component of the weight of the lowest internal vertex of .

Every diweighted *n*-tree can be written uniquely in the form

(6.1)

where with *p*+*q* = *n*, and *m* is the integer component of the weight of the lowest internal vertex of . Every diweighted *n*-tree for *n* ≥ 2 can be obtained from *n* copies of the 1-tree by iterating the operations _{l}, _{r}, and [*m*] (*m* ≥ 0).

**Enveloping algebras**

Let (*V*, *αV* ) be a Hom-module. Consider the module

(6.2)

where is a copy of A generator will be abbreviated to. Define two bilinear operations by setting

Define a linear map by the rules:

Note that is the analogue of *F _{H N As}*(

Let (L, [−,−], α) be a Hom-Leibniz algebra. Define an increasing sequence of two-sided ideals,

as follows. Set *I*^{1} to be the two-sided ideal in generated by the subset consisting of:

for . In (6), *L* is regarded as a submodule of via the inclusion map , and . The first five types of generators in *I*^{1} correspond to the five Hom-dialgebra axioms (5.2). Inductively, set

(6.3)

We are now ready for the Leibniz analogue of the enveloping Hom-associative algebra functor *U _{H Lie}*.

**Theorem 3**. *Let* (*L*, [−,−], *α*)* be a Hom-Leibniz algebra. Then:*

1. *I*^{∞} (6.3)* is a two-sided ideal in * *that is closed under αF* .

2. *The quotient module* *equipped with the induced maps of* , , *and* *αF , is a Hom-dialgebra*.

3. *The functor U _{ H Leib}* :

Since this Theorem can be proved by arguments that are essentially identical to those in Section 4, we will omit the proof.

The author would like to thank the referee for helpful comments and suggestions.

- HartwigJT, LarssonD, SilvestrovSD(2006) Deformations of Lie algebras usingσ-derivations. J Algebra 295: 314-361.
- LodayJL(1992) Cyclichomology.Grundl Math WissBd 301, Springer-Verlag.
- Loday JL(1993) Une version non commutative des alg`ebres de Lie: les alg`ebres de Leibniz. Ens Math 39: 269-293.
- Loday JL(2001) Dialgebras.In “Dialgebras and related operads”. Lect. Notes Math.1763, Springer-Verlag 7-66.
- LodayJL,PirashviliT(1992) Universal enveloping algebras of Leibniz algebras and (co)homology. Math Ann 296: 139-158.
- Loday JL, Pirashvili T (1998)The tensor category of linear maps and Leibniz algebras. Georgian Math J 5: 263-276.
- LodayJL, QuillenD(1984)Cyclic homology and the Lie algebra homology of matrices. Comm Math Helv 59: 565-591.
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