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Journal of Generalized Lie Theory and Applications
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Enveloping algebras of Hom-Lie algebras

Donald YAU

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

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Abstract

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

Introduction

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(L) of a Hom-Lie algebra L. In other words, U H Lie is the left adjoint functor of H Lie. This is analogous to the fact that the functor Lie admits a left adjoint U, the enveloping algebra functor. The construction of U H Lie(L) makes use of the combinatorial objects of weighted binary trees, i.e., planar binary trees in which the internal vertices are equipped with weights of non-negative integers.

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 right Loday algebras) [2,3,4,5,6] are non-skew-symmetric versions of Lie algebras in which the bracket satisfies a variant of the Jacobi identity. In particular, Lie algebras are examples of Leibniz algebras. In the Leibniz setting, the objects that play the role of associative algebras are called dialgebras, which were introduced by Loday in [4]. Dialgebras have two associative binary operations that satisfy three additional associative-type axioms. Leibniz algebras were extended to Hom-Leibniz algebras in [8]. Extending some of the work of Loday [4], we will introduce Hom-dialgebras and construct the adjoint pair (U H Lieb, H Leib) of functors. In a Hom-dialgebra, there are two binary operations that satisfy five α-twisted associative-type axioms.

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.

Equation

Moreover, the commutativity, Equation, 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 Equation for a Lie algebra L, where Equation is the ground field. It is reasonable to expect that our enveloping algebra functors U H Lie and U H Leib, or slight variations of these functors, will play similar roles for Hom-Lie and Hom-Leibniz (co)homology, respectively.

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 Equation. 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, Equation 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).

Preliminaries on binary trees

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 Tn denote the set of planar binary trees with n leaves and one root. Below are the first four sets Tn.

Equation

Each dot represents an internal vertex. From now on, an element of Tn will simply be called an n-tree. In each n-tree, there are (n−1) internal vertices, the lowest one of which is connected to the root. Note that the cardinality of the set Tn+1 is the nth Catalan number Cn = (2n)!/n!(n+1)!.

Grafting of trees

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

Equation

Note that grafting is a nonassociative operation.

Conversely, by cutting the two upward branches from the lowest internal vertex, each n-tree Equation can be uniquely represented as the grafting of two trees, say, Equation, 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 Equation, in which:

1. Equation is an n-tree and

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

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

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

Equation    (2.1)

Grafting of weighted trees

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

Equation

The grafting can be pictured as

Equation

The +m operation

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

Equation

Pictorially, if

Equation

then

Equation

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

Equation    (2.2)

where Equation with p + q = n, and r is the weight of the lowest internal vertex of Equation. The same process can be applied to Equation and Equation 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 Equation and [r] (r ≥ 0). For example, denoting the 1-tree by i, the weighted 4-tree in (2.1) can be written as

Equation

Hom-modules and Hom-nonassociative algebras

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 Equation denote a field of characteristic 0. Unless otherwise specified, modules, Equation, Hom, and End (linear endomorphisms) are all meant over Equation.

Hom-modules

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

1. a module V and

2. a linear endomorphism Equation.

A morphism Equation of Hom-modules is a linear map Equation such that Equation. 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: Equation is a bilinear map,

3. Equation.

A morphism Equation is a linear map Equation such that Equation and Equation. 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 Equation A. In general, given elements x1, . . . , xn Equation A, there are #Tn = Cn−1 ways to parenthesize the monomial x1 · · · xn to obtain an element in A. Indeed, given an n-tree Equation, one can label the n leaves of Equation from left to right as x1, . . . , xn. Starting from the top, at each internal vertex v of Equation, we multiply the two elements represented by the two upward branches connected to v. For example, for the five 4-trees in T4 as displayed in Section 2.1, the corresponding parenthesized monomials of x1x2x3x4 are x1(x2(x3x4)), x1((x2x3)x4), (x1x2)(x3x4), (x1(x2x3))x4, ((x1x2)x3)x4. Conversely, every parenthesized monomial x1 · · · xn corresponds to an n-tree.

Ideals

Let (A, μ, α) be a Hom-nonassociative algebra, and let Equation be a non-empty subset of elements of A. Then the two-sided ideal Equation generated by S is the smallest sub-Equation- module of A containing S such that Equation and Equation, but Equation is not necessarily closed under α. The two-sided ideal generated by S always exists and can be constructed as the sub-Equation-module of A spanned by all the parenthesized monomials x1 · · · xn in A with n ≥ 1 such that at least one xj lies in S. This notion of two-sided ideals will be used below (first in Section 4.2) in the constructions of the enveloping Hom-algebras and the free Hom-associative algebras (Section 4.3).

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

Equation     (3.1)

inductively via the rules:

1. (x)i = x for x Equation A, where i denotes the 1-tree.

2. If Equation as in (2.2), then Equation

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

Equation

for Equation. Note that

Equation

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 FH N As : HomMod → HomNonAs defined as

Equation

Equation

Proof. First we equip FH N As (V ) with the structure of a Hom-nonassociative algebra. For elements Equation, a generatorEquation will be abbreviated toEquation. The binary operation Equation is defined as

Equation

The linear map Equation is defined by the rules:

Equation

Let Equation denote the obvious inclusion map.

To show that FH N As is the left adjoint of E, let (A, μA, αA) be a Hom-nonassociative algebra, and let f : VA be a morphism of Hom-modules. We must show that there exists a unique morphism g : FH N As(V ) → A of Hom-nonassociative algebras such that Equation. Define a map g : FH N As(V ) → A by

Equation    (3.2)

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

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

Equation

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

Equation

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 Equation can be obtained from Equation by repeatedly taking products μF and applying αF . Indeed, if Equation as in (2.2), then Equation. The same argument then applies to Equation, 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 Equation the free Hom-nonassociative algebra of Equation HomMod. This object is the analogue in the Hom-nonassociative setting of the non-unital tensor algebra Equation Other free Hom-algebras can be obtained from the free Hom-nonassociative algebra. One such example is given in Section 4.3.

Enveloping algebras of Hom-Lie algebras

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 Equation

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 Equation. 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 Equation- module andEquation [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 UHLie.

Enveloping algebras

Let (L, [−,−], α) be a Hom-Lie algebra. Consider the free Hom-nonassociative algebra (FHN As(L), μF , αF ) and the sequence of two-sided ideals, Equation defined as follows. Let I1 be the two-sided ideal

Equation

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

Equation

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

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

To show that the quotient FHN As(L)/I, equipped with the induced maps of μF and αF , is a Hom-associative algebra, we must show that α-associativity (4.1) holds. Let x, y, and z be elements in FH N As(L). Consider the diagram of Equation-modules,

Equation

The first projection map sends the element Equation to 0, since it is in the image of the map Equation and, therefore, in I1. It follows that the image of the element (αF (x)(yz) − (xy)αF (z)) in the quotient FH N As(L)/I is 0 as well. This shows that FH N As(L)/I is a Hom-associative algebra.

From now on, we will denote the Hom-associative algebra (FH N As(L)/I, μF , αF) of Lemma 1 by (UH Lie(L), μ, α). This defines a functor UH Lie : HomLie → HomAs.

Theorem 2. The functor UH Lie : HomLie → HomAs is left adjoint to the functor H Lie.

Proof. Let (L, [−,−], αL) be a Hom-Lie algebra, and let j : L UH Lie(L) be the composition of the maps Equation. Let (A, μA, αA) be a Hom-associative algebra, and let f : L H Lie(A) be a morphism of Hom-Lie algebras. In other words, f : L A is a linear map such that Equation and

f([x, y]) = f(x)f(y) − f(y)f(x)

for Equation. We must show that there exists a unique morphism h: UH Lie(L) → A of Homassociative algebras such that Equation (as morphisms of Equation-modules).

By Theorem 1, there exists a unique morphism g : FH N As(L) → A of Hom-nonassociative algebras such that Equation. The map g is defined in (3.2). We claim that g(I) = 0. It suffices to show that g(In) = 0 for all n ≥ 1. To see this, first note that g(z) = 0 for any element z in the image of the map Equation, since g commutes with both μ and α and A satisfies α-twisted associativity (4.1). Moreover, for elements Equation, we have that

Equation

It follows that g(I1) = 0, again because g commutes with μ. By induction, if g(In) = 0, then g(αF (In)) = αA(g(In)) = 0 as well. Therefore, g(In+1) = 0, which finishes the induction step. Since g(In) = 0 for all n ≥ 1, it follows that g(I) = 0, as claimed.

The previous paragraph shows that g factors through FH N As(L)/I= UH Lie(L). In other words, there exists a linear map h: UH Lie(L) → A such that Equation . Since the operations μ and α on UH Lie(L) are induced by the ones on FH NAs(L), it follows that h is also compatible with μ and α. In other words, h is a morphism of Hom-associative algebras such that

Equation

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 UH Lie can be slightly modified to obtain the free Hom-associative algebra functor. Indeed, all we need to do is to redefine the ideals In as follows. Let (V, α) be a Hom-module. Define

Equation

Essentially the same argument as above shows that Jis a two-sided ideal that is closed under α. Moreover, the quotient module

Equation     (4.2)

equipped with the induced maps of μF and αF , is the free Hom-associative algebra of (V, α). In other words, FH As : HomMod → HomAs is the left adjoint of the forgetful functor HomAs → HomMod. The functor FH As gives us a way to construct a Hom-associative algebra starting with just a Hom-module.

Conversely, if (A, μ, α) is a Hom-associative algebra, then the adjoint of the identity map on A is a surjective morphism Equation of Hom-associative algebras. The kernel of g is a two-sided ideal in FH As(A) that is closed under α. This allows us to write any given Hom-associative algebra A as a quotient of a free Hom-associative algebra,

Equation    (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.

Hom-dialgebras and Hom-Leibniz algebras

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 Equation-module equipped with two bilinear maps Equation, satisfying the following five axioms:

Equation     (5.1)

for Equation. 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 Equation, where D is a Equation-module, Equation are bilinear maps, and Equation, such that the following five axioms are satisfied for Equation:

Equation     (5.2)

We will often denote such a Hom-dialgebra by D. A morphism Equation of Hom-dialgebras is a linear map that is compatible with α and the products Equation and Equation . 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, Equation, α) and (D, Equation, α) are Hom-associative algebras.

Examples of Hom-dialgebras

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

2. If (D, Equation, Equation) is a dialgebra, then (D, Equation, Equation, α = IdD) 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:

Equation

such that the following three conditions hold for Equation

Equation

A morphism f : MN 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 Equation

For example, if g : AB 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 Equation. In particular, the identity map IdA makes A into a Hom-A-bimodule, and g : AB becomes a morphism of Hom-A-bimodules.

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

Equation

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

Equation

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 Equation-module,Equation, and [−,−] : Equation is a bilinear map, that satisfies the Hom-Leibniz identity,

Equation    (5.3)

for Equation 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 = IdL in a Hom-Leibniz algebra (L, [−,−], α), then (L, [−,−]) is called a Leibniz algebra [2-6], which is a non-skew-symmetric version of a Lie algebra. In [4, Proposition 4.2], Loday showed that a dialgebra gives rise to a Leibniz algebra via a version of the commutator bracket (see (5.4) below). The result below is the Hom-algebra analogue of that result.

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

Equation    (5.4)

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

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

Equation

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: HomDiHomLeib     (5.5)

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

Enveloping algebras of Hom-Leibniz algebras

The purpose of this section is to construct the left adjoint UH Leib of the functor H Leib. On the one hand, this is the Leibniz analogue of the functor UH 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 UH Lie, the construction of UH Leib depends on a suitable notion of trees, which we discuss next.

Diweighted trees

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

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

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

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

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

Equation

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

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

Equation

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

Equation

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

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

Equation      (6.1)

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

Enveloping algebras

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

Equation     (6.2)

where Equation is a copy of Equation A generatorEquation will be abbreviated toEquation. Define two bilinear operationsEquation by setting

Equation

Define a linear map Equation by the rules:

Equation

Note that Equation is the analogue of FH N As(V ) (Theorem 1) with two bilinear operations. In Equation, the two-sided ideal Equation generated by a non-empty subset S is the smallest sub-Equation-module of Equation containing S such that Equation whenever Equation and Equation, but Equation is not necessarily closed under αF . It can be constructed as the sub-Equation-module of Equation spanned by all the parenthesized monomials x1 ¤ · · · ¤ xn with n ≥ 1 and Equation such that at least one xj lies in S.

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

Equation

as follows. Set I1 to be the two-sided ideal in Equation generated by the subset consisting of:

Equation

for Equation. In (6), L is regarded as a submodule of Equation via the inclusion map Equation, and Equation. The first five types of generators in I1 correspond to the five Hom-dialgebra axioms (5.2). Inductively, set

Equation      (6.3)

We are now ready for the Leibniz analogue of the enveloping Hom-associative algebra functor UH Lie.

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

1. I (6.3) is a two-sided ideal in Equation that is closed under αF .

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

3. The functor U H Leib : HomLeib → HomDi is left adjoint to the functor H Leib (5.5).

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

Acknowledgement

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

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