Enveloping algebras of color hom-Lie algebrasjournals.tubitak.gov.tr/math/issues/mat-19-43-1/mat-43-1-26-1808-96… · ARMAKAN et al./Turk J Math presented as an introduction to color

Turk J Math(2019) 43: 316 – 339© TÜBİTAKdoi:10.3906/mat-1808-96

Turkish Journal of Mathematics

http :// journa l s . tub i tak .gov . t r/math/

Research Article

Enveloping algebras of color hom-Lie algebras

Abdoreza ARMAKAN1 , Sergei SILVESTROV2, Mohammad Reza FARHANGDOOST1,∗

1Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran2Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University,

Västerås, Sweden

Received: 19.08.2018 • Accepted/Published Online: 12.12.2018 • Final Version: 18.01.2019

Abstract: In this paper, the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the freeinvolutive hom-associative color algebra on a hom-module is described and applied to obtain the universal envelopingalgebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincaré–Birkhoff–Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra.

Key words: Color hom-Lie algebra, enveloping algebra

1. IntroductionThe investigations of various quantum deformations (or q -deformations) of Lie algebras started a period ofrapid expansion in 1980s, stimulated by the introduction of quantum groups motivated by applications tothe quantum Yang–Baxter equation, quantum inverse scattering methods, and constructions of the quantumdeformations of universal enveloping algebras of semisimple Lie algebras. Since then, several other versionsof q -deformed Lie algebras have appeared, especially in physical contexts such as string theory, vertex modelsin conformal field theory, quantum mechanics and quantum field theory in the context of deformations ofinfinite-dimensional algebras, primarily the Heisenberg algebras, oscillator algebras, and Witt and Virasoroalgebras [3, 16–19, 21–23, 28, 30, 41–43]. In these pioneering works, it has been discovered in particularthat in these q -deformations of Witt and Visaroro algebras and some related algebras, some interesting q -deformations of Jacobi identities, extending Jacobi identity for Lie algebras, are satisfied. This has been oneof the initial motivations for the development of general quasideformations and discretizations of Lie algebrasof vector fields using more general σ -derivations (twisted derivations) in [25], and introduction of abstractquasi-Lie algebras and subclasses of quasi-hom-Lie algebras and hom-Lie algebras as well as their generalcolored (graded) counterparts in [25, 36, 37, 39, 58]. These generalized Lie algebra structures with (graded)twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within thesame algebraic framework such quasideformations and discretizations of Lie algebras of vector fields using σ -derivations, describing general descritizations and deformations of derivations with twisted Leibniz rule, andthe well-known generalizations of Lie algebras such as color Lie algebras, which are the natural generalizationsof Lie algebras and Lie superalgebras.

Quasi-Lie algebras are nonassociative algebras for which the skew-symmetry and the Jacobi identity∗Correspondence: [email protected] AMS Mathematics Subject Classification: 17B75, 17B99, 17B35, 17B37, 17D99, 16W10, 16W55

This work is licensed under a Creative Commons Attribution 4.0 International License.316

https://orcid.org/0000-0002-3245-3114

ARMAKAN et al./Turk J Math

are twisted by several deforming twisting maps and also the Jacobi identity in quasi-Lie and quasi-hom-Liealgebras in general contains six twisted triple bracket terms. Hom-Lie algebras is a special class of quasi-Lie algebras with the bilinear product satisfying the nontwisted skew-symmetry property as in Lie algebras,whereas the Jacobi identity contains three terms twisted by a single linear map, reducing to the Jacobi identityfor ordinary Lie algebras when the linear twisting map is the identity map. Subsequently, hom-Lie admissiblealgebras were considered in [45], where the hom-associative algebras were also introduced and shown to behom-Lie admissible natural generalizations of associative algebras corresponding to hom-Lie algebras. In [45],moreover, several other interesting classes of hom-Lie admissible algebras generalizing some nonassociativealgebras, as well as examples of finite-dimentional hom-Lie algebras were described. Since these pioneeringworks [25, 36, 37, 39, 40, 45], hom-algebra structures have become a popular area with increasing number ofpublications in various directions.

Hom-Lie algebras, hom-Lie superalgebras, and color hom-Lie algebras are important special classes ofcolor (Γ -graded) quasi-Lie algebras introduced first by Larsson and Silvestrov in [37, 39]. Hom-Lie algebras andhom-Lie superalgebras were studied further in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yauand other authors [12, 38, 44–48, 51, 52, 57, 59–62, 64–66, 68, 69], and color hom-Lie algebras were considered,for example, in [1, 13, 14, 68]. In [4], the constructions of hom-Lie and quasi-hom-Lie algebras based on twisteddiscretizations of vector fields [25] and hom-Lie admissible algebras were extended to hom-Lie superalgebras,a subclass of graded quasi-Lie algebras [37, 39]. We also wish to mention that Z3 -graded generalizationsof supersymmetry, Z3 -graded algebras, ternary structures, and related algebraic models for classifications ofelementary particles and unification problems for interactions, quantum gravity and noncommutative gaugetheories [2, 31–34] also provide interesting examples related to hom-associative algebras, graded hom-Liealgebras, twisted differential calculi, and n -ary hom-algebra structures. It would be a project of great interestto extend and apply all the constructions and results in the present paper in the relevant contexts of the articles[2, 4, 6, 8–10, 31–33, 37, 39, 45].

An important direction with many fundamental open problems in the theory of (color) quasi-Lie algebrasand in particular (color) quasi-hom-Lie algebras and (color) hom-Lie algebras is the development of comprehen-sive fundamental theory, explicit constructions, examples and algorithms for enveloping algebraic structures,expanding the corresponding more developed fundamental theory and constructions for enveloping algebrasof Lie algebras, Lie superalgebras, and general color Lie algebras [11, 20, 29, 49, 50, 53–56]. Several authorshave tried to construct the enveloping algebras of hom-Lie algebras. For instance, Yau has constructed theenveloping hom-associative algebra UHLie(L) of a hom-Lie algebra L in [64] as the left adjoint functor of HLieusing combinatorial objects of weighted binary trees, i.e. planar binary trees in which the internal vertices areequipped with weights of nonnegative integers. This is analogous to the fact that the functor Lie admits aleft adjoint U , the enveloping algebra functor. He also introduced construction of the counterpart functorsHLie and UHLie for hom-Leibniz algebras. In [26], for hom-associative algebras and hom-Lie algebras, theenvelopment problem, operads, and the Diamond Lemma, and Hilbert series for the hom-associative operadand free algebra were studied. Recently, making use of free involutive hom-associative algebras, the authors in[24] have found an explicit constructive way to obtain the universal enveloping algebras of hom-Lie algebras inorder to prove the Poincaré–Birkhoff–Witt theorem.

In this paper, we will give a brief review of well-known facts about hom-Lie algebras and their envelopingalgebras. We will then present some new results, in the hope that they may eventually have a bearing onrepresentation theory of color hom-Lie algebras. In Section 2, some necessary notions and definitions are

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presented as an introduction to color hom-Lie algebras. In Section 3 the notion of universal enveloping algebrasof color hom-Lie algebras is given and several useful result about involutive color hom-Lie algebras are proven.In Section 4, we prove the analogous of the well known Poincaré–Birkhoff–Witt theorem for color hom-Liealgebras, using the definitions and results of Sections 2 and 3. Finally, due to the importance of hom-Liesuperalgebras, we present the most important results of the paper in hom-Lie superalgebras case.

2. Basic concepts on hom-Lie algebras and color quasi-Lie algebras

We start by recalling some basic concepts from [24, 45, 47]. We use k to denote a commutative unital ring (forexample a field).

Definition 2.1 (i) A hom-module is a pair (M,α) consisting of a k-module M and a linear operatorα :M →M .

(ii) A hom-associative algebra is a triple (A, ., α) consisting of a k-module A , a linear map · : A⊗A→ A calledthe multiplication and a multiplicative linear operator α : A → A which satisfies the hom-associativitycondition, namely

α(x) · (y · z) = (x · y) · α(z),

for all x, y, z ∈ A .

(iii) A hom-associative algebra or a hom-module is called involutive if α2 = id .

(iv) Let (M,α) and (N, β) be two hom-modules. A k-linear map f : M → N is called a morphism ofhom-modules if

f(α(x)) = β(f(x)),

for all x ∈M .

(v) Let (A, ·, α) and (B, •, β) be two hom-associative algebras. A k-linear map f : A → B is called amorphism of hom-associative algebras if

(1) f(x · y) = f(x) • f(y),

(2) f(α(x)) = β(f(x)), for all x, y ∈ A .

(vi) If (A, ·, α) is a hom-associative algebra, then B ⊆ A is called a hom-associative subalgebra of A if it isclosed under the multiplication · and α(B) ⊆ B . A submodule I is called a hom-ideal of A if x · y ∈ I

and x · y ∈ I for all x ∈ I and y ∈ A , and also α(I) ⊆ I .

One can find various examples of hom-associative algebras and their properties in [45, 47].

Definition 2.2 Let (M,α) be an involutive hom-module. A free involutive hom-associative algebra on M

is an involutive hom-associative algebra (FM , ⋆, β) together with a morphism of hom-modules j : M → FM

with the property that for any involutive hom-associative algebra A together with a morphism f : M → A ofhom-modules, there is a unique morphism f̄ : FM → A of hom-associative algebras such that f = f̄ ◦ j .

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Our next goal is to recall the definition of an involutive hom-associative algebra on an involutive hom-module (M,α) from [24], which is known as the hom-tensor algebra and is denoted here by T (M) . Note thatas an R -module, T (M) is the same as the tensor algebra, i.e.

T (M) =⊕i≥1

M⊗i,

on which we have the following multiplication in order to obtain a hom-associative algebra. First, the linearmap α on M is extended to a linear map αT on M⊗i by the tensor multiplicativity, i.e.

αT (x) = αT (x1 ⊗ · · · ⊗ xi) := α(x1)⊗ · · · ⊗ α(xi),

for all pure tensors x := x1 ⊗ · · · ⊗ xi ∈M⊗i , i ≥ 1 . One can see that αT has the following properties:

(i) αT (x⊗ y) = αT (x)⊗ αT (y) , for all x ∈M⊗i , y ∈M⊗j .

(ii) α2T = id .

Now, the binary operation on T (M) is defined as follows:

x⊙ y := αi−1T (x)⊗ y1 ⊗ · · ·αT (y2 ⊗ · · · ⊗ yi), (2.1)

for all x ∈M⊗i and y ∈M⊗j .

Theorem 2.3 [24] Let (M,α) be an involutive hom-module. Then,

(i) The triple T (M) := (T (M),⊙, αT ) is an involutive hom-associative algebra.

(ii) The quadruple (T (M),⊙, αT , iM ) is the free involutive hom-associative algebra on M .

It is now convenient to recall some definitions for hom-Lie algebras [25, 36, 37, 39, 45].

Definition 2.4 A hom-Lie algebra is a triple (g, [, ], α) , where g is a vector space equipped with a skew-symmetric bilinear map [, ] : g× g → g and a linear map α : g → g such that

[α(x), [y, z]] + [α(y), [z, x]] + [α(z), [x, y]] = 0,

for all x, y, z ∈ g , which is called hom-Jacobi identity.

A hom-Lie algebra is called a multiplicative hom-Lie algebra if α is an algebraic morphism, i.e. for anyx, y ∈ g ,

α([x, y]) = [α(x), α(y)].

We call a hom-Lie algebra regular if α is an automorphism. Moreover, it is called involutive if α2 = id .A subvector space h ⊂ g is a hom-Lie subalgebra of (g, [, ], α) if α(h) ⊂ h and h is closed under the

bracket operation, i.e.[x1, x2]g ∈ h,

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for all x1, x2 ∈ h. Let (g, [, ], α) be a multiplicative hom-Lie algebra. Let αk denote the k -times compositionof α by itself, for any nonnegative integer k , i.e.

αk = α ◦ ... ◦ α (k − times),

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

α−k = α−1 ◦ ... ◦ α−1 (k − times).

We now recall the notion of a color hom-Lie algebra step by step in order to indicate them as ageneralization of Lie color algebras.

Definition 2.5 [11, 39, 49, 53, 54] Given a commutative group Γ (referred to as the grading group), acommutation factor on Γ with values in the multiplicative group K \ {0} of a field K of characteristic 0is a map

ε : Γ× Γ → K \ {0},

satisfying three properties:

(i) ε(α+ β, γ) = ε(α, γ)ε(β, γ),

(ii) ε(α, γ + β) = ε(α, γ)ε(α, β),

(iii) ε(α, β)ε(β, α) = 1.

A Γ-graded ε-Lie algebra (or a Lie color algebra) is a Γ-graded linear space

X =⊕γ∈Γ

Xγ ,

with a bilinear multiplication (bracket) [., .] : X ×X → X satisfying the following properties:

(i) Grading axiom: [Xα, Xβ ] ⊆ Xα+β ,

(ii) Graded skew-symmetry: [a, b] = −ε(α, β)[b, a],

(iii) Generalized Jacobi identity:ε(γ, α)[a, [b, c]] + ε(β, γ)[c, [a, b]] + ε(α, β)[b, [c, a]] = 0,

for all a ∈ Xα, b ∈ Xβ , c ∈ Xγ and α, β, γ ∈ Γ . The elements of Xγ are called homogenous of degree γ , for allγ ∈ Γ .

Analogous to the other kinds of definitions of hom-algebras, the definition of a color hom-Lie algebra canbe given as follows [1, 13, 14, 39, 68].

Definition 2.6 A color hom-Lie algebra is a quadruple (g, [., .], ε, α) consisting of a Γ-graded vector space g ,a bi-character ε , an even bilinear mapping

[., .] : g× g → g,

(i.e. [ga, gb] ⊆ ga+b , for all a, b ∈ Γ) and an even homomorphism α : g → g such that for homogeneouselements x, y, z ∈ g , we have

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1. ε-skew symmetric: [x, y] = −ε(x, y)[y, x],

2. ε-Hom-Jacobi identity:∑

cyclic{x,y,z} ε(z, x)[α(x), [y, z]] = 0.

Color hom-Lie algebras are a special class of general color quasi-Lie algebras (Γ -graded quasi-Lie algebras)defined first by Larsson and Silvestrov in [39].

Let g =⊕

γ∈Γ gγ and h =⊕

γ∈Γ hγ be two Γ -graded color Lie algebras. A linear mapping f : g → h issaid to be homogenous of the degree µ ∈ Γ if

f(gγ) ⊆ hγ+µ,

for all γ ∈ Γ . If in addition, f is homogenous of degree zero, i.e.

f(gγ) ⊆ hγ ,

holds for any γ ∈ Γ , then f is said to be even.Let (g, [, ], ε, α) and (g′, [, ]′, ε′, α′) be two color hom-Lie algebras. A homomorphism of degree zero

f : g → g′ is said to be a morphism of color hom-Lie algebras if

1. [f(x), f(y)]′ = f([x, y]) , for all x, y ∈ g,

2. f ◦ α = α′ ◦ f.

In particular, if α is a morphism of color Lie algebras, then we call (g, [, ], ε, α) , a multiplicative colorhom-Lie algebra.

Example 2.7 [1] As in case of hom-associative and hom-Lie algebras, examples of multiplicative color hom-Lie algebras can be constructed for example by the standard method of composing multiplication with algebramorphism.

Let (g, [, ], ε) be a color Lie algebra and α be a color Lie algebra morphism. Then (g, [, ]α := α◦ [., .], ε, α)is a multiplicative color hom-Lie algebra.

Definition 2.8 A hom-associative color algebra is a triple (V, µ, α) consisting of a color space V , an evenbilinear map µ : V × V → V and an even homomorphism α : V → V satisfying

µ(α(x), µ(y, z)) = µ(µ(x, y), α(z)),

for all x, y, z ∈ V .

A hom-associative color algebra or a color hom-Lie algebra is said to be involutive if α2 = id .As in the case of an associative algebra and a Lie algebra, a hom-associative color algebra (V, µ, α) gives

a color hom-Lie algebra by antisymmetrization. We denote this color hom-Lie algebra by (A, [, ]A, βA) , whereβA = α and [x, y]A = xy − yx , for all x, y ∈ A .

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3. The universal enveloping algebraIn this section, we introduce the notion of the universal enveloping algebra of a color hom-Lie algebra. Moreover,we prove a new result on the free involutive hom-associative color algebra on an involutive hom-module.

Definition 3.1 Let (V, αV ) be an involutive hom-module. A free involutive hom-associative color algebra onV is an involutive hom-associative color algebra (F (V ), ∗, αF ) together with a morphism of hom-modules

jV : (V, αV ) → (F (V ), αF ),

with the property that, for any involutive hom-associative color algebra (A, ., αA) together with a morphismf : (V, αV ) → (A,αA) of hom-modules, there is a unique morphism f : (F (V ), ∗, αF ) → (A, ., αA) of hom-associative color algebras such that f = f ◦ jV .

Definition 3.2 Let (g, [, ], α) be a color hom-Lie algebra. A universal enveloping hom-associative color algebraof g is a hom-associative color algebra

U(g) := (U(g), µU , αU ),

together with a morphism φg : g → U(g) of color hom-Lie algebras such that for any hom-associative coloralgebra (A,µ, αA) and any color hom-Lie algebra morphism ϕ : (g, [, ]g, βg) , there exists a unique morphismϕ̄ : U(g) → A of hom-associative color algebras such that ϕ̄ ◦ φg = ϕ .

The following lemma shows an easy way to construct the universal algebra when we have an involutivecolor hom-Lie algebra.

Lemma 3.3 Let (g, [, ]g, βg) be an involutive color hom-Lie algebra.

(i) Let (A, ·, αA) be a hom-associative algebra. Let

f : (g, [, ]g, βg) → (A, [, ]A, βA)

be a morphism of color hom-Lie algebras and let B be the hom-associative subcolor algebra of A generatedby f(g) . Then B is involutive.

(ii) The universal enveloping hom-associative algebra (U(g), φg) of (g, [, ]g, βg) is involutive.

(iii) In order to verify the universal property of (U(g), φg) in Definition 3.2, we only need to consider involutivehom-associative algebras A := (A, ·A, αA) .

Proof

(i) LetS = {x ∈ A|α2

A(x) = x}.

One can easily check that S is a submodule. Also, for x, y ∈ S , we have xy ∈ S , since α2A(xy) =

α2A(x)α

2A(y) = xy . Moreover, we have

α2A(αA(x)) = αA(α

2A(x)) = αA(x),

which shows that αA(x) ∈ S , for all x ∈ S . Thus, S is a hom-associative subalgebra of A since f is amorphism.

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(ii) Since U(g) is generated by φg(g) as a hom-associative algebra, the statement follows from (i) .

(iii) We should prove that, assuming that the universal property of U(g) holds for involutive hom-associativealgebras, then it holds for all hom-associative algebras. Let (A, ·, αA) be a hom-associative algebra andlet

ψ : (g, [, ]g, βg) → (A, [, ]A, βA)

be a morphism of color hom-Lie algebras. Let

S = {x ∈ A|α2A(x) = x}

be the involutive hom-associative subalgebra of A defined in the proof of (i) . Since im(ψ) is contained inS , ψ is the composition of a morphism ψS : (g, [, ]g, βg) → (S, [, ]S , βS) of hom-associative color algebraswith the inclusion i : S → A . By assumption, there is a morphism ψ̄S : U(g) → A of hom-associativecolor algebras such that ψ̂S ◦ φg = ψS . Then composing with the inclusion i : B → A , we obtain amorphism ψ̄ : U(g) → A of hom-associative color algebras such that ψ̄ ◦ φg = ψ .Now, let ψ̄′ : U(g) → A be another morphism of hom-associative color algebras such that ψ̄′ ◦ φg = ψ .By (ii) , im(ψ̄′) is involutive. So ψ̄′ is the composition of a morphism ψ̄′

S : U(g) → S with the inclusioni : S → A and ψ̄′

S◦φg = ψS . Since S is involutive, the morphisms ψ̄′S and ψ̄S coincide. As a consequence,

ψ̄′ and ψ̄ coincide, which completes the proof.

2

We can now give the construction of the universal enveloping hom-associative color algebra of an involutivecolor hom-Lie algebra.

Theorem 3.4 Let g := (g, [, ]g, βg) be an involutive color hom-Lie algebra. Let

T (g) := (T (g),⊙, αT )

be the free hom-associative algebra on the hom-module underlying g obtained in Theorem 2.3. Let I be thehom-ideal of T (g) generated by the set

{a⊗ b− ε(a, b)b⊗ a− [a, b]} (3.1)

and let

U(g) =T (g)

I

be the quotient hom-associative algebra. Let ψ be the composition of the natural inclusion i : g → T (g) with thequotient map π : T (g) → U(g) . Then (U(g), ψ) is a universal enveloping hom-associative algebra of g . Also,the universal enveloping hom-associative algebra of g is unique up to isomorphism.

Proof The multiplication in U(g) is denoted by ∗ . The map ψ is a morphism of hom-modules since it is thecomposition of two hom-module morphisms. We have

ψ([x, y]g) =π([x, y]g) = π(x⊗ y − ε(x, y)y ⊗ x)

=π(x⊙ y − ε(x, y)y ⊙ x) = π(x) ∗ π(y)− ε(x, y)π(y) ∗ π(x)

=ψ(x) ∗ ψ(y)− ε(x, y)ψ(y) ∗ ψ(x) = [ψ(x), ψ(y)]g,

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for all x, y ∈ g , since x⊗ y − ε(x, y)y ⊗ x− [x, y]g is in I = ker(π) . Therefore, ψ is a morphism of color hom-Lie algebras. Now, using Lemma 3.3 (iii) , we consider an arbitrary involutive hom-associative color algebraA := (A, ·A, αA) . Let ξ : (g, [, ]g, βg) → (A, [, ]A, βA) be a morphism of color hom-Lie algebras. Since T (g) isthe free involutive hom-associative color algebra on the underlying hom-module of g , according to Theorem2.3, there exists a hom-associative color algebra morphism ξ̃ : T (g) → A of hom-associative color algebras suchthat ξ̃ ◦ ig = ξ . We have

ξ̃(x⊗ y − ε(x, y)y ⊗ x) =ξ̃(x⊙ y − ε(x, y)y ⊙ x)

=ξ̃(x) ·A ξ̃(y)− ε(x, y)ξ̃(y) ·A ξ̃(x)

=ξ(x) ·A ξ(y)− ε(x, y)ξ(y) ·A ξ(x) = [ξ(x), ξ(y)]A

=ξ([x, y]A) = ξ̃([x, y]A).

So I is contained in ker(ξ̃) and ξ̃ induces a morphism ξ̄ : U(g) → A of hom-associative color algebras suchthat ξ̃ = ξ̄ ◦ π . Therefore, ξ̄ ◦ ψ = ξ̄ ◦ π ◦ i = ξ̃ ◦ i = ξ .

Let ξ̄′ : U(g) → A be another morphism of hom-associative color algebras such that ξ̄′ ◦ψ = ξ . We mustshow that ξ̄(u) = ξ̄′(u) , for all u ∈ U(g) . It is sufficient to show that ξ̄π(a) = ξ̄′π(a) , for a ∈ g⊗i with i ≥ 1 ,since T (g) =

⊕i≥1 g

⊗i . We do this by using the induction on i ≥ 1 . For i = 1 , we have

(ξ̄ ◦ π)(a) = (ξ̄ ◦ π ◦ i)(a) = (ξ̃ ◦ i)(a) = ξ(a) = (ξ̄′ ◦ ψ)(a) = (ξ̄′ ◦ π)(a).

Now, assume that the statement holds for i ≥ 1 . Let a = a′ ⊗ ai+1 ∈ g⊗(i+1) , where a′ ∈ g⊗i . We have

(ξ̄ ◦ π)(a) =(ξ̄ ◦ π)(a′ ⊙ ai+1) = ξ̄(π(a′)) ·A ξ̄(π(ai+1))

=ξ̄′(π(a′)) ·A ξ̄′(π(ai+1)) = (ξ̄′ ◦ π)(a′ ⊙ ai+1) = (ξ̄′ ◦ π)(a).

Thus, the uniqueness of ξ̄ makes (U(g), ψ) a universal enveloping algebra of g .The only thing left to prove is the uniqueness of U(g) up to isomorphism. Let (U(g)1, ψ1) be another

universal enveloping algebra of g . By the definition of universal algebra, there exist homomorphisms

f : U(g) → U(g)1

andf1 : U(g)1 → U(g)

of hom-associative color algebras such that f ◦ ψ = ψ1 and f1 ◦ ψ1 = ψ . Therefore,

f1 ◦ f ◦ ψ = ψ = idU(g) ◦ ψ.

Since id and f1 ◦ f are both hom-associative homomorphisms, by the uniqueness in the universal property of(U(g), ψ) , we have f1 ◦ f = idU(g) , and hence

f ◦ f1 = idU(g)1 ,

which completes the proof. 2

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4. The Poincaré–Birkhoff–Witt theoremIn this section we prove a Poincaré–Birkhoff–Witt like type theorem for involutive color hom-Lie algebras.

Let g be a Lie color algebra with an ordered basis

X = {xn|n ∈ H},

where H is a well- and totally ordered set. Let I be the ideal of the free associative algebra T (g) on g , whichwas given in Theorem 3.4, so that U(g) is the universal enveloping algebra of g . The Poincaré–Birkhoff–Witttheorem states that the linear subspace I of T (g) has a canonical linear complement which has a basis givenby

W := {xn1⊗ · · · ⊗ xni

|n1 ≥ · · · ≥ ni, i ≥ 0}, (4.1)

called the Poincaré–Birkhoff–Witt basis of U(g) .To simplify the notations, we denote x̄ := βg(x) for x ∈ g and x̄ := αT (x) for x ∈ g⊗i , i ≥ 1 . There

exists a linear operator which is introduced in [24].

θ : g⊗i → g⊗i, (4.2)

which maps every x := x1 ⊗ · · · ⊗ xi to x̃1 ⊗ · · · ⊗ x̃i , where

x̃n =

{x̄n if n = 2k + 1 and k ≥ 1,xn otherwise.

Now, we can study linear generators of the hom-ideal I and express them in terms of the tensor product.Since βg is involutive, it is also bijective, so

βg(g) = g,

and we have the same argument for θ :θ(g⊗i) = g⊗i.

Let us now review some properties of the linear operator αT and the multiplication ⊙ which was statedin [24]. First we have

αjT (g

⊗i) = g⊗i, j ≥ 0, i ≥ 0.

Then for any natural numbers r, s ≥ 1 , we have

g⊗r ⊙ g⊗s = αs−1T (g⊗r)⊗ g⊗ αT (g

⊗(s−1)) = g⊗r+s.

In the following lemma, for a = a1 ⊗ · · · ⊗ ai ∈ g⊗i and i ≥ 1 , we denote l(a) = i .

Lemma 4.1 Let u = u1 ⊗ · · · ⊗ uj ∈ g⊗j , v = v1 ⊗ · · · ⊗ vk ∈ g⊗k and w = w1 ⊗ · · · ⊗ wl ∈ g⊗l . Let θ bedefined as in (4.2). Then

(i) θ(u) = u1 ⊗ u2 ⊗ αT (u3)⊗ α2T (u4) · · · ⊗ αj−2

T (uj) = u1 ⊗⊗jk=2α

k−2T (uk) ,

(ii) θ(αT (u)) = αT (θ(u)) ,

(iii) θ(u⊗w) = θ(u)⊗ αj−1T (w1)⊗ · · · ⊗ αj+l−2

T (wl)

325


(iv) If l(v) ≥ 1 and l(u) = l(v) + 1 , i.e. j = k + 1 , then there is c ∈ g⊗j such that

θ(u⊗w) = θ(u)⊗ c and θ(v⊗w) = θ(v)⊗ αT (c).

Proof Straightforward calculations. 2

Theorem 4.2 Let g be an involutive color hom-Lie algebra. Let θ : T (g) → T (g) be as defined in (4.2). Let Ibe the hom-ideal of T (g) as defined in Theorem 3.4. Let

J =∑

n,m≥0

∑a∈g⊗n

b∈g⊗m

∑a,b∈g

(a⊗ (a⊗ b− ε(a, b)b⊗ a)⊗ b− αT (a)⊗ [a, b]g ⊗ αT (b)). (4.3)

Then

(i) I =∑

n,m≥0

∑a,b∈g(g

⊗n ⊙ (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ g⊗m

(ii) θ(I) = J

Proof

(i) Since the hom-ideal I is generated by the elements of the form

a⊗ b− ε(a, b)b⊗ a− [a, b]g,

for all a, b ∈ g , the right-hand side is contained in the left-hand side. To prove the opposite, we justneed to prove that the right-hand side is a hom-ideal of T (g) , i.e. it is closed under the left and rightmultiplication and the operator αT . Therefore, we should check these one by one. For any natural numberk ≥ 0 , we have

((g⊗n ⊙ (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ g⊗m)⊙ g⊗k

=αT (g⊗n ⊙ (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ (g⊗m ⊙ αT (g

⊗k))

=αT (g⊗n ⊙ (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ g⊗m+k

=(αT (g⊗n)⊙ αT (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ g⊗m+k

=(g⊗n ⊙ (αT (a)⊗ αT (b)− ε(a, b)αT (b)⊗ αT (a)− [αT (a), αT (b)]g))

⊙ g⊗m+k,

which is seen to be contained in∑x,y∈g

(g⊗n ⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ g⊗m+k.

Thus the right-hand side is closed under the right multiplication. We can also get

αT ((g⊗n ⊙ (a⊗ b− ε(a, b)b⊗ a− [a, b]g))⊙ g⊗m)

=(g⊗n ⊙ (αT (a)⊗ αT (b)− ε(a, b)αT (b)⊗ αT (a)− [αT (a), αT (b)]g))

⊙g⊗m,

326


which is contained in ∑x,y∈g

(g⊗n ⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ g⊗m.

So the right-hand side is a hom-ideal of (T (g),⊙, αT ) , which also contains the elements of the form

(x⊗ y − ε(x, y)y ⊗ x− [x, y]g)

for all x, y ∈ g . Therefore, it contains the left-hand side.

(ii) We first prove that θ(I) is contained in J . By the first part of the proposition, we just need to verifythat the element

θ((a⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ b

is contained in J , for x, y ∈ g and a, b in g⊗n, g⊗m , respectively, n,m ≥ 0 .We have

(a⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ b

=(αT (a)⊗ (x⊗ αT (y)− ε(x, y)y ⊗ αT (x))− a⊗ [x, y]g)⊙ b

=αm−1T (αT (a)⊗ (x⊗ αT (y)− ε(x, y)y ⊗ αT (x))⊗ b1

⊗αT (b2 ⊗ · · · ⊗ bm)

−αm−1T (a⊗ [x, y]g)⊗ b1 ⊗ αT (b2 ⊗ · · · ⊗ bm),

by the definition of ⊙ . Furthermore, according to Lemma 4.1 (iv), there exists c ∈ g⊗n such that

θ((a⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ b)

=θ(αm−1T (αT (a)⊗ (x⊗ αT (y)− ε(x, y)y ⊗ αT (x))))⊗c

−θ(αm−1T (a⊗ [x, y]g))⊗ αT (c)

=θ(αm−1T (a)⊗ (αm−1

T (x)⊗ (x⊗ αT (y)− ε(x, y)y ⊗ αT (x))))⊗ c

−θ(αm−1T (a)⊗ [αm−1

T (x), αm−1T (y)]g))⊗ αT (c)

=θ(αmT (a))⊗ (αn−1

T (αm−1T (x))⊗ αn

T (αmT (y))− ε(x, y)αn−1

T (αm−1T (y))

⊗ αnT (α

mT (x)))⊗ c− θ(αm−1

T (a)⊗ [αm−1T (x), αm−1

T (y)]g))⊗ αT (c)

=θ(αmT (a))⊗ (αn+m

T (x)⊗ αn+mT (y)− ε(x, y)αn+m

T (y)⊗ αn+mT (x))⊗ c

−αT (θ(αmT (a)))⊗ [αn+m

T (x), αn+mT (y)]g))⊗ αT (c).

This is an element in∑u∈g⊗n

v∈g⊗m

∑s,t∈g

u⊗ (s⊗ t− ε(s, t)t⊗ s)⊗ v− αT (u)⊗ [s, t]g ⊗ αT (v))

if we simply take u := θ(αmT (a)) , s := αm+n

T (x) and t := αm+nT (y) . Therefore, θ(I) is contained in J .

Conversely, since θ and αT are bijective, the above argument shows that any term

u⊗ (s⊗ t− ε(s, t)t⊗ s)⊗ v− αT (u)⊗ [s, t]g ⊗ αT (v)),

327


in the previous sum can be expressed in the form

θ((a⊙ (x⊗ y − ε(x, y)y ⊗ x− [x, y]g))⊙ b),

which shows the surjectivity of θ and completes the proof.

2

In the next theorem, we suppose that g is an involutive color hom-Lie algebra with a basis X = {xn|n ∈ ω}for a well-ordered set ω .

Theorem 4.3 Let g := (g, [, ]g, βg) be an involutive color hom-Lie algebra such that βg(X) = X . Let W bethe one defined in (4.1) and let µ ∈ k be given. If we define

Jµ :=∑

n,m≥0

∑a∈g⊗n

b∈g⊗m

∑a,b∈g

(a⊗ (a⊗ b− ε(a, b)b⊗ a)⊗ b− µn+mαT (a)⊗ [a, b]g ⊗ αT (b)), (4.4)

then we can have the linear decompositionT (g) = Jµ ⊕ kW.

Proof Let us first introduce some notations. For i ≥ 2 let

x := xn1⊗ xn2

⊗ · · · ⊗ xni∈ X⊗i ⊆ g⊗i.

Define the index of x to bed :=| {(r, s)|r < s, nr < ns, 1 ≤ r, s ≤ i} | .

Let gi,d be the linear span of all pure tensors x of degree i and index d . Then we have

g⊗i =⊕d⩾0

gi,d.

In particular, gi,0 = kW (i) , where

W (i) := {xn1⊗ xn2

⊗ · · · ⊗ xni∈ X⊗i|n1 ⩾ n2 ⩾ · · · ⩾ ni}.

In order to prove T (g) = Jµ ⊕ kW , we need to prove that

g⊗i ⊆ Jµ ⊕∑

1⩽q⩽i

kW (q),

using induction on i ≥ 1 . For i = 1 , we have kW (1) = g . So g ⊆ Jµ⊕kW (1) . Suppose that the above equationis true for n ≥ 1 . Since g⊗(i+1) =

∑d≥0 gi+1,d , we just need to prove that

gi+1,d ⊆ Jµ ⊕∑

1⩽q⩽i+1

kW (q),

for all d ≥ 0 . We use induction on d . For d = 0 , we have gi+1,0 = kW (i+1) . Suppose for l ≥ 0 we have

gi+1,l ⊆ Jµ ⊕∑

1⩽q⩽i+1

kW (q).

328


Let x = xn1 ⊗ xn2 ⊗ · · · ⊗ xni ∈ X⊗(i+1) ∩ qi+1,l+1 . Since l + 1 ≥ 1 , we can choose an integer 1 ≤ r ≤ i suchthat nr ≤ nr+1 . Let

x′ = xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · ⊗ xni+1

be the pure tensor formed by interchanging xnrby xnr+1

in x . Then

x′ ∈ gi+1,l ⊆ Jµ ⊕∑

1⩽q⩽i+1

kW (q).

Since the definition of Jµ gives

x− x′ ≡ µi−1αT (xn1 ⊗ · · · ⊗ xni)⊗ [xnr , xnr+1 ]g ⊗ αT (xnr+2 ⊗ · · · ⊗ xni+1)(mod Jµ),

by the induction hypothesis on i , we have

x ∈ Jµ ⊕∑

1⩽q⩽i+1

kW (q) ⊕∑

1⩽q⩽i

kW (q).

So x is in Jµ ⊕∑

1⩽q⩽i+1 kW(q) . This proves that gi+1,l+1 ⊆ Jµ ⊕

∑1⩽q⩽i+1 kW

(q) . Hence, g⊗i+1 ⊆

Jµ ⊕∑

1⩽q⩽i+1 kW(q) which completes the induction steps on i .

Now, we want to show that Jµ ∩ kW = 0 . Let S be an operator on T (g) such that

(i) S(t) = t, for all t ∈W.

(ii) if p ≥ 2, 1 ≤ s ≤ p− 1, and ns < ns+1, then (4.5)

S(xn1 ⊗ · · · ⊗ xns ⊗ xns+1 ⊗ · · · ⊗ xnp)

= S(xn1⊗ · · · ⊗ xns+1

⊗ xns⊗ · · · ⊗ xnp

)

+S(µp−2αT (xn1⊗ · · · ⊗ xns−1

)⊗ [xns, xns+1

]g ⊗ αT (xns+2⊗ · · · ⊗ xnp

)).

We define S on∑

1⩽q⩽i g⊗q by induction on i . For i = 1 , we define S := Idg . Let n ≥ 2 and let S be an

operator on∑

1⩽q⩽i g⊗q satisfying (4.5) for all tensors of degree i . Note that g⊗i+1 =

∑d⩽0 gi+1,d . For a pure

tensorx = xn1 ⊗ xn2 ⊗ · · · ⊗ xni+1 ∈ X⊗i+1 ⊆ g⊗i+1,

we use induction again on d which is the index of x , in order to extend S to an operator on∑

1⩽q⩽i+1 g⊗q .

For d = 0 , define S(x) = x . For l ≥ 0 , suppose that S(x) has been defined for

x ∈∑

1⩽p⩽l

gi+1,p.

Let x ∈ gi+1,l+1 . Let 1 ⩽ r ⩽ i be an integer such that nr < nr+1 . Then

S(x) := S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · ⊗ xni+1

)

+ S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g ⊗ αT (xnr+2⊗ · · · ⊗ xni+1

)).

329


We should show that S is well-defined and it is independent of the choice of r . Therefore, let r′ be anotherinteger, 1 ≤ r′ ≤ i , such that nr′ < nr′+1 . Consider

u :=S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · ⊗ xni+1

)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g ⊗ αT (xnr+2⊗ · · · ⊗ xni+1

)),

and

v :=S(xn1 ⊗ · · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1 ⊗ · · · ⊗ xnr′−1)⊗ [xnr′ , xnr′+1

]g ⊗ αT (xnr′+2⊗ · · · ⊗ xni+1)).

We check that u = v . There appear two cases:Case 1: If |r − r′| ≥ 2 , without losing the generality, we assume r − r′ ≥ 2 . Since u, v ∈

∑0⩽p⩽l gi+1,p +∑

1⩽q⩽i g⊗q , we have

u =S(xn1 ⊗ · · · ⊗ xnr+1 ⊗ xnr ⊗ · · · ⊗ xnr′ ⊗ xnr′+1⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1 ⊗ · · · ⊗ xnr−1)⊗ [xnr , xnr+1 ]g ⊗ αT (xnr+2 ⊗ · · · ⊗ xni+1))

=S(xn1 ⊗ · · · ⊗ xnr+1 ⊗ xnr ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1 ⊗ · · · ⊗ xnr−1 ⊗ xnr ⊗ · · · )⊗ [xnr′ , xnr′+1]g

⊗ αT (· · · ⊗ xnr+1))

+S(µi−1αT (xn1 ⊗ · · · )⊗ [xnr , xnr+1 ]g

⊗ αT (· · · ⊗ xnr′ ⊗ xnr′+1⊗ · · · ⊗ xni+1)),

v =S(xn1 ⊗ · · · ⊗ xnr ⊗ xnr+1 ⊗ · · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1 ⊗ · · · ⊗ xnr ⊗ xnr−1 ⊗ · · · )⊗ [xnr′ , xnr′+1]g

⊗ αT (xnr+2 ⊗ · · · ⊗ xni+1))

=S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · ⊗ xnr′+1

⊗ xnr′ ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr

, xnr+1]g

⊗ αT (· · · ⊗ xnr′+1⊗ xnr′+1

⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

⊗ xnr+1⊗ · · · )⊗ [xnr′ , xnr′+1

]g

⊗ αT (· · · ⊗ xni+1)),

by the induction hypothesis. Now, since βg(X) = X and

xnr̸= xnr+1

, xnr′ ̸= xnr′+1,

we get αT (xnr), αT (xnr+1

), αT (xnr′ ), αT (xnr′+1) ∈ X and

αT (xnr ) ̸= αT (xnr+1), αT (xnr′ ) ̸= αT (xnr′+1).

330


This leads to four different cases among which we consider only the case of αT (xnr ) > αT (xnr+1) andαT (xnr′ ) < αT (xnr′+1

) without loosing the generality. We obtain

S(µi−1αT (xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · )⊗ [xnr′ , xnr′+1

]g

⊗ αT (· · · ⊗ xni+1))

= S(µi−1αT (xn1⊗ · · · ⊗ xnr

⊗ xnr+1⊗ · · · )⊗ [xnr′ , xnr′+1

]g

⊗ αT (· · · ⊗ xni+1))

+ S(µ2i−3(xn1⊗ · · · ⊗ [αT (xnr+1

), αT (xnr)]g

⊗ · · · ⊗ [αT (xnr′ ), αT (xnr′+1)]g ⊗ · · · ⊗ xni+1

)),

and

S(µi−1αT (xn1 ⊗ · · · )⊗ [xnr , xnr+1 ]g

⊗ αT (· · · ⊗ xnr′ ⊗ xnr′+1⊗ · · · ⊗ xni+1))

=S(µi−1αT (xn1 ⊗ · · · )⊗ [xnr , xnr+1 ]g

⊗ αT (· · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1))

+S(µ2i−3(xn1 ⊗ · · · ⊗ [αT (xnr ), αT (xnr+1)]g

⊗ · · · ⊗ [αT (xnr′ ), αT (xnr′+1)]g ⊗ · · · ⊗ xni+1)).

If we combine the two former expressions for u , we get

u =S(xn1 ⊗ · · · ⊗ xnr+1 ⊗ xnr ⊗ · · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1 ⊗ · · · ⊗ xnr ⊗ xnr+1 ⊗ · · · )⊗ [xnr′ , xnr′+1]g

⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1 ⊗ · · · )⊗ [xnr , xnr+1 ]g

⊗ αT (· · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1))

+S(µ2i−3(xn1 ⊗ · · · ⊗ [αT (xnr+1), αT (xnr )]g ⊗ · · ·

⊗ [αT (xnr′ ), αT (xnr′+1)]g ⊗ · · · ⊗ xni+1))

+S(µ2i−3(xn1 ⊗ · · · ⊗ [αT (xnr ), αT (xnr+1)]g ⊗ · · ·

⊗ [αT (xnr′ ), αT (xnr′+1)]g ⊗ · · · ⊗ xni+1))

=S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ · · · ⊗ xnr′+1

⊗ xnr′ ⊗ · · ·xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

⊗ xnr+1⊗ · · · )⊗ [xnr′ , xnr′+1

]g

⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr

, xnr+1]g

⊗ αT (· · · ⊗ xnr′+1⊗ xnr′ ⊗ · · · ⊗ xni+1

)) = v,

331


by the skew-symmetry condition of the bracket.Case 2: If |r − r′| = 1 , once again, without loosing the generality, let us suppose r′ = r + 1 . This leads tonr < ni+1 < ni+2 . We obtain

u =S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr⊗ xnr+2

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g

⊗ αT (xnr+2⊗ · · · ⊗ xni+1

))

=S(xn1⊗ · · · ⊗ xnr+1

⊗ xnr+2⊗ xnr

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+1

)⊗ [xnr, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g

⊗ αT (xnr+2⊗ · · · ⊗ xni+1

))

=S(xn1⊗ · · · ⊗ xnr+2

⊗ xnr+1⊗ xnr

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr+1

, xnr+2]g ⊗ αT (xnr

⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+1

)⊗ [xnr, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g

⊗ αT (xnr+2⊗ · · · ⊗ xni+1

)),

and

v =S(xn1⊗ · · · ⊗ xnr

⊗ xnr+2⊗ xnr+1

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

=S(xn1⊗ · · · ⊗ xnr+2

⊗ xnr⊗ xnr+1

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr

, xnr+2]g ⊗ αT (xnr+1

⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

=S(xn1 ⊗ · · · ⊗ xnr+2 ⊗ xnr+1 ⊗ xnr ⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+2

)⊗ [xnr, xnr+1

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr


⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1)).

Moreover, for any a < b , t1 ∈ g⊗m , t2 ∈ g⊗k , m+ k = i− 2 , we have

S(t1 ⊗ a⊗ b⊗ t2)− S(t1 ⊗ b⊗ a⊗ t2) = S(µm+kαT (t1)⊗ [a, b]⊗ αT (t2)).

332


So the sum of the last three terms of the previous expression of u is

S(µi−1αT (xn1⊗ · · · )⊗ [xnr+1

, xnr+2]g ⊗ αT (xnr

⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+1

)⊗ [xnr, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ [xnr, xnr+1

]g ⊗ αT (xni+2⊗ · · · ⊗ xni+1

))

=S(µi−1αT (xn1⊗ · · · )⊗ αT (xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µ2i−3(xn1⊗ · · · ⊗ [[xnr+1

, xnr+2]g, αT (xnr

)]g ⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr


)⊗ αT (· · · ⊗ xni+1))

+S(µ2i−3xn1⊗ · · · ⊗ xnr−1

⊗ [αT (xnr+1), [xnr

, xnr+2]g]g ⊗ · · · ⊗ xni+1

)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr−1

)⊗ αT (xnr+2)⊗ [xnr

, xnr+1]g

⊗ αT (· · · ⊗ xni+1))

+S(µ2i−3xn1⊗ · · · ⊗ xnr−1

⊗ [[xnr, xnr+1

]g, αT (xnr+2)]g ⊗ · · · ⊗ xni+1

)

=S(µi−1αT (xn1⊗ · · · ⊗ xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr

, xnr+2]g ⊗ αT (xni+1

⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+2

)⊗ [xnr, xnr+1

]g ⊗ αT (· · · ⊗ xni+1))

using the hom-Jacobi identity. Thus, we can obtain

u =S(xn1⊗ · · · ⊗ xnr+2

⊗ xnr+1⊗ xnr

⊗ · · · ⊗ xni+1)

+S(µi−1αT (xn1⊗ · · · ⊗ xnr

)⊗ [xnr+1, xnr+2

]g ⊗ αT (· · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · )⊗ [xnr

, xnr+2]g ⊗ αT (xni+1

⊗ · · · ⊗ xni+1))

+S(µi−1αT (xn1⊗ · · · ⊗ xnr+2

)⊗ [xnr, xnr+1

]g ⊗ αT (· · · ⊗ xni+1))

=v.

Now that u = v in either cases, let x ∈ Jg,β ∩ kW . Then S(x) = x and S(x) = 0 . Therefore x = 0 and we getthat Jg,β ∩ kW = 0 which completes the proof. 2

We are now ready to prove the Poincaré–Birkhoff–Witt theorem for involutive color hom-Lie algebras inthe second part of the next theorem.

Theorem 4.4 Let k be a field whose characteristic is not 2. Let g := (g, [, ]g, βg) be an involutive color hom-Liealgebra on k . Let θ : T (g) → T (g) be as described in (4.2). Let I be the hom-ideal of T (g) generated by thecommutators defined in Theorem 3.4. Let J be as defined in (4.4). Then there is a well-ordered basis X of g

such that forW =WX = {xi1 ⊗ · · · ⊗ xin |i1 ≥ · · · ≥ in, n ≥ 0},

the following statements hold.

(i) T (g) = J ⊕ kH,

333


(ii) θ(W ) is a basis of U(g) .

Proof

(i) Let g+ and g− be the eigenspaces of 1 and –1 of g , respectively. Then we have the decomposition

g = g+ ⊕ g−.

Let B+ and B− be some basis for g+ and g− , respectively. There are two different cases:If the cardinality of B+ is more than the cardinality of B− , fix an injection ι : B− → B+ . Then thefollowing set is a basis of g

X := {ι(x) + x, ι(x)− x|x ∈ B−} ∪ (B+ \B−),

and βg(X) = X . Let W be defined with a well order on X and choose µ = 1 in (4.4). Then we have

T (g) = Jg,β,1 ⊕ kW.

If the cardinality of B+ is not more than the cardinality of B+ , it suffices to assume γ := −βg and takeµ = −1 as it is in the last case.

(ii) follows directly from Theorem.4.3 and (i) .

2

5. Hom-Lie superalgebras caseThe study of hom-Lie superalgebras has been widely in the center of interest these last years. The motivationcame from the generalization of Lie superalgebras, or in some cases, the generalization of hom-Lie algebras.However, here, we deal with them as a special case of color hom-Lie algebras. One can simply put Γ =

Z2 in Definition 2.6 and define ε in such a way that ε(x, y) = (−1)|x||y| to get the following definition[1, 7, 13, 14, 39, 68].

Definition 5.1 A hom-Lie superalgebra is a triple (g, [, ], α) consisting of a superspace g , a bilinear map[, ] : g× g → g and a superspace homomorphism α : g → g , both of which are degree zero satisfying

1. [x, y] = −(−1)|x||y|[y, x],

2. (−1)|x||z|[α(x), [y, z]] + (−1)|y||x|[α(y), [z, x]] + (−1)|z||y|[α(z), [x, y]] = 0,

for all homogeneous elements x, y, z ∈ g .

In particular, one can easily recall from the first section, the notions of a multiplicative hom-Lie superalgebra,a morphism of hom-Lie superalgebras, and a hom-associative superalgebra. Moreover, a hom-associativesuperalgebra or a hom-Lie superalgebra is said to be involutive if α2 = id .

Again, as in the previous cases, a hom-associative superalgebra (V, µ, α) gives a hom-Lie superalgebraby antisymmetrization. We denote this hom-Lie superalgebra again by (A, [, ]A, βA) , where βA = α , [x, y]A =

xy − yx , for all x, y ∈ A .

334


Let (V, αV ) be an involutive hom-module. A free involutive hom-associative color algebra on V is an in-volutive hom-associative super algebra (F (V ), ∗, αF ) together with a morphism of hom-modules jV : (V, αV ) →(F (V ), αF ) with the property that, for any involutive hom-associative superalgebra (A, ., αA) together with amorphism f : (V, αV ) → (A,αA) of hom-modules, there is a unique morphism f : (F (V ), ∗, αF ) → (A, ., αA)

of hom-associative superalgebras such that f = f ◦ jV .The definition of the universal enveloping algebra, as can be predicted, is just a modification of the

Definition 3.2.

Definition 5.2 Let (g, [, ], α) be a hom-Lie superalgebra. A universal enveloping hom-associative superalgebraof g is a hom-associative superalgebra

U(g) := (U(g), µU , αU ),

together with a morphism φg : g → U(g) of hom-Lie superalgebras such that for any hom-associative superalgebra(A,µ, αA) and any hom-Lie superalgebra morphism ϕ : (g, [, ]g, βg) , there exists a unique morphism ϕ̄ : U(g → A)

of hom-associative superalgebras such that ϕ̄ ◦ φg = ϕ .

The following lemma shows an easy way to construct the universal algebra when we have an involutivehom-Lie superalgebra.

Lemma 5.3 Let (g, [, ]g, βg) be an involutive hom-Lie superalgebra.

(i) Let (A, ·, αA) be a hom-associative algebra, f : (g, [, ]g, βg) → (A, [, ]A, βA) be a morphism of hom-Liesuperalgebras and B be the hom-associative subsuperalgebra of A generated by f(g) . Then B is involutive.

(ii) The universal enveloping hom-associative algebra (U(g), φg) of (g, [, ]g, βg) is involutive.

(iii) In order to verify the universal property of (U(g), φg) in Definition 5.2, we need to consider only theinvolutive hom-associative algebras A := (A, ·A, αA) .

We can now give the construction of the universal enveloping hom-associative superalgebra of an involutivehom-Lie superalgebra.

Theorem 5.4 Let g := (g, [, ]g, βg) be an involutive hom-Lie superalgebra. Let

T (g) := (T (g),⊙, αT )

be the free hom-associative algebra on the hom-module underlying g . Let I be the hom-ideal of T (g) generatedby the set

{a⊗ b− (−1)|a||b|b⊗ a− [a, b]} (5.1)

and let

U(g) =T (g)

I

be the quotient hom-associative algebra. Let ψ be the composition of the natural inclusion i : g → T (g) with thequotient map π : T (g) → U(g) . Then (U(g), ψ) is a universal enveloping hom-associative algebra of g . Also,the universal enveloping hom-associative algebra of g is unique up to isomorphism.

335


We can also have a Poincaré–Birkhoff–Witt-like theorem for involutive hom-Lie superalgebras as a specialcase of color hom-Lie algebras, i.e. if g is a Lie superalgebra with an ordered basis X = {xn|n ∈ H} where His a well- and totally ordered set, let I be the ideal of the free associative algebra T (g) on g , which was givenin Theorem 5.4, so that U(g) is the universal enveloping algebra of g . We also suppose that g is an involutivehom-Lie superalgebra with a basis X = {xn|n ∈ ω} for a well-ordered set ω . The next theorem, which isa combination of theorems in the previous section, will help us give the Poincaré–Birkhoff–Witt theorem forinvolutive hom-Lie superalgebras.

Theorem 5.5 Let g := (g, [, ]g, βg) be an involutive hom-Lie superalgebra such that βg(X) = X . Letθ : T (g) → T (g) be as defined in (4.2), I be the hom-ideal of T (g) as defined in Theorem 5.4, let W belike the one defined in (4.1) and let µ ∈ k be given. Moreover, let

Jµ :=∑

n,m≥0

∑a∈g⊗n

b∈g⊗m

∑a,b∈g

(a⊗ (a⊗ b− (−1)|a||b|b⊗ a)⊗ b

− µn+mαT (a)⊗ [a, b]g ⊗ αT (b)), (5.2)

Then

(i) I =∑

n,m≥0

∑a,b∈g(g

⊗n ⊙ (a⊗ b− (−1)|a||b|b⊗ a− [a, b]g))⊙ g⊗m.

(ii) θ(I) = J

(iii) We can have the linear decomposition T (g) = Jµ ⊕ kW .

Finally, we give the Poincaré–Birkhoff–Witt theorem for involutive hom-Lie superalgebras.

Theorem 5.6 Let k be a field whose characteristic is not 2. Let g := (g, [, ]g, βg) be an involutive hom-Liesuperalgebra on k . Let θ : T (g) → T (g) be as described in (4.2). Let I be the hom-ideal of T (g) generated bythe commutators defined in Theorem 5.4. Let J be as defined in (5.2). Then there is a well-ordered basis X ofg such that for

W =WX = {xi1 ⊗ · · · ⊗ xin |i1 ≥ · · · ≥ in, n ≥ 0}

the following statements hold.

(i) T (g) = J ⊕ kH,

(ii) θ(W ) is a basis of U(g) .

Acknowledgments

This paper is supported by grant no. 92grd1m82582 of Shiraz University, Shiraz, Iran. The first authorArmakan is grateful to the Mathematics and Applied Mathematics research environment MAM, Division ofApplied Mathematics, School of Education, Culture and Communication at Mälardalen University, Västerås,Sweden for creating an excellent research environment during his visit from September 2016 to March 2017.

336


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