# Equivalent Description of Hom-Lie Algebroids

Authors: Zhen Xiong
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

We study representations of Hom-Lie algebroids, give some properties of Hom-Lie algebroids, and discuss equivalent statements of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other under some conditions.


---

## Body

## 1. Introduction

The notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [1] as a part of a study of deformations of the Witt and the Virasoro algebras. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. Some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [1, 2]. Because of close relations to discrete and deformed vector fields and differential calculus [1, 3, 4], more people pay special attention to this algebraic structure. For a party of k-cochains on Hom-Lie algebras, called k-Hom-cochains, there is a series of coboundary operators [5]; for regular Hom-Lie algebras, [6] gives a new coboundary operator on k-cochains, and there are many works have been done by the special coboundary operator [6, 7]. In [8], there is a series of coboundary operators, and the author generalizes the result “If k is a Lie algebra, ρ:k→gl(V) is a representation if and only if there is a degree-1 operator D on Λk∗⊗V satisfying D2=0, and (1)Dξ∧η⊗u=dkξ∧η⊗u+-1kξ∧Dη⊗u,∀ξ∈∧kk∗,η∈∧lk∗,u∈V,where dk:∧kg∗→∧k+1g∗ is the coboundary operator associated with the trivial representation.”Geometric generalizations of Hom-Lie algebras are given in [7, 9]. In [9], C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in [10], based on Hom-Lie algebroids from [9], the authors study representation of Hom-Lie algebroids. In [7], the authors make small modifications to the definition of Hom-Lie algebroids and give a new definition of Hom-Lie algebroids; based on the new definition of Hom-Lie algebroids, definitions of Hom-Lie bialgebroids and Hom-Courant algebroids are given.In this article, we first study representations of Hom-Lie algebroids, give equivalent statements of Hom-Lie algebroids, and prove that different definitions of Hom-Lie algebroids are given by the same Hom-Lie algebras and their representations.The paper is organized as follows. In Section2, we recall some basic notions. In Section 3, first, we study representations of Hom-Lie algebroids and give some properties of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other (Theorem 7, Theorem 8).


## 2. Preliminaries

### 2.1. Hom-Lie Algebras and Their Representations

The notion of a Hom-Lie algebra was introduced in [1]; see also [11, 12] for more information.Definition 1.
( 1 ) A Hom-Lie algebra is a triple (g,[·,·],α) consisting of a vector space g, a skew-symmetric bilinear map (bracket) [·,·]:∧2g→g and a linear transformation α:g→g satisfying α[x,y]=[α(x),α(y)], and the following Hom-Jacobi identity:(2)αx,y,z+αy,z,x+αz,x,y=0,∀x,y,z∈g.
A Hom-Lie algebra is called a regular Hom-Lie algebra ifα is a linear automorphism.
( 2 ) A subspace h⊂g is a Hom-Lie subalgebra of (g,[·,·],α) if α(h)⊂h and h is closed under the bracket operation [·,·], i.e., for all x,y∈h, [x,y]∈h.
( 3 ) A morphism from the Hom-Lie algebra (g,[·,·]g,α) to the Hom-Lie algebra (h,[·,·]h,γ) is a linear map ψ:g→h such that ψ([x,y]g)=[ψ(x),ψ(y)]h and ψ∘α=γ∘ψ.Representation and cohomology theories of Hom-Lie algebra are systematically introduced in [13, 14]. See [15] for homology theories of Hom-Lie algebras.Definition 2.
A representation of the Hom-Lie algebra(g,[·,·],α) on a vector space V with respect to β∈gl(V) is a linear map ρ:g→gl(V), such that, for all x,y∈g, the following equalities are satisfied: (3)ραx∘β=β∘ρx;ρx,y∘β=ραx∘ρy-ραy∘ρx.Let(g,[·,·],α) be a Hom-Lie algebra, V be a vector space, and ρ:g→gl(V) be a representation of (g,[·,·],α) on the vector space V with respect to β∈gl(V).The set ofk-cochains on g with values in V, which we denote by Ck(g;V), is the set of skew-symmetric k-linear maps from g×⋯×g (k-times) to V: (4)Ckg;V≔η:∧kg→Vis  a  linear  map.In [8], when β∈GL(V), there is a series of operators ds:Ck(g;V)→Ck+1(g;V) which is given by (5)dsηx1,…,xk+1=∑i=1k+1-1i+1βk+1+sρxiβ-k-2-sηαx1,…,x^i,…,αxk+1+∑i<j-1i+jηxi,xj,αx1,…,xi,j^,…,αxk+1,where β-1 is the inverse of β, η∈Ck(g;V), and the authors have the results: ds∘ds=0.


### 2.2. Hom-Lie Algebroids

Now, we introduce two kinds of definitions of Hom-Lie algebroids; they are from [7, 9]. For more about Hom-Lie algebroids, please see [7, 9].Definition 3 (see [9]).
A Hom-Lie algebroid is a quintuple(A,φ,[·,·],ρA,αA), where A is a vector bundle over a manifold M,φ:M→M is a smooth map, [·,·]:Γ(A)⊗Γ(A)→Γ(A) is a bilinear map, called bracket, ρA:φ!A→φ!TM is a vector bundle morphism, called anchor, and αA:Γ(A)→Γ(A) is a linear endomorphism of Γ(A), for X,Y∈Γ(A),f∈C∞(M) such that (1)
αA(fX)=φ∗(f)αA(X);(2)
the triple(Γ(A),[·,·],αA) is a Hom-Lie algebra;(3)
the following Hom-Leibniz identity holds:(6)X,fY=φ∗fX,Y+ρAXfαAY;(4)
ρA is a representation of Hom-Lie algebra (Γ(A),[·,·],αA) on C∞(M) with respect to φ∗.In fact, according to Definition3, for X,Y∈Γ(A),f,g∈C∞(M), we have the following properties:(a)
αA([X,Y])=[αA(X),αA(Y)];(b)
φ∗:C∞(M)→C∞(M), defined by φ∗(f)=f∘φ,φ∗ is a morphism of C∞(M);(c)
ρA(X)(fg)=φ∗(f)ρA(X)(g)+φ∗(g)ρA(X)(f). According to (3) in Definition, (7)ρAXfg=X,fg=X,fαAg+X,gαAf=αAgρAXf+αAfρAXgwhere·,· is defined in Definition 3.1 of [9];(d)
αA(f)=φ∗(f), when α=id, then φ∗=id, and Hom-Lie algebroid (A,φ,[·,·],ρA,αA) is just a Lie algebroid;(e)
ρA(fX)=φ∗(f)ρA(X). It follows from (8)ρAfXg=fX,g=αAfX,g=φ∗fρAXg.Definition 4 (see [7]).
A Hom-Lie algebroid is a quintuple(B,φ,[·,·],ρB,αB), where B is a vector bundle over a manifold M,φ:M→M is a smooth map, [·,·]:Γ(B)⊗Γ(B)→Γ(B) is a bilinear map, called bracket, ρB:B→φ!TM is a bundle map, called anchor, and αB:Γ(B)→Γ(B) is a linear endomorphism of Γ(B), for X,Y∈Γ(B),f∈C∞(M) such that (1)
αB(fX)=φ∗(f)αB(X);(2)
the triple(Γ(B),[·,·],αB) is a Hom-Lie algebra;(3)
the following Hom-Leibniz identity holds:(9)X,fY=φ∗fX,Y+ρBαBXfαBY;(4)
ρB is a representation of Hom-Lie algebra (Γ(B),[·,·],αB) on C∞(M) with respect to φ∗.From Definition4, for X,Y∈Γ(A),f,g∈C∞(M), we have(a)
αB([X,Y])=[αB(X),αB(Y)];(b)
φ∗:C∞(M)→C∞(M), defined by φ∗(f)=f∘φ,φ∗ is a morphism of C∞(M);(c)
ρB(X)(fg)=φ∗(f)ρB(X)(g)+φ∗(g)ρB(X)(f);(d)
αB(f)=φ∗(f);(e)
ρB(fX)=fρB(X).WhenαB and φ are invertible, Hom-Lie bialgebroids and Hom-Courant algebroids are given in [7].


## 2.1. Hom-Lie Algebras and Their Representations

The notion of a Hom-Lie algebra was introduced in [1]; see also [11, 12] for more information.Definition 1.
( 1 ) A Hom-Lie algebra is a triple (g,[·,·],α) consisting of a vector space g, a skew-symmetric bilinear map (bracket) [·,·]:∧2g→g and a linear transformation α:g→g satisfying α[x,y]=[α(x),α(y)], and the following Hom-Jacobi identity:(2)αx,y,z+αy,z,x+αz,x,y=0,∀x,y,z∈g.
A Hom-Lie algebra is called a regular Hom-Lie algebra ifα is a linear automorphism.
( 2 ) A subspace h⊂g is a Hom-Lie subalgebra of (g,[·,·],α) if α(h)⊂h and h is closed under the bracket operation [·,·], i.e., for all x,y∈h, [x,y]∈h.
( 3 ) A morphism from the Hom-Lie algebra (g,[·,·]g,α) to the Hom-Lie algebra (h,[·,·]h,γ) is a linear map ψ:g→h such that ψ([x,y]g)=[ψ(x),ψ(y)]h and ψ∘α=γ∘ψ.Representation and cohomology theories of Hom-Lie algebra are systematically introduced in [13, 14]. See [15] for homology theories of Hom-Lie algebras.Definition 2.
A representation of the Hom-Lie algebra(g,[·,·],α) on a vector space V with respect to β∈gl(V) is a linear map ρ:g→gl(V), such that, for all x,y∈g, the following equalities are satisfied: (3)ραx∘β=β∘ρx;ρx,y∘β=ραx∘ρy-ραy∘ρx.Let(g,[·,·],α) be a Hom-Lie algebra, V be a vector space, and ρ:g→gl(V) be a representation of (g,[·,·],α) on the vector space V with respect to β∈gl(V).The set ofk-cochains on g with values in V, which we denote by Ck(g;V), is the set of skew-symmetric k-linear maps from g×⋯×g (k-times) to V: (4)Ckg;V≔η:∧kg→Vis  a  linear  map.In [8], when β∈GL(V), there is a series of operators ds:Ck(g;V)→Ck+1(g;V) which is given by (5)dsηx1,…,xk+1=∑i=1k+1-1i+1βk+1+sρxiβ-k-2-sηαx1,…,x^i,…,αxk+1+∑i<j-1i+jηxi,xj,αx1,…,xi,j^,…,αxk+1,where β-1 is the inverse of β, η∈Ck(g;V), and the authors have the results: ds∘ds=0.


## 2.2. Hom-Lie Algebroids

Now, we introduce two kinds of definitions of Hom-Lie algebroids; they are from [7, 9]. For more about Hom-Lie algebroids, please see [7, 9].Definition 3 (see [9]).
A Hom-Lie algebroid is a quintuple(A,φ,[·,·],ρA,αA), where A is a vector bundle over a manifold M,φ:M→M is a smooth map, [·,·]:Γ(A)⊗Γ(A)→Γ(A) is a bilinear map, called bracket, ρA:φ!A→φ!TM is a vector bundle morphism, called anchor, and αA:Γ(A)→Γ(A) is a linear endomorphism of Γ(A), for X,Y∈Γ(A),f∈C∞(M) such that (1)
αA(fX)=φ∗(f)αA(X);(2)
the triple(Γ(A),[·,·],αA) is a Hom-Lie algebra;(3)
the following Hom-Leibniz identity holds:(6)X,fY=φ∗fX,Y+ρAXfαAY;(4)
ρA is a representation of Hom-Lie algebra (Γ(A),[·,·],αA) on C∞(M) with respect to φ∗.In fact, according to Definition3, for X,Y∈Γ(A),f,g∈C∞(M), we have the following properties:(a)
αA([X,Y])=[αA(X),αA(Y)];(b)
φ∗:C∞(M)→C∞(M), defined by φ∗(f)=f∘φ,φ∗ is a morphism of C∞(M);(c)
ρA(X)(fg)=φ∗(f)ρA(X)(g)+φ∗(g)ρA(X)(f). According to (3) in Definition, (7)ρAXfg=X,fg=X,fαAg+X,gαAf=αAgρAXf+αAfρAXgwhere·,· is defined in Definition 3.1 of [9];(d)
αA(f)=φ∗(f), when α=id, then φ∗=id, and Hom-Lie algebroid (A,φ,[·,·],ρA,αA) is just a Lie algebroid;(e)
ρA(fX)=φ∗(f)ρA(X). It follows from (8)ρAfXg=fX,g=αAfX,g=φ∗fρAXg.Definition 4 (see [7]).
A Hom-Lie algebroid is a quintuple(B,φ,[·,·],ρB,αB), where B is a vector bundle over a manifold M,φ:M→M is a smooth map, [·,·]:Γ(B)⊗Γ(B)→Γ(B) is a bilinear map, called bracket, ρB:B→φ!TM is a bundle map, called anchor, and αB:Γ(B)→Γ(B) is a linear endomorphism of Γ(B), for X,Y∈Γ(B),f∈C∞(M) such that (1)
αB(fX)=φ∗(f)αB(X);(2)
the triple(Γ(B),[·,·],αB) is a Hom-Lie algebra;(3)
the following Hom-Leibniz identity holds:(9)X,fY=φ∗fX,Y+ρBαBXfαBY;(4)
ρB is a representation of Hom-Lie algebra (Γ(B),[·,·],αB) on C∞(M) with respect to φ∗.From Definition4, for X,Y∈Γ(A),f,g∈C∞(M), we have(a)
αB([X,Y])=[αB(X),αB(Y)];(b)
φ∗:C∞(M)→C∞(M), defined by φ∗(f)=f∘φ,φ∗ is a morphism of C∞(M);(c)
ρB(X)(fg)=φ∗(f)ρB(X)(g)+φ∗(g)ρB(X)(f);(d)
αB(f)=φ∗(f);(e)
ρB(fX)=fρB(X).WhenαB and φ are invertible, Hom-Lie bialgebroids and Hom-Courant algebroids are given in [7].


## 3. Representations of Hom-Lie Algebroids

In this section, we assume that mapφ:M→M is involution; i.e., φ2=id.Let(E,φ,[·,·],ρE,αE) be a Hom-Lie algebroid, whence (ρE,φ∗) is a representation of (Γ(E),[·,·],αE) on C∞(M), where E is A or B. We define ds:Ck(Γ(E);C∞(M))→Ck+1(Γ(E);C∞(M)), s=0,1,…, by setting (10)dsηX1,…,Xk+1=∑i=1k+1-1i+1φ∗k+1+sρEXiφ∗-k-2-sηαEX1,…,X^i,…,αEXk+1+∑i<j-1i+jηXi,Xj,αEX1,…,X^i,…,X^j,…,αEXk+1,where Ck(Γ(E);C∞(M))={η:∧kΓ(E)→C∞(M)isaRlinear  map}, Xi∈Γ(E).We define mapαE∗:Ck(Γ(E);C∞(M))→Ck(Γ(E);C∞(M)) by (11)αE∗ηX1,…,Xk=φ∗∘ηαEX1,…,αEXk,Xi∈ΓE. When f∈C0(Γ(E);C∞(M))=C∞(M), we have αE∗(f)=φ∗(f).LetΓ(∧kE∗)={η∈Ck(Γ(E);C∞(M))|f∈C∞(M),η(fX1,…,Xk)=fη(X1,…,Xk)}. Then, Γ(∧kE∗) is a subset of Ck(Γ(E);C∞(M)). Let αE∗ acts on Γ(∧kE∗), we have (12)αE∗:Γ∧kE∗→Γ∧kE∗. Actually, when ξ∈Γ(E∗), for f∈C∞(M), we have (13)αE∗ξfX=φ∗∘ξαEfX=φ∗∘ξφ∗fαEX=φ∗2fφ∗∘ξαEX=fαE∗ξX, so αE∗(ξ)∈Γ(E∗), and we have (14)αE∗ξ1∧⋯ξk=αE∗ξ1∧⋯αE∗ξk,ξ1,…,ξk∈ΓE∗.Let Γ(∧•E∗)=⊕kΓ(∧kE∗) and C•(Γ(E);C∞(M))=⊕kCk(Γ(E);C∞(M)); we have that Γ(∧•E∗) is a subset of C•(Γ(E);C∞(M)). So, if ξ∈Γ(∧•E∗), we have dsξ∈C•(Γ(E);C∞(M)).At the same time,φ∗ can induce a map φ∗¯:Ck(Γ(E);C∞(M))→Ck(Γ(E);C∞(M)), which is defined by (15)φ∗¯ηX1,…,Xk=φ∗∘ηX1,…,Xk,Xi∈ΓE.Then, we have (16)φ∗¯η1∧⋯ηk=φ∗¯η1∧⋯φ∗¯ηk.Proposition 5.
With above notations, forη1∈Ck(Γ(E);C∞(M)), η2∈Cl(Γ(E);C∞(M)), we have (17)dsη1∧η2=ds+lη1∧φ∗¯∘αE∗η2+-1kφ∗¯∘αE∗η1∧ds+kη2.Proof.
First letk=1, η1∧η2∈Cl+1(Γ(E);C∞(M)); we have (18)dsη1∧η2X1,…,Xl+2=∑i=1-1i+1φ∗l+2+sρEXiφ∗-l-3-sη1∧η2αEx1,…,X^i,…,αExl+2+∑i<j-1i+jη1∧η2Xi,Xj,αEX1,…,Xi,j^,…,αEXl+2=∑i<q-1q+i+1η2αEX1,…,Xi,q^,…,αEXl+2φ∗l+2+sρEXiφ∗-l-3-sη1αEXq+∑q<i-1i+qη1αEXqφ∗l+2+sρEXiφ∗-l-3-sη2αEX1,…,Xq,i^,…,αEXl+2+∑i<j-1i+jη1(Xi,Xjη2αEX1,…,Xi,j^,…,αEXl+2+∑i<q-1q+i+1η1αExqφ∗l+2+sρEXiφ∗-l-3-sη2αEX1,…,Xi,q^,…,αEXl+2+∑q<i-1i+qη2αEX1,…,Xq,i^,…,αEXl+2φ∗l+2+sρEXiφ∗-l-3-sη1αEXq+∑q<i<j-1q+i+jη1αEXqη2Xi,Xj,αEX1,…,xq,i,j^,…,αEXl+2+∑i<q<j-1q+i+j+1η1αEXqη2Xi,Xj,αEX1,…,xi,q,j^,…,αEXl+2+∑i<j<q-1q+i+jη1αEXqη2Xi,Xj,αEX1,…,xi,j,q^,…,αEXl+2=ds+lη1∧φ∗¯∘αE∗η2X1,…,Xl+2+-1φ∗¯∘αE∗η1∧ds+1η2X1,…,Xl+2. So, when k=1, we have (19)dsη1∧η2=ds+lη1∧φ∗¯∘αE∗η2+-1φ∗¯∘αE∗η1∧ds+1η2. By induction on k, assume that when k=n, we have (20)dsη1∧η2=ds+lη1∧φ∗¯∘αE∗η2+-1nφ∗¯∘αE∗η1∧ds+nη2. For any η3∈C1(Γ(E);C∞(M)), then η1∧η3∈Cn+1(Γ(E);C∞(M)); we have (21)dsη1∧η3∧η2=dsη1∧η3∧η2=ds+l+1η1∧φ∗¯∘αE∗η3∧η2+-1nφ∗¯∘αE∗η1∧ds+nη3∧η2=ds+l+1η1∧φ∗¯∘αE∗η3∧φ∗¯∘αE∗η2+-1nφ∗¯∘αE∗η1∧ds+n+lη3∧φ∗¯∘αE∗η2+-1φ∗¯∘αE∗η3∧ds+n+1η2=ds+l+1η1∧φ∗¯∘αE∗η3+-1nφ∗¯∘αE∗η1∧ds+n+lη3∧φ∗¯∘αE∗η2+-1n+1φ∗¯∘αE∗η1∧φ∗¯∘αE∗η3∧ds+n+1η2=ds+lη1∧η3∧φ∗¯∘αE∗η2+-1n+1φ∗¯∘αE∗η3∧η1∧ds+n+1η2.The proof is completed.Proposition 6.
With above notations, we have(22)αE∗∘ds=ds∘αE∗;φ∗¯∘ds=ds+1∘φ∗¯.Proof.
With straightforward computations, for anyη∈Ck(Γ(E);C∞(M)), we have (23)αE∗∘dsηX1,…,Xk+1=φ∗∘dsηαEX1,…,αEXk+1=∑i=1-1i+1φ∗k+2+sρEαEXiφ∗-k-2-sηαE2X1,…,X^i,…,αE2Xk+1+∑i<j-1i+jφ∗∘ηαEXi,αEXj,αE2X1,…,Xi,j^,…,αE2Xk+1=∑i=1-1i+1φ∗k+3+sρEXiφ∗-k-4-sαE∗ηαEX1,…,X^i,…,αEXk+1+∑i<j-1i+jαE∗ηXi,Xj,αEX1,…,Xi,j^,…,αEXk+1=ds∘αE∗ηX1,…,Xk+1.At the same time, we have (24)φ∗¯∘dsηX1,…,Xk+1=∑i=1-1i+1φ∗k+2+sρEXiφ∗-k-2-sηαEX1,…,X^i,…,αEXk+1+∑i<j-1i+jφ∗∘ηXi,Xj,αEX1,…,Xi,j^,…,αEXk+1=∑i=1-1i+1φ∗k+2+sρEXiφ∗-k-3-sφ∗¯ηαEX1,…,X^i,…,αEXk+1+∑i<j-1i+jφ∗¯ηXi,Xj,αEX1,…,Xi,j^,…,αEXk+1=ds+1∘φ∗¯ηX1,…,Xk+1. We complete this proof.Now, we revisited representations of Hom-Lie algebroids, respectively, based on Definitions3 and 4.Theorem 7.
LetA be a vector bundle over manifold M,φ:M→M is a smooth map and φ2=id, and αA:Γ(A)→Γ(A) is a linear endomorphism of Γ(A), i.e., for f∈C∞(M),X∈Γ(A),αA(fX)=φ∗(f)αA(X). Then (A,φ,[·,·],ρA,αA) is a Hom-Lie algebriod defined by Definition 3 if and only if there is a series of operators ds:Ck(Γ(A);C∞(M))→Ck+1(Γ(A);C∞(M)),s=0,1,…, such that (i)
ds∘ds=0;(ii)
for anyη1∈Ck(Γ(A);C∞(M)),η2∈Cl(Γ(A);C∞(M)), we have (25)dsη1∧η2=ds+lη1∧φ∗¯∘αA∗η2+-1kφ∗¯∘αA∗η1∧ds+kη2;(iii)
αA∗∘ds=ds∘αA∗; φ∗¯∘ds=ds+1∘φ∗¯;(iv)
forf∈C∞(M)=C0(Γ(A);C∞(M)), we have: d0f∈Γ(A∗);(v)
forf∈C∞(M),ξ∈Γ(A∗), we have (26)d0ξfX1,X2=φ∗fd0ξX1,X2.Proof.
For necessity, with above propositions which we proved, we just need to prove (iv) and (v). For Hom-Lie algebriod(A,φ,[·,·],ρA,αA) and f,g∈C∞(M)=C0(Γ(E);C∞(M)),X∈Γ(A), by the definition of ds, we have (27)d0fgX=φ∗ρAgXf=gφ∗ρAXf=gd0fX.
Forf∈C∞(M),ξ∈Γ(A∗), we have (28)d0ξfX1,X2=ρAfX1φ∗ξαAX2-ρAX2φ∗ξαAfX1-ξfX1,X2=φ∗fρAX1φ∗ξαAX2-φ∗fρAX2φ∗ξαAX1-ξαAX1ρAX2f-ξφ∗fX1,X2-ρAX2fαAX1=φ∗fd0ξX1,X2.So, we proved the necessity of this theorem. Now, we prove the adequacy of this theorem.
Sept 1, we defineρA:φ!A→φ!TM by(29)φ∗ρAXf=d0fX,X∈ΓA,f∈C∞M.Then, by (iv), for g∈C∞(M), φ∗ρA(gX)f=d0f(gX)=gd0f(X)=gφ∗ρA(X)f, we have(30)ρAgX=φ∗gρAX.On the other hand, we have (31)d0fgX=φ∗ρAXfg. By (ii), for f,g∈C0(Γ(A);C∞(M))=C∞(M), we have (32)d0fg=d0fg+fd0g.Then, we have(33)ρAXfg=φ∗gρAXf+φ∗fρAXg.The definition of ρA is reasonable.
By (iii), we have(34)αA∗∘d0fX=φ∗∘d0fαAX=φ∗∘φ∗ρAαAXf=d0αA∗fX=φ∗∘ρAXφ∗f. We find the result:(35)ρAαAXφ∗=φ∗ρAX.Sept 2, for any ξ∈Γ(A∗),X,Y∈Γ(A), we define [·,·]:Γ(A)∧Γ(A)→Γ(A) by(36)ξX,Y=ρAXφ∗ξαAY-ρAYφ∗ξαAX-d0ξX,Y.So, by (36), we have (37)αA∗ξX,Y=ρAXφ∗αA∗ξαAY-ρAYφ∗αA∗ξαAX-d0αA∗ξX,Y=ρAXξαA2Y-ρAYξαA2X-d0αA∗ξX,Y. By αA∗(ξ)([X,Y])=φ∗ξ(αA[X,Y]), (35), (iii), and (36), we have (38)ξαAX,Y=φ∗ρAXξαA2Y-φ∗ρAYξαA2X-φ∗∘αA∗∘d0ξX,Y=ρAαAXφ∗ξαA2Y-ρAαAYφ∗ξαA2X-d0ξαAX,αAY=ξαAX,αAY.We have the following:(39)αAX,Y=αAX,αAY.For any f∈C∞(M), by (i), (29), and (36), we have (40)0=d0∘d0fX,Y=ρAXφ∗d0fαAY-ρAYφ∗d0fαAX-d0fX,Y=ρAXρAαAYf-ρAYρAαAXf-φ∗ρAX,Yf.We get (41)ρAXρAαAY-ρAYρAαAX=φ∗ρAX,Yφ∗ρAXρAαAYφ∗-φ∗ρAYρAαAXφ∗=ρAX,Yφ∗(42)ρAαAXρAY-ρAαAYρAX=ρAX,Yφ∗.Sept 3, by (36), (30), (33), and (v), we have (43)ξX,fY=ρAXφ∗ξαAfY-ρAfYφ∗ξαAX-d0ξX,fY=ρAXfφ∗ξαAY-φ∗fρAYφ∗ξαAX-φ∗fd0ξX,Y=φ∗fξX,Y+ξρAXfαAY. So, for f∈C∞(M),X,Y∈Γ(A), we have(44)X,fY=φ∗fX,Y+ρAXfαAY.Sept 4, by (iii), we have (45)d1=φ∗¯∘d0∘φ∗¯.For η1,η2∈C1(Γ(A);C∞(M)), by (ii), we have (46)d0η1∧η2=d1η1∧φ∗¯∘αA∗η2-φ∗¯∘αA∗η1∧d1η2=φ∗¯d0φ∗¯η1∧αA∗η2-αA∗η1∧d0φ∗¯η2.By (36), (47)d0η1∧η2X,Y,Z=φ∗ρAXη1αAYη2αAZ-φ∗ρAXη1αAZη2αAY+η1αAYφ∗ρAXη2αAZ-η1αAZφ∗ρAXη2αAY-φ∗ρAYη1αAXη2αAZ+φ∗ρAYη1αAZη2αAX-η1αAXφ∗ρAYη2αAZ+η1αAZφ∗ρAYη2αAX+φ∗ρAZη1αAXη2αAY-φ∗ρAZη1αAYη2αAX+η1αAXφ∗ρAZη2αAY-η1αAYφ∗ρAZη2αAX-η1X,Yη2αAZ+η1αAZη2X,Y+η1X,Zη2αAY-η1αAYη2X,Z-η1Y,Zη2αAX+η1αAXη2Y,Z=φ∗ρAXη1∧η2αAY,αAZ-φ∗ρAYη1∧η2αAX,αAZ+φ∗ρAZη1∧η2αAX,αAY-η1∧η2X,Y,αAZ+η1∧η2X,Z,αAY-η1∧η2Y,Z,αAX. Then, for any η∈C2(Γ(A);C∞(M)), we have (48)d0ηX,Y,Z=φ∗ρAXηαAY,αAZ-φ∗ρAYηαAX,αAZ+φ∗ρAZηαAX,αAY-ηX,Y,αAZ+ηX,Z,αAY-ηY,Z,αAX. Sept 5, for ξ∈Γ(A∗), by (36), (42), and (i), we have (49)0=d0∘d0ξX,Y,Z=φ∗ρAXd0ξαAY,αAZ-φ∗ρAYd0ξαAX,αAZ+φ∗ρAZd0ξαAX,αAY-d0ξX,Y,αAZ+d0ξX,Z,αAY-d0ξY,Z,αAX=ξX,Y,αAZ+Y,Z,αAX+Z,X,αAY. So, we have(50)X,Y,αAZ+Y,Z,αAX+Z,X,αAY=0.
By (39) and (50), (Γ(A),[·,·],αA) is a Hom-Lie algebra.
By (44), we have [X,fY]=φ∗(f)[X,Y]+ρA(X)fαA(Y).
By (35) and (42), ρA is a representation of Hom-Lie algebra (Γ(A),[·,·],αA) on C∞(M) with respect to φ∗.
The proof is completed.Theorem 8.
LetB be a vector bundle over manifold M,φ:M→M is a smooth map and φ2=id, and αB:Γ(B)→Γ(B) is a linear endomorphism of Γ(B), i.e., for f∈C∞(M), X∈Γ(B), αB(fX)=φ∗(f)αB(X). Then (B,φ,[·,·],ρB,αB) is a Hom-Lie algebriod define by Definition 4 if and only if there is a series of operators ds:Ck(Γ(B);C∞(M))→Ck+1(Γ(B);C∞(M)),s=0,1,…, such that(1)
ds∘ds=0;(2)
for anyη1∈Ck(Γ(B);C∞(M)),η2∈Cl(Γ(B);C∞(M)), we have (51)dsη1∧η2=ds+lη1∧φ∗¯∘αB∗η2+-1kφ∗¯∘αB∗η1∧ds+kη2;(3)
αB∗∘ds=ds∘αB∗; φ∗¯∘ds=ds+1∘φ∗¯;(4)
forf∈C∞(M)=C0(Γ(B);C∞(M)), we have d1f∈Γ(B∗);(5)
forf∈C∞(M),ξ∈Γ(B∗), we have (52)d1ξfX1,X2=φ∗fd1ξX1,X2.Proof.
For necessity, with above Propositions which we proved, we just need to prove(4) and (5).
For Hom-Lie algebriod(B,φ,[·,·],ρB,αB) and f,g∈C∞(M)=C0(Γ(B);C∞(M)),X∈Γ(B), by the definition of ds, we have (53)d1fgX=ρBgXφ∗f=gρBXφ∗f=gd1fX.
For Hom-Lie algebriod(B,φ,[·,·],ρB,αB) and f∈C∞(M),ξ∈Γ(B∗), we have (54)d1ξfX1,X2=φ∗ρBfX1ξαBX2-φ∗ρBX2φ∗ξαBfX1-ξfX1,X2=φ∗fφ∗ρBX1ξαBX2-φ∗fρBX2ξαBX1+φ∗ξαBX1ρBX2φ∗f-ξφ∗fX1,X2-ρBαBX2fαBX1=φ∗fd1ξX1,X2. So, we proved the necessity of this theorem. The sufficiency of this theorem is similar to Theorem 7.Remark 9.
(i) ForE is a vector bundle over M, we hope that we are able to get a pull-back diagram: (55)E⟶φ!E⟶E↓↓↓M⟶φM⟶φMSo, we assume that φ2=id.
(ii) For given Hom-Lie algebriodE, we have operators ds, then, by operators ds, we can get the other definitions of Hom-Lie algebriod.



---

*Source: 8417516-2018-09-19.xml*