Nonabelian omni-Lie algebroids

Abstract

In this paper, we study the structure of nonabelian omni-Lie algebroids. Firstly, taking Lie algebroid (E,[\cdot ,\cdot {]}_E, {\rho }_E) as the starting point, a nonabelian omni-Lie algebroid is defined on direct sum bundle \mathfrak{D} E⨁\mathfrak{J}E, where \mathfrak{D} E and \mathfrak{J}E are, respectively, the gauge Lie algebroid and the jet bundle of vector bundle E, and study its properties. Furthermore, it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid, and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.

Keywords

Nonabelian omni-Lie algebroid / omni-Lie algebroid / trivial deformation / matched pair of Leibniz algebroids

1 Introduction

In [14], Weinstein introduced the concept of omni-Lie algebra by defining the specific algebraic structure on \mathfrak{g}\mathfrak{l}(V)\oplus V of vector space V to study the linearization of standard Courant algebroid. Note that Dirac structures of the omni-Lie algebra \mathfrak{g}\mathfrak{l}(V)⨁ V characterize all Lie algebra structures on V, even if \mathfrak{g}\mathfrak{l}(V)⨁V is not a Lie algebra. In [4], Chen and Liu defined the omni-Lie algebroid structure on direct sum bundle \mathfrak{D}E⨁\mathfrak{J}E, where \mathfrak{D}E is the gauge Lie algebroid, \mathfrak{J}E is the jet bundle of a vector bundle E. If rank(E) \ge 2 (or a line bundle E), then there is a one-to-one correspondence between Lie algebroid structures (or local Lie algebra structures) on E and Dirac structures of omni-Lie algebroid. An omni-Lie algebroid is reduced to an omni-Lie algebra if the base manifold is a point. In [12], Loday introduced the concept of Leibniz algebra, which becomes a Lie algebra when its bracket is skew-symmetric. In [10], Lang, Sheng and Xu introduced the notion of a nonabelian omni-Lie algebra associated to a Lie algebra, which is a trivial deformation of omni-Lie algebra, as well as the double of a matched pair of Leibniz algebras \mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)} and \mathfrak{g}.

In this paper, with the previous results of omni-Lie algebroids and nonabelian omni-Lie algebras, we run a further study on the nonabelian condition of nonabelian omni-Lie algebroids. Our main research content is nonabelian omni-Lie algebroid structures on the direct sum bundle \mathfrak{D} E⨁\mathfrak{J}E, where \mathfrak{D}E and \mathfrak{J}E are its gauge Lie algebroid and its jet bundle on the vector bundle E, respectively. We introduce the concept of a nonabelian omni-Lie algebroid from a Lie algebroid, and study its propositions. The nonabelian omni-Lie algebroid is a trivial deformation of an omni-Lie algebroid, and form a matched pair of Leibniz algebroids. The paper is organized as follows. In Section 2, we review some basic concepts and its related knowledges. In Section 3, we introduce the concept of a nonabelian omni-Lie algebroid on the direct sum bundle \mathfrak{D}E⨁\mathfrak{J}E, and show its geometric propositions by reconstructing Dorfman bracket \{\cdot ,\cdot {\}}_{\pi } of a nonabelian omni-Lie algebroid. It is proved that the nonabelian omni-Lie algebroid is a trivial deformation of an omni-Lie algebroid. By reviewing the basic concept of a matched pair of Leibniz algebroids, we show that a nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids. In Section 4, we have a conclusion of the full paper.

2 Preliminary

In this section, firstly, we review some basic concepts.

Definition 2.1 [11]　The standard Courant algebroid on M is a quadruple (TM⨁T^{\ast }M,(\cdot ,\cdot {)}_{+},\{\cdot ,\cdot \} ,\rho ), where the anchor map \rho is a projection from TM⨁T^{\ast }M to TM, the nondegenerate symmetric pairing is defined by

(X+\alpha ,Y+\beta {)}_{+}= i_X\beta +i_Y\alpha ,\mathrm{\forall } X+\alpha ,Y+\beta \in {\mathfrak{X}}^1(M) \oplus {\mathrm{\Omega }}^1(M) ,

and \{\cdot ,\cdot \} is the Dorfman bracket, which is given by

\{X+ \alpha ,Y+\beta \}= [X,Y ]+L_X\beta -i_{Y}d \alpha .

Definition 2.2 [10, 14]　An omni-Lie algebra associated to a vector space V is a triple (\mathfrak{g}\mathfrak{l}(V)⨁V, (\cdot ,\cdot {)}_{+},\{\cdot ,\cdot \}), for any A,B\in \mathfrak{g}\mathfrak{l}( V),u ,v\in V, where

(i) The nondegenerate symmetric pairing (\cdot ,\cdot {)}_{+} is given by

(2.1)(A+u ,B+v{)}_{+}=Av+Bu;

(ii) The bilinear bracket operation \{\cdot , \cdot \} is given by

\{A+ u,B+v \}=[A,B ]+Av ;

(\cdot ,\cdot {)}_{+} and \{\cdot , \cdot \} are compatible in the sense that,

(\{ A+u,B +v\} ,C+w{)}_{+}+(B+v ,\{A +u,C+ w\}{)}_{+}= A(B+ v,C+w {)}_{+}.

Definition 2.3 [10]　A nonabelian omni-Lie algebra associated to the Lie algebra (\mathfrak{g},[\cdot ,\cdot {]}_{\mathfrak{g}}) is a triple (\mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)}\oplus \mathfrak{g} ,(\cdot , \cdot {)}_{+}, \{\cdot , \cdot {\}}_{\mathfrak{g}}), where (\cdot ,\cdot {)}_{+} is the symmetric V-valued pairing given by (2.1) and \{\cdot ,\cdot {\}}_{\mathfrak{g}} is the bilinear bracket operation given by

\{A+ u,B+v {\}}_{\mathfrak{g}}= [A,B ]+[A,a{d}_v]+[ a{d}_u,B]-a{d}_{Av}+Av +[u, v{]}_{\mathfrak{g}}.

The anchor map {\rho }_{\mathfrak{g}}:\mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)}\oplus \mathfrak{g} \to \mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)} is defined by

{\rho }_{\mathfrak{g}}(A+u)= A+a{d}_u.

Proposition 2.1　 With the above notations, a nonabelian omni-Lie algebra satisfies the following properties:

(i) (\mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)}\oplus \mathfrak{g},\{ \cdot ,\cdot {\}}_{\mathfrak{g}}) is a Leibniz algebra;

(ii) the pairing (\cdot ,\cdot {)}_{+} and the bracket\{\cdot ,\cdot {\}}_{\mathfrak{g}} are compatible in the sense that:

(\{ e_1,e_2{\}}_{\mathfrak{g}},e_3{)}_{+}+(e_2, \{e_1,e_3{\}}_{\mathfrak{g}}{)}_{+}= {\rho }_{\mathfrak{g}}(e_1)(e_2,e_3),

for any e_1, e_2,e_3\in \mathfrak{g}\mathfrak{l}\mathfrak{(}\mathfrak{g}\mathfrak{)}\oplus \mathfrak{g}.

Next, we review the concept of omni-Lie algebroid. For a vector bundle E, its 1-jet vector bundle \mathfrak{J}E and its gauge Lie algebroid \mathfrak{D}E have two corresponding exact sequences. One is given by

(2.2)0⟶Hom(TM ,E)\stackrel{\mathbb{e}}{⟶}\mathfrak{J}E\stackrel{\mathbb{p}}{⟶}E⟶0,

and the other is given by

(2.3)0⟶\mathfrak{g}\mathfrak{l}( E)\stackrel{\mathbb{i}}{⟶}\mathfrak{D}E\stackrel{\alpha }{⟶}TM ⟶0,

which is usually called the Atiyah sequence. Since the gauge Lie algebroid \mathfrak{D} E has a natural representation on E, the standard Lie algebroid cohomology comes from the complex (\mathrm{\Gamma }(Hom( {\wedge }^{\bullet }\mathfrak{D}E, E)),\mathbb{d}), where \mathbb{d}:\mathrm{\Gamma } (Hom ({\wedge }^n\mathfrak{D} E,E) )\to \mathrm{\Gamma }( Hom( {\wedge }^{n+1}\mathfrak{D} E,E) ) is the coboundary operator. When n=0, one can check that

\mathbb{d}u=[u]\in \mathrm{\Gamma }( \mathfrak{J}E)\subset \mathrm{\Gamma }( Hom( \mathfrak{D}E,E)),\mathrm{\forall }u \in \mathrm{\Gamma }(E) .

Moreover, \mathrm{\Gamma }( \mathfrak{J}E) is an invariant subspace of the Lie derivative {\mathfrak{L}}_{\mathfrak{d}}, for any \mathfrak{d}\in \mathrm{\Gamma }( \mathfrak{D}E). Here {\mathfrak{L}}_{\mathfrak{d}} is defined by

⟨{\mathfrak{L}}_{\mathfrak{d}}\mu ,{\mathfrak{d}}^{\mathrm{\prime }}{⟩}_E= \mathfrak{d}⟨\mu ,{\mathfrak{d}}^{\mathrm{\prime }}{⟩}_E- ⟨\mu ,[\mathfrak{d},{\mathfrak{d}}^{\mathrm{\prime }}{]}_{\mathfrak{D}}{⟩}_E,\mu \in \mathrm{\Gamma }( \mathfrak{J}E), {\mathfrak{d}}^{\mathrm{\prime }}\in \mathrm{\Gamma }(\mathfrak{D}E).

Definition 2.4 [4]　Let \mathcal{E}=\mathfrak{D} E⨁\mathfrak{J}E, the quadruple ( \mathcal{E},\{\cdot ,\cdot \},(\cdot ,\cdot {)}_E,\rho ) is called an omni-Lie algebroid, where \rho is the projection of \mathcal{E} onto \mathfrak{D}E, the Dorfman bracket \{\cdot , \cdot \}: \mathrm{\Gamma }( \mathcal{E}) \times \mathrm{\Gamma }(\mathcal{E})\to \mathrm{\Gamma }( \mathcal{E}) is given by

(2.4)\{\mathfrak{d} +\mu ,\mathfrak{r} +\nu \}=[\mathfrak{d} ,\mathfrak{r}{]}_{\mathfrak{D}}+ {\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r} {⟩}_E,

and a nondegenerate, symmetric, E-valued 2-form(\cdot ,\cdot {)}_E on \mathcal{E} is defined by

(2.5)(\mathfrak{d}+\mu ,\mathfrak{r}+\nu {)}_E=⟨\mathfrak{d},\nu {⟩}_E+⟨\mathfrak{r},\mu {⟩}_E.

Comparing with Courant algebroids, an omni-Lie algebroid has some similar properties as follows:

Proposition 2.2 [4]　 With the notation above, an omni-Lie algebroid satisfies the following properties:

(i) (\mathrm{\Gamma }(\mathcal{E}),\{\cdot ,\cdot \}) is a Leibniz algebra;

(ii) \rho \{X, Y\}= [\rho (X), \rho (Y){]}_{\mathfrak{D}};

(iii) \{X,f Y\}= f\{X ,Y\} +(\alpha \circ \rho (X)) (f)Y(where \alpha is the projection from \mathfrak{D}E to TM);

(iv) \{X,X \}=\frac{1}{2}\mathbb{d}(X,X{)}_E;

(v) \rho (X)(Y, Z{)}_E= (\{X,Y\},Z{)}_E+(Y,\{ X,Z\}{)}_E,

for all X,Y,Z \in \mathrm{\Gamma }(\mathcal{E}),f\in C^{\mathrm{\infty }}(M).

3 Nonabelian omni-Lie algebroids

In this section, on the basis of the omni-Lie algebra be the linearization of the Courant algebroid and the nonabelian omni-Lie algebra be a trivial deformation of the omni-Lie algebra, we will introduce the notion of a nonabelian omni-Lie algebroids and study its algebraic and geometric properties [4,6-7,10,14].

Given a Lie algebroid (E,[\cdot ,\cdot {]}_E,{\rho }_E), one can construct the bundle map \pi :\mathfrak{J}E\to \mathfrak{D}E,

{\pi }^{\mathrm{♯}}(\mathbb{d}u)(v )=[u,v{]}_E,

for all u,v\in \mathrm{\Gamma }(E).

Lemma 3.1 [2]　 On the section space \mathrm{\Gamma } (\mathfrak{J}E), the bracket [\cdot ,\cdot {]}_{\pi }:\mathrm{\Gamma }( \mathfrak{J}E)\times \mathrm{\Gamma }( \mathfrak{J}E)\to \mathrm{\Gamma } (\mathfrak{J}E) is defined by

(3.1)[\mu ,\nu {]}_{\pi }= L_{{\pi }^{\mathrm{♯}}(\mu )}\nu -L_{{\pi }^{\mathrm{♯}}(\nu )}\mu -\mathbb{d}\pi (\mu ,\nu ).

For all \mu ,\nu \in \mathrm{\Gamma }( \mathfrak{J}E), the bundle map holds:

(3.2){\pi }^{\mathrm{♯}}[\mu ,\nu {]}_{\pi }=[{\pi }^{\mathrm{♯}}(\mu ) ,{\pi }^{\mathrm{♯}}(\nu ) {]}_{\mathfrak{D}}.

The triple (\mathfrak{J}E, [\cdot ,\cdot {]}_{\pi },\alpha \circ {\pi }^{\mathrm{♯}}) is a Lie algebroid, where the bundle map \alpha is given by the exact sequence (2.3).

Given a Lie algebroid(E,[\cdot ,\cdot {]}_E,{\rho }_E), there is a natural E-Lie bialgebroid( \mathfrak{D}E,\mathfrak{J}E) [3].

Definition 3.1　With the notation above, the coboundary operator {\mathbb{d}}_{\pi }:Hom({\wedge }^n\mathfrak{J} E,E) \to Hom({\wedge }^{n+1}\mathfrak{J}E, E) is defined by

(3.3)

for all \mathrm{\Delta }\in H om({\wedge }^n\mathfrak{J}E, E),{\mu }_1,{\mu }_2, \cdots ,{\mu }_n\in \mathrm{\Gamma }( \mathfrak{J}E). We can check that {\mathbb{d}}_{\pi }^2 =0.

When n=1, the map {\mathbb{d}}_{\pi }: \mathrm{\Gamma }( \mathfrak{D}E)\to Hom ({\wedge }^2\mathfrak{J} E,E) is given by

({\mathbb{d}}_{\pi }\mathfrak{d} )({\mu }_1,{\mu }_2)={\pi }^{\mathrm{♯}}({\mu }_1)⟨ \mathfrak{d},{\mu }_2{⟩}_E-{\pi }^{\mathrm{♯}}({\mu }_2)⟨\mathfrak{d},{\mu }_1{⟩}_E- ⟨\mathfrak{d} ,[{\mu }_1,{\mu }_2{]}_{\pi } ⟩.

where \mathfrak{d} \in \mathrm{\Gamma }(\mathfrak{D}E),{\mu }_1, {\mu }_2\in \mathrm{\Gamma } (\mathfrak{J}E).

The contraction operator {\mathfrak{i}}_{\mu } is defined by:

(3.4){\mathfrak{i}}_{\mu }\mathfrak{d}=⟨\mathfrak{d},\mu {⟩}_E.

Where \mu \in \mathrm{\Gamma }(\mathfrak{J}E),\mathfrak{d}\in \mathrm{\Gamma }( \mathfrak{D}E).

Definition 3.2　The Lie derivative {\mathfrak{L}}_{\mu }^{\pi }:Hom({\wedge }^n\mathfrak{J}E, E)\to Hom({\wedge }^n\mathfrak{J} E,E) is defined by

(3.5)({\mathfrak{L}}_{\mu }^{\pi }\mathrm{\Delta })({\mu }_1, \cdots ,{\mu }_n)={\pi }^{\mathrm{♯}}(\mu )(\mathrm{\Delta }( {\mu }_1,\cdots ,{\mu }_n)) -\sum _{i=1}^n\mathrm{\Delta }({\mu }_1, \cdots ,[\mu ,{\mu }_i{]}_{\pi } ,\cdots ,{\mu }_n).

Where \mathrm{\Delta }\in Hom({\wedge }^n\mathfrak{J}E,E),\mu , {\mu }_1,{\mu }_2, \cdots ,{\mu }_n\in \mathrm{\Gamma }( \mathfrak{J}E).

Proposition 3.1　For all \mu ,\nu \in \mathrm{\Gamma } (\mathfrak{J}E), the contraction operator {\mathfrak{i}}_{\mu } and the Lie derivative {\mathfrak{L}}_{\mu }^{\pi } have the following properties:

(i) {\mathfrak{i}}_{[\mu ,\nu {]}_{\pi }}={\mathfrak{L}}_{\mu }^{\pi }{\mathfrak{i}}_{\nu }-{\mathfrak{i}}_{\nu }{\mathfrak{L}}_{\mu }^{\pi };

(ii) {\mathfrak{L}}_{[\mu ,\nu {]}_{\pi }}^{\pi }={\mathfrak{L}}_{\mu }^{\pi } {\mathfrak{L}}_{\nu }^{\pi }-{\mathfrak{L}}_{\nu }^{\pi } {\mathfrak{L}}_{\mu }^{\pi };

(iii) {\mathfrak{L}}_{\mu }^{\pi }={\mathfrak{i}}_{\mu }{\mathbb{d}}_{\pi }+{\mathbb{d}}_{\pi }{\mathfrak{i}}_{\mu };

(iv) {\mathfrak{L}}_{\mu }^{\pi }{\mathbb{d}}_{\pi }={\mathbb{d}}_{\pi }{\mathfrak{L}}_{\mu }^{\pi }.

Proof　For all \mathrm{\Delta } \in Hom({\wedge }^n\mathfrak{J} E,E) ,\mu ,\nu ,{\mu }_1,{\mu }_2,\cdots ,{\mu }_n\in \mathrm{\Gamma }( \mathfrak{J}E),

(i) According to Lemma 3.1 and equations (3.4) and (3.5), it can be obtained

\begin{array}{rl}({\mathfrak{L}}_{\mu }^{\pi }\circ {\mathfrak{i}}_{\nu }\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_{n-1})& ={\pi }^{\mathrm{♯}}(\mu )({\mathfrak{i}}_{\nu }\mathrm{\Delta }({\mu }_1,\cdots ,{\mu }_{n-1}))\\ & -\sum _{i=1}^{n-1}{\mathfrak{i}}_{\nu }\mathrm{\Delta }({\mu }_1,\cdots ,[\mu ,{\mu }_i{]}_{\pi },\cdots ,{\mu }_{n-1})\\ & ={\pi }^{\mathrm{♯}}(\mu )(\mathrm{\Delta }(\nu ,{\mu }_1,\cdots ,{\mu }_{n-1}))\\ & -\sum _{i=1}^{n-1}\mathrm{\Delta }(\nu ,{\mu }_1,\cdots ,[\mu ,{\mu }_i{]}_{\pi },\cdots ,{\mu }_{n-1}).\end{array}

Similarly, it can be calculated

\begin{array}{rl}({\mathfrak{i}}_{\nu }\circ {\mathfrak{L}}_{\mu }^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_{n-1})& ={\mathfrak{L}}_{\mu }^{\pi }\mathrm{\Delta }(\nu ,{\mu }_1,\cdots ,{\mu }_{n-1})\\ & ={\pi }^{\mathrm{♯}}(\mu )(\mathrm{\Delta }(\nu ,{\mu }_1,\cdots ,{\mu }_{n-1}))-\mathrm{\Delta }([\mu ,\nu {]}_{\pi },{\mu }_1,\cdots ,{\mu }_{n-1})\\ & -\sum _{i=1}^{n-1}\mathrm{\Delta }(\nu ,{\mu }_1,\cdots ,[\mu ,{\mu }_i{]}_{\pi },\cdots ,{\mu }_{n-1});\end{array}

then

(( {\mathfrak{L}}_{\mu }^{\pi }\circ {\mathfrak{i}}_{\nu }- {\mathfrak{i}}_{\nu }\circ {\mathfrak{L}}_{\mu }^{\pi })\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_{n-1})= {\mathfrak{i}}_{[\mu ,\nu {]}_{\pi }}\mathrm{\Delta } ({\mu }_1, \cdots ,{\mu }_{n-1}).

(ii) We first calculate the left-hand side of the equation, then we can get

\begin{array}{rl}& ({\mathfrak{L}}_{[\mu ,\nu {]}_{\pi }}^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_n)\\ & ={\pi }^{\mathrm{♯}}([\mu ,\nu {]}_{\pi })(\mathrm{\Delta }({\mu }_1,\cdots ,{\mu }_n))-\sum _{i=1}^n\mathrm{\Delta }({\mu }_1,\cdots ,[[\mu ,\nu {]}_{\pi },{\mu }_i{]}_{\pi },\cdots ,{\mu }_n).\end{array}

Similarly, we can calculate the right-hand side of the equation,

\begin{array}{rl}& (({\mathfrak{L}}_{\mu }^{\pi }{\mathfrak{L}}_{\nu }^{\pi }-{\mathfrak{L}}_{\nu }^{\pi }{\mathfrak{L}}_{\mu }^{\pi })\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_n)\\ & ={\pi }^{\mathrm{♯}}(\mu )(({\mathfrak{L}}_{\nu }^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_n))-{\pi }^{\mathrm{♯}}(\nu )(({\mathfrak{L}}_{\mu }^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_n))\\ & -\sum _{i=1}^n({\mathfrak{L}}_{\nu }^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,[\mu ,{\mu }_i{]}_{\pi },\cdots ,{\mu }_n)+\sum _{i=1}^n({\mathfrak{L}}_{\mu }^{\pi }\mathrm{\Delta })({\mu }_1,\cdots ,[\nu ,{\mu }_i{]}_{\pi },\cdots ,{\mu }_n)\\ & =([{\pi }^{\mathrm{♯}}(\mu ),{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}\mathrm{\Delta })({\mu }_1,\cdots ,{\mu }_n)+\sum _{i=1}^n\mathrm{\Delta }({\mu }_1,\cdots ,[\nu ,[\mu ,{\mu }_i{]}_{\pi }{]}_{\pi },\cdots ,{\mu }_n)\\ & -\sum _{i=1}^n\mathrm{\Delta }({\mu }_1,\cdots ,[\mu ,[\nu ,{\mu }_i{]}_{\pi }{]}_{\pi },\cdots ,{\mu }_n),\end{array}

and it can be proved that [\cdot ,\cdot {]}_{\pi } satisfies Jacobi identity and identity (3.2).

(iii) By identity (3.3), we can get

(iv) By {\mathbb{d}}_{\pi }^2= 0, we can get

{\mathfrak{L}}_{\mu }^{\pi }\circ {\mathbb{d}}_{\pi }= {\mathfrak{i}}_{\mu }\circ {\mathbb{d}}_{\pi }^2 +{\mathbb{d}}_{\pi }\circ {\mathfrak{i}}_{\mu }\circ {\mathbb{d}}_{\pi }={\mathbb{d}}_{\pi }({\mathfrak{i}}_{\mu }\circ {\mathbb{d}}_{\pi }+{\mathbb{d}}_{\pi }\circ {\mathfrak{i}}_{\mu })={\mathbb{d}}_{\pi }\circ {\mathfrak{L}}_{\mu }^{\pi }.

□

Definition 3.3　The nonabelian omni-Lie algebroid associated to a Lie algebroid ( \mathfrak{J}E,[ \cdot ,\cdot {]}_{\pi },{\pi }^{\mathrm{♯}}) is a quadruple (\mathfrak{D}E⨁ \mathfrak{J}E,\{\cdot ,\cdot {\}}_{\pi },(\cdot ,\cdot {)}_E,{\rho }_{\pi }), where the nondegenerate symmetric E-valued pair (\cdot ,\cdot {)}_E is identity (2.5), the Dorfman bracket\{\cdot , \cdot {\}}_{\pi } is defined by

(3.6)\{\mathfrak{d} +\mu ,\mathfrak{r} +\nu {\}}_{\pi }=[\mathfrak{d} ,\mathfrak{r}{]}_{\mathfrak{D}}+ {\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}-{\mathfrak{i}}_{\nu } {\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{i}}_{\mathfrak{r}}\mathbb{d}\mu +[\mu , \nu {]}_{\pi },

and the anchor map {\rho }_{\pi }:\mathfrak{D}E⨁ \mathfrak{J}E\to \mathfrak{D} E is defined by

(3.7){\rho }_{\pi }(\mathfrak{d} +\mu )=\mathfrak{d}+{\pi }^{\mathrm{♯}}(\mu ),

for all \mathfrak{d},\mathfrak{r}\in \mathrm{\Gamma }( \mathfrak{D}E), \mu ,\nu \in \mathrm{\Gamma }( \mathfrak{J}E).

Comparing with an omni-Lie algebroid, we will show that a nonabelian omni-Lie algebroid has some similar properties as follows.

Theorem 3.1　 With the notation above, a nonabelian omni-Lie algebroid satisfies the following properties:

(i) (\mathrm{\Gamma }(\mathfrak{D}E⨁ \mathfrak{J}E), \{\cdot , \cdot {\}}_{\pi } ) is a Leibniz algebra;

(ii) {\rho }_{\pi }\{X ,Y{\}}_{\pi }=[{\rho }_{\pi }(X),{\rho }_{\pi } (Y){]}_{\mathfrak{D}};

(iii) \{X,f Y{\}}_{\pi } =f\{ X,Y{\}}_{\pi }+(\alpha \circ {\rho }_{\pi }(X) )(f)Y(where \alpha is the projection from \mathfrak{D}E to TM);

(iv) \{X,X {\}}_{\pi }= \frac{1}{2}(\mathbb{d}+{\mathbb{d}}_{\pi })(X,X{)}_E;

(v) {\rho }_{\pi }(X)(Y ,Z{)}_E=(\{X,Y {\}}_{\pi }, Z{)}_E+ (Y,\{X, Z{\}}_{\pi } {)}_E,

where X,Y,Z\in \mathrm{\Gamma }( \mathfrak{D}E⨁\mathfrak{J}E),f\in C^{\mathrm{\infty }}(M).

Proof　(i) It can be obtained through the direct calculation.

(ii) For all \mathfrak{d},\mathfrak{r}\in \mathrm{\Gamma }( \mathfrak{D}E), \mu ,\nu \in \mathrm{\Gamma }( \mathfrak{J}E), by equations (3.6) and (3.7), we have

\begin{array}{rl}{\rho }_{\pi }\{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\pi }& ={\rho }_{\pi }([\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}-{\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{i}}_{\mathfrak{r}}\mathbb{d}\mu +[\mu ,\nu {]}_{\pi })\\ & =[\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}-{\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{i}}_{\mathfrak{r}}\mathbb{d}\mu +[\mu ,\nu {]}_{\pi }).\end{array}

Next, we can calculate the right-hand side of the equation. It can be obtained

\begin{array}{rl}[{\rho }_{\pi }(\mathfrak{d}+\mu ),,{\rho }_{\pi }(\mathfrak{r}+\nu ){]}_{\mathfrak{D}}& =[\mathfrak{d}+{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}+{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}\\ & =[\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mathfrak{d}}{\pi }^{\mathrm{♯}}(\nu )-{\mathfrak{L}}_{\mathfrak{r}}{\pi }^{\mathrm{♯}}(\mu )+[{\pi }^{\mathrm{♯}}(\mu ),{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}.\end{array}

Since

\begin{array}{rl}{\mathfrak{L}}_{\mathfrak{d}}{\pi }^{\mathrm{♯}}(\nu )=({\mathfrak{L}}_{\mathfrak{d}}\pi )(\nu )+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu )=[\mathfrak{d},\pi {]}_{\mathfrak{D}}(\nu )+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu );& \\ -{\mathfrak{L}}_{\mathfrak{r}}{\pi }^{\mathrm{♯}}(\mu )=-({\mathfrak{L}}_{\mathfrak{r}}\pi )(\mu )-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu )=-[\mathfrak{r},\pi {]}_{\mathfrak{D}}(\mu )-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu );& \\ {\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}={\mathfrak{i}}_{\mu }{\mathbb{d}}_{\pi }\mathfrak{r}+{\mathbb{d}}_{\pi }{\mathfrak{i}}_{\mu }\mathfrak{r}=[\pi ,\mathfrak{r}{]}_{\mathfrak{D}}(\mu )-{\pi }^{\mathrm{♯}}(\mathbb{d}(\mu ,\mathfrak{r}{)}_E);& \\ -{\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d}=[\mathfrak{d},\pi {]}_{\mathfrak{D}}(\nu );{\pi }^{\mathrm{♯}}(-{\mathfrak{i}}_{\mathfrak{r}}\mathbb{d}\mu )=-{\pi }^{\mathrm{♯}}((\mathbb{d}\mu )(\mathfrak{r})),& \end{array}

we have

{\rho }_{\pi }\{\mathfrak{d} +\mu ,\mathfrak{r} +\nu {\}}_{\pi }-[ {\rho }_{\pi }(\mathfrak{d} +\mu ),{\rho }_{\pi }( \mathfrak{r}+\nu ) {]}_{\mathfrak{D}}={\pi }^{\mathrm{♯}}[\mu ,\nu {]}_{\pi } -[{\pi }^{\mathrm{♯}}(\mu ),{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}.

Since {\pi }^{\mathrm{♯}}[\mu ,\nu {]}_{\pi }=[{\pi }^{\mathrm{♯}}(\mu ),{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}, we can get {\rho }_{\pi }\{\mathfrak{d} +\mu ,\mathfrak{r} +\nu {\}}_{\pi }=[{\rho }_{\pi }(\mathfrak{d} +\mu ),{\rho }_{\pi }( \mathfrak{r}+\nu ) {]}_{\mathfrak{D}}.

(iii) For all \mathfrak{d},\mathfrak{r}\in \mathrm{\Gamma }( \mathfrak{D}E), \mu ,\nu \in \mathrm{\Gamma }( \mathfrak{J}E), f\in C^{\mathrm{\infty }}(M), we have

\begin{array}{rl}\{\mathfrak{d}+\mu ,f(\mathfrak{r}+\nu ){\}}_{\pi }& =[\mathfrak{d},f\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mu }^{\pi }f\mathfrak{r}-{\mathfrak{i}}_{f\nu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}f\nu -{\mathfrak{i}}_{f\mathfrak{r}}\mathbb{d}\mu +[\mu ,f\nu {]}_{\pi }\\ & =f[\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+(\alpha \circ \mathfrak{d})(f)\mathfrak{r}+f{\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}\\ & +(\alpha \circ {\pi }^{\mathrm{♯}})(\mu )(f)\mathfrak{r}-f{\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d}+f{\mathfrak{L}}_{\mathfrak{d}}\nu +(\alpha \circ \mathfrak{d})(f)\nu \\ & -f{\mathfrak{i}}_{\nu }\mathbb{d}\mu +f[\mu ,\nu {]}_{\pi }+(\alpha \circ {\pi }^{\mathrm{♯}}(\mu ))(f)\nu \\ & =f([\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}-{\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{i}}_{\nu }\mathbb{d}\mu +[\mu ,\nu {]}_{\pi })\\ & +((\alpha \circ \mathfrak{d})(f)\mathfrak{r}+(\alpha \circ {\pi }^{\mathrm{♯}}(\mu ))(f)\mathfrak{r}\\ & +(\alpha \circ \mathfrak{d})(f)\nu +(\alpha \circ {\pi }^{\mathrm{♯}}(\mu ))(f)\nu )\\ & =f\{(\mathfrak{d}+\mu ),(\mathfrak{r}+\nu ){\}}_{\pi }+((\alpha \circ \mathfrak{d})+(\alpha \circ {\pi }^{\mathrm{♯}}(\mu )))(f)(\mathfrak{r}+\nu )\\ & =f\{(\mathfrak{d}+\mu ),(\mathfrak{r}+\nu ){\}}_{\pi }+(\alpha \circ {\rho }_{\pi })(\mathfrak{d}+\mu )(f)(\mathfrak{r}+\nu ).\end{array}

(iv) According to Definition 3.3, it can be obtained

\begin{array}{rl}\{\mathfrak{d}+\mu ,\mathfrak{d}+\mu {\}}_{\pi }& =[\mathfrak{d},\mathfrak{d}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mu }^{\pi }\mathfrak{d}-{\mathfrak{i}}_{\mu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}\mu -{\mathfrak{i}}_{\mathfrak{d}}\mathbb{d}\mu +[\mu ,\mu {]}_{\pi }\\ & ={\mathfrak{L}}_{\mu }^{\pi }\mathfrak{d}-{\mathfrak{i}}_{\mu }{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{\mathfrak{d}}\mu -{\mathfrak{i}}_{\mathfrak{d}}\mathbb{d}\mu \\ & ={\mathbb{d}}_{\pi }{\mathfrak{i}}_{\mu }\mathfrak{d}+\mathbb{d}{\mathfrak{i}}_{\mathfrak{d}}\mu .\end{array}

From formula (2.5), we can get \{\mathfrak{d} +\mu ,\mathfrak{d} +\mu {\}}_{\pi }=\frac{1}{2}(\mathbb{d}+ {\mathbb{d}}_{\pi })(\mathfrak{d}+\mu ,\mathfrak{d} +\mu {)}_E.

(v) For all {\mathfrak{d}}_1,{\mathfrak{d}}_2, {\mathfrak{d}}_3\in \mathrm{\Gamma } (\mathfrak{D}E),{\mu }_1, {\mu }_2,{\mu }_3\in \mathrm{\Gamma }( \mathfrak{J}E), we have

\begin{array}{rl}& (\{{\mathfrak{d}}_1+{\mu }_1,{\mathfrak{d}}_2+{\mu }_2{\}}_{\pi },{\mathfrak{d}}_3+{\mu }_3{)}_E\\ & =([{\mathfrak{d}}_1,{\mathfrak{d}}_2{]}_{\mathfrak{D}}+{\mathfrak{L}}_{{\mu }_1}^{\pi }{\mathfrak{d}}_2-{\mathfrak{i}}_{{\mu }_2}{\mathbb{d}}_{\pi }\mathfrak{d}+{\mathfrak{L}}_{{\mathfrak{d}}_1}{\mu }_2-{\mathfrak{i}}_{{\mathfrak{d}}_2}\mathbb{d}{\mu }_1+[{\mu }_1,{\mu }_2{]}_{\pi },{\mathfrak{d}}_3+{\mu }_3{)}_E\\ & =⟨[{\mathfrak{d}}_1,{\mathfrak{d}}_2{]}_{\mathfrak{D}}+{\mathfrak{L}}_{{\mu }_1}^{\pi }{\mathfrak{d}}_2-{\mathfrak{i}}_{{\mu }_2}{\mathbb{d}}_{\pi }{\mathfrak{d}}_1,{\mu }_3{⟩}_E+⟨{\mathfrak{d}}_3,{\mathfrak{L}}_{{\mathfrak{d}}_1}{\mu }_2-{\mathfrak{i}}_{{\mathfrak{d}}_2}\mathbb{d}{\mu }_1+[{\mu }_1,{\mu }_2{]}_{\pi }{⟩}_E.\end{array}

Similarly, it can be obtained by calculation

\begin{array}{rl}& ({\mathfrak{d}}_2+{\mu }_2,\{{\mathfrak{d}}_1+{\mu }_1,{\mathfrak{d}}_3+{\mu }_3{\}}_{\pi }{)}_E\\ & =⟨{\mathfrak{d}}_2,{\mathfrak{L}}_{{\mathfrak{d}}_1}{\mu }_3-{\mathfrak{i}}_{{\mathfrak{d}}_3}\mathbb{d}{\mu }_1+[{\mu }_1,{\mu }_3{]}_{\pi }{⟩}_E+⟨[{\mathfrak{d}}_1,{\mathfrak{d}}_3{]}_{\mathfrak{D}}+{\mathfrak{L}}_{{\mu }_1}^{\pi }{\mathfrak{d}}_3-{\mathfrak{i}}_{{\mu }_3}{\mathbb{d}}_{\pi }{\mathfrak{d}}_1,{\mu }_2{⟩}_E.\end{array}

By straight calculation, we can get

\begin{array}{rl}⟨[{\mathfrak{d}}_1,{\mathfrak{d}}_2{]}_{\mathfrak{D}},{\mu }_3{⟩}_E+⟨{\mathfrak{d}}_2,{\mathfrak{L}}_{{\mathfrak{d}}_1}{\mu }_3{⟩}_E=⟨{\mathfrak{d}}_1,⟨{\mathfrak{d}}_2,{\mu }_3{⟩}_E{⟩}_E;& \\ ⟨{\mathfrak{d}}_3,{\mathfrak{L}}_{{\mathfrak{d}}_1}{\mu }_2{⟩}_E+⟨[{\mathfrak{d}}_1,{\mathfrak{d}}_3{]}_{\mathfrak{D}},{\mu }_2{⟩}_E=⟨{\mathfrak{d}}_1,⟨{\mathfrak{d}}_3,{\mu }_2{⟩}_E{⟩}_E;& \\ ⟨-{\mathfrak{i}}_{{\mu }_2}{\mathbb{d}}_{\pi }{\mathfrak{d}}_1,{\mu }_3{⟩}_E+⟨-{\mathfrak{i}}_{{\mu }_3}{\mathbb{d}}_{\pi }{\mathbb{d}}_{\pi }{\mathfrak{d}}_1,{\mu }_2{⟩}_E=0;& \\ ⟨{\mathfrak{d}}_3,-{\mathfrak{i}}_{{\mathfrak{d}}_2}\mathbb{d}{\mu }_1{⟩}_E+⟨{\mathfrak{d}}_2,-{\mathfrak{i}}_{{\mathfrak{d}}_3}\mathbb{d}{\mu }_1{⟩}_E=0;& \\ ⟨{\mathfrak{L}}_{{\mu }_1}^{\pi }{\mathfrak{d}}_2,{\mu }_3{⟩}_E+⟨{\mathfrak{d}}_2,[{\mu }_1,{\mu }_3{]}_{\pi }{⟩}_E=⟨{\pi }^{\mathrm{♯}}({\mu }_1),⟨{\mathfrak{d}}_2,{\mu }_3{⟩}_E{⟩}_E;& \\ ⟨{\mathfrak{d}}_3,[{\mu }_1,{\mu }_2{]}_{\pi }{⟩}_E+⟨{\mathfrak{L}}_{{\mu }_1}^{\pi }{\mathfrak{d}}_3,{\mu }_2{⟩}_E=⟨{\pi }^{\mathrm{♯}}({\mu }_1),⟨{\mu }_2,{\mathfrak{d}}_3{⟩}_E{⟩}_E.& \end{array}

Therefore, it can be proved that

\begin{array}{rl}{\rho }_{\pi }({\mathfrak{d}}_1+{\mu }_1)({\mathfrak{d}}_2+{\mu }_2,{\mathfrak{d}}_3+{\mu }_3{)}_E& =(\{{\mathfrak{d}}_1+{\mu }_1,{\mathfrak{d}}_2+{\mu }_2{\}}_{\pi },{\mathfrak{d}}_3+{\mu }_3{)}_E\\ & +({\mathfrak{d}}_2+{\mu }_2,\{{\mathfrak{d}}_1+{\mu }_1,{\mathfrak{d}}_3+{\mu }_3{\}}_{\pi }{)}_E.\end{array}

□

Next, we will explain that the nonabelian omni-Lie algebroid can be regarded as a trivial deformation of the omni-Lie algebroid.

Definition 3.4 [9]　Let (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathfrak{L}}, {\rho }_{\mathfrak{L}}) be a Leibniz algebroid, \mathcal{N} be a bundle mapping of Leibniz algebroid \mathfrak{L}, which is defined by

\{e_1,e_2{\}}_{\mathcal{N}}=\{\mathcal{N} e_1,e_2{\}}_{\mathfrak{L}}+\{e_1, \mathcal{N}{e}_2{\}}_{\mathfrak{L}}-\mathcal{N}\{e_1, e_2{\}}_{\mathfrak{L}},

and

T\mathcal{N}(e_1, e_2)=\{\mathcal{N} e_1,\mathcal{N} e_2{\}}_{\mathfrak{L}}- \mathcal{N}\{e_1, e_2{\}}_{\mathcal{N}}.

If T \mathcal{N}=0, i.e. \mathcal{N}\{e_1, e_2{\}}_{\mathcal{N}}=\{\mathcal{N} e_1,\mathcal{N} e_2{\}}_{\mathfrak{L}}, then \mathcal{N} is called a Nijenhuis operator.

Let \{\cdot ,\cdot {\}}_{\lambda } be a bilinear operator, and \omega be a bilinear map from \mathfrak{L}\otimes \mathfrak{L} to \mathfrak{L}. Consider a \lambda-parametrized family of bilinear operations

\{e_1,e_2{\}}_{\lambda }=\{ e_1,e_2{\}}_{\mathfrak{L}}+\lambda \omega (e_1,e_2).

When all the brackets \{\cdot , \cdot {\}}_{\lambda } endow \mathfrak{L} with Leibniz algebras, a deformation of the Leibniz algebra is generated by \omega. If there exists a linear operator \mathcal{N} :\mathfrak{L}\to \mathfrak{L} such that for T_{\lambda }=id+\lambda \mathcal{N} there holds

T_{\lambda }\{e_1,e_2{\}}_{\lambda }=\{T_{\lambda }{e}_1,T_{\lambda }{e}_2{\}}_{\mathfrak{L}}.

Here the deformation \omega (e_1,e_2)\triangleq \{e_1,e_2{\}}_{\mathcal{N}} is said to be trivial.

Let \mathcal{N} be a Nijenhuis operator on the Leibniz algebra. Then \mathcal{N} \{e_1,e_2{\}}_{\mathcal{N}}=\{\mathcal{N} e_1,\mathcal{N} e_2{\}}_{\mathfrak{L}}. The Nijenhuis operator can give a obvious trivial deformation of the Leibniz algebroid [5, 13].

Proposition 3.2 [9]　 Let \mathcal{N} be a Nijenhuis operator on the Leibniz algebroid (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathfrak{L}}, {\rho }_{\mathfrak{L}}), then we have

(i) (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathcal{N}},{\rho }_{\mathcal{N}}) is a Leibniz algebra (where {\rho }_{\mathcal{N}}= {\rho }_{\mathfrak{L}}\circ \mathcal{N});

(ii) (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathfrak{L}}+ \{\cdot , \cdot {\}}_{\mathcal{N}},{\rho }_{\mathfrak{L}}+ {\rho }_{\mathcal{N}}) is a Leibniz algebroid;

(iii) For ( \mathfrak{L},\{ \cdot ,\cdot {\}}_{\mathcal{N}}, {\rho }_{\mathcal{N}}) and (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathfrak{L}}, {\rho }_{\mathfrak{L}}), \mathcal{N} is a morphism of Leibniz algebroids from (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathcal{N}},{\rho }_{\mathcal{N}}) to (\mathfrak{L},\{\cdot ,\cdot {\}}_{\mathfrak{L}}, {\rho }_{\mathfrak{L}}).

( \mathfrak{J}E,[ \cdot ,\cdot {]}_{\pi },\alpha \circ {\pi }^{\mathrm{♯}}) is a Lie algebroid, and the bundle map \mathcal{N}:\mathfrak{D} E\oplus \mathfrak{J}E\to \mathfrak{D}E\oplus \mathfrak{J}E is defined by

(3.8)\mathcal{N}( \mathfrak{d}+\mu ) ={\pi }^{\mathrm{♯}}(\mu ) \mathrm{\forall } \mathfrak{d}\in \mathrm{\Gamma }(\mathfrak{D}E),\mu \in \mathrm{\Gamma }( \mathfrak{J}E),

where the bundle map \alpha is given by the exact sequence (2.3).

Theorem 3.2　 The nonabelian omni-Lie algebroid ( \mathfrak{D}E\oplus \mathfrak{J} E,\{ \cdot ,\cdot {\}}_{\pi },(\cdot ,\cdot {)}_E,{\rho }_{\pi }) is a trivial deformation of the omni-Lie algebroid ( \mathfrak{D}E\oplus \mathfrak{J} E,\{ \cdot ,\cdot \},( \cdot ,\cdot {)}_E,\rho ).

Proof　According to Definition 3.4 and formula (3.8), we have

\begin{array}{rl}& \{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\mathcal{N}}\\ & =\{{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}+\nu \}+\{\mathfrak{d}+\mu ,{\pi }^{\mathrm{♯}}(\nu )\}-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)\\ & =[{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu +[\mathfrak{d},{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}-{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ){⟩}_E\\ & -{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)\\ & =([{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}{]}_{\mathfrak{D}}+[\mathfrak{d},{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E))\\ & +({\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu -{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ){⟩}_E),\end{array}

for all \mathfrak{d},\mathfrak{r}\in \mathrm{\Gamma }( \mathfrak{D}E), \mu ,\nu \in \mathrm{\Gamma }( \mathfrak{J}E), and then

\mathcal{N}\{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\mathcal{N}}= {\pi }^{\mathrm{♯}}({\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu -{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ){⟩}_E).

According to formula (3.1), we have {\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu -{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ) {⟩}_E= [\mu ,\nu {]}_{\pi }. Therefore \mathcal{N}\{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\mathcal{N}}= {\pi }^{\mathrm{♯}}([\mu ,\nu {]}_{\pi } ).

According to formula (3.2), we get

{\pi }^{\mathrm{♯}}([\mu ,\nu {]}_{\pi } )=[{\pi }^{\mathrm{♯}}(\mu ) ,{\pi }^{\mathrm{♯}}(\nu ) {]}_{\mathfrak{D}},

and we have \mathcal{N}\{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\mathcal{N}}= \{\mathcal{N}( \mathfrak{d}+\mu ) ,\mathcal{N}(\mathfrak{r}+\nu )\}. Now we have shown that \mathcal{N} is a Nijenhuis operator.

By formula (2.4) and Definition 3.4, we have

\begin{array}{rl}& \{\mathfrak{d}+\mu ,\mathfrak{r}+\nu \}+\{\mathfrak{d}+\mu ,\mathfrak{r}+\nu {\}}_{\mathcal{N}}\\ & =[\mathfrak{d},\mathfrak{r}{]}_{\mathfrak{D}}+{\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E+([{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}{]}_{\mathfrak{D}}+[\mathfrak{d},{\pi }^{\mathrm{♯}}(\nu ){]}_{\mathfrak{D}}\\ & -{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu -{\mathfrak{L}}_{\mathfrak{r}}\mu +\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E))+({\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu -{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ){⟩}_E).\end{array}

By calculation, we have

\begin{array}{rl}& [{\pi }^{\mathrm{♯}}(\mu ),\mathfrak{r}{]}_{\mathfrak{D}}+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu -\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)\\ & =-{\mathfrak{L}}_{\mathfrak{r}}{\pi }^{\mathrm{♯}}(\mu )+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu -\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)\\ & =-({\mathfrak{L}}_{\mathfrak{r}}\pi )(\mu )-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu )+{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{r}}\mu -\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)\\ & =[\pi ,\mathfrak{r}{]}_{\mathfrak{D}}(\mu )-{\pi }^{\mathrm{♯}}(\mathbb{d}⟨\mu ,\mathfrak{r}{⟩}_E)={\mathfrak{i}}_{\mu }{\mathbb{d}}_{\pi }\mathfrak{r}+{\mathbb{d}}_{\pi }{\mathfrak{i}}_{\mu }\mathfrak{r}={\mathfrak{L}}_{\mu }^{\pi }\mathfrak{r}.\end{array}

Similarly, it can be obtained by calculation

[\mathfrak{d},{\pi }^{\mathrm{♯}}(\nu )]-{\pi }^{\mathrm{♯}}({\mathfrak{L}}_{\mathfrak{d}}\nu )=- {\mathfrak{i}}_{\nu }{\mathbb{d}}_{\pi }\mathfrak{d} .

Since

{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\mu )}\nu -{\mathfrak{L}}_{{\pi }^{\mathrm{♯}}(\nu )}\mu +\mathbb{d}⟨\mu ,{\pi }^{\mathrm{♯}}(\nu ) {⟩}_E= [\mu ,\nu {]}_{\pi }.

it can be obtained:

\{\mathfrak{d} +\mu ,\mathfrak{r} +\nu {\}}_{\pi }=\{\mathfrak{d}+\mu ,\mathfrak{r} +\nu \}+\{ \mathfrak{d}+\mu ,\mathfrak{r} +\nu {\}}_{\mathcal{N}}.

Remark 3.1　In special, we have shown that ( \mathfrak{D}E\oplus \mathfrak{J} E,\{ \cdot ,\cdot {\}}_{\pi },{\rho }_{\pi }) is a Leibniz algebroid.

Lemma 3.2 [4]　 Let ( \mathfrak{J}E,[ \cdot ,\cdot {]}_{\pi },\alpha \circ {\pi }^{\mathrm{♯}}) be the Lie algebroid. The following statements are equivalent:

(i) \alpha \circ \pi \circ \mathbb{d}:\mathrm{\Gamma }(E) \to \mathcal{X} (M) induces a bundle map E\to TM;

(ii) \alpha \circ \pi \circ \mathbb{e}=0;

(iii) Hom(T M,E) is an ideal of \mathfrak{J}E;

(iv) there is a reduced Lie algebroid structure on E\cong \mathfrak{J}E/Im(\mathbb{e}) given by:

{\rho }_E=\alpha \circ \pi \circ \mathbb{d}, [u ,v{]}_E=\mathbb{p}[\mathbb{d}u, \mathbb{d}v{]}_{\pi } =\pi (\mathbb{d}u)v ,\mathrm{\forall } u,v\in \mathrm{\Gamma }(E).

where the bundle map \mathbb{p} is given by the exact sequence (2.2).

Lemma 3.3 [4, 6]　 Given a Lie algebroid (E,[\cdot ,\cdot {]}_E,{\rho }_E), \pi (\mathbb{d}u)=[ u,\cdot {]}_E can construct a nonabelian omni-Lie algebroid ( \mathfrak{D}E⨁\mathfrak{J}E, \{\cdot , \cdot {\}}_{\pi } ,(\cdot , \cdot {)}_E, {\rho }_{\pi }), here u\in \mathrm{\Gamma } (E).

Theorem 3.3　 If rank( E)\ge 2, then there is a one-to-one correspondence between Lie algebroid structures on E and the nonabelian omni-Lie algebroids.

Proof　Let ( \mathfrak{J}E,[ \cdot ,\cdot {]}_{\pi },\alpha \circ {\pi }^{\mathrm{♯}}) be a Lie algebroid, where \pi : \mathfrak{D}E\to \mathfrak{J} E is a bundle map. According to [4], one can define the nonabelian omni-Lie algebroid. When rank(E) \ge 2, it can be obtained a Lie algebroid on E by Lemma 3.2.

Obviously, the opposite direction holds true with the previous narration and Lemma 3.3.□

Example 3.1　Given a Lie algebroid (E,[\cdot ,\cdot ], \rho ), rank(E )\ge 2, there is a nonabelian omni-Lie algebroid \mathfrak{D}E\oplus \mathfrak{J}E, for a bundle map N:E\to E, the Nijenhuis torsion of N is defined by

T_N(u,v):= N[u, v{]}_N- [Nu, Nv],

where [u,v {]}_N=[Nu,v]+[ u,Nv]-N[u,v ], for all u,v\in \mathrm{\Gamma }(E). We define a twisted bundle map \pi \circ \hat{N}-a{d}_N\circ \pi :\mathfrak{J}E\to \mathfrak{D}E, where \hat{N} is the lift of N, a twisted bundle map is given by

\hat{N}:\mathfrak{J} E\to \mathfrak{J} E,[u ]\mapsto [Nu].

\pi \circ \hat{N}-a{d}_N\circ \pi ={\pi }_N, the following statements are equivalent:

(i) a quadruple (\mathfrak{D}E⨁ \mathfrak{J}E,\{\cdot ,\cdot {\}}_{{\pi }_N},(\cdot ,\cdot {)}_E,{\rho }_{{\pi }_N}) is a nonabelian omni-Lie algebroid;

(ii) (E,[\cdot ,\cdot {]}_N,\rho \circ N) is a Lie algebroid;

(iii) [T_N(u,v ),w]+T_N([u ,v], w)+c .p.=0 ,\mathrm{\forall } u,v,w \in \mathrm{\Gamma }(E).

The nonabelian omni-Lie algebroid \mathfrak{D}E⨁\mathfrak{J}E can be regarded as a matched pair of Leibniz algebroids \mathfrak{D}E and \mathfrak{J} E, which will be explained (refer to [8]).

Definition 3.5　 ( \mathfrak{L},[\cdot ,\cdot {]}_{\mathfrak{L}}, {\rho }_1),(\mathfrak{M},[\cdot ,\cdot {]}_{\mathfrak{M}},{\rho }_2) are Leibniz algebroids, if the direct sum bundle \mathfrak{L}⨁\mathfrak{M} has a Leibniz algebroid structure (\mathfrak{L}⨁\mathfrak{M} ,[\cdot , \cdot {]}_{\mathfrak{L}\bowtie \mathfrak{M}},\rho ), and \mathfrak{L},\mathfrak{M} are Leibniz subalgebroids of \mathfrak{L}⨁\mathfrak{M}, then ( \mathfrak{L},\mathfrak{M}) is a matched pair of Leibniz algebroids \mathfrak{L} and \mathfrak{M}.

Under the identification \mathrm{\Gamma }( \mathfrak{L}⨁\mathfrak{M}) \cong \mathrm{\Gamma }(\mathfrak{L}) ⨁\mathrm{\Gamma }(\mathfrak{M}), Definition 3.5 implies that \rho (g,y )={\rho }_1(g) +{\rho }_2(y), [g,h{]}_{\mathfrak{L}\bowtie \mathfrak{M}}=[ g,h{]}_{\mathfrak{L}}, and that [x,y{]}_{\mathfrak{L}\bowtie \mathfrak{M}}=[ x,y{]}_{\mathfrak{M}}, for all g,h\in \mathrm{\Gamma }( \mathfrak{L}),x,y\in \mathrm{\Gamma }(\mathfrak{M}).

{\pi }_1 represents the projection from \mathrm{\Gamma }( \mathfrak{L}⨁\mathfrak{M}) \cong \mathrm{\Gamma }(\mathfrak{L}) ⨁\mathrm{\Gamma }(\mathfrak{M}) to \mathrm{\Gamma }( \mathfrak{L}), {\pi }_2 represents the projection from \mathrm{\Gamma }( \mathfrak{L}⨁\mathfrak{M}) \cong \mathrm{\Gamma }(\mathfrak{L}) ⨁\mathrm{\Gamma }(\mathfrak{M}) to \mathrm{\Gamma }( \mathfrak{M}), they induce the following linear mappings respectively:

\begin{array}{rl}{\rho }_2^L:\mathrm{\Gamma }(\mathfrak{L})\times \mathrm{\Gamma }(\mathfrak{M})\to \mathrm{\Gamma }(\mathfrak{M});& \\ {\rho }_2^R:\mathrm{\Gamma }(\mathfrak{M})\times \mathrm{\Gamma }(\mathfrak{L})\to \mathrm{\Gamma }(\mathfrak{M});& \\ {\rho }_1^L:\mathrm{\Gamma }(\mathfrak{M})\times \mathrm{\Gamma }(\mathfrak{L})\to \mathrm{\Gamma }(\mathfrak{L});& \\ {\rho }_1^R:\mathrm{\Gamma }(\mathfrak{L})\times \mathrm{\Gamma }(\mathfrak{M})\to \mathrm{\Gamma }(\mathfrak{L}),& \end{array}

where

\begin{array}{rl}{\rho }_2^L(x,h)={\mathrm{\Pi }}_2[x,h{]}_{\mathfrak{L}\bowtie \mathfrak{M}},{\rho }_2^R(h,x)={\mathrm{\Pi }}_2[h,x{]}_{\mathfrak{L}\bowtie \mathfrak{M}};& \\ {\rho }_1^L(g,y)={\mathrm{\Pi }}_1[g,y{]}_{\mathfrak{L}\bowtie \mathfrak{M}},{\rho }_1^R(y,g)={\mathrm{\Pi }}_1[y,g{]}_{\mathfrak{L}\bowtie \mathfrak{M}}.& \end{array}

Proposition 3.3 [1]　 If ( \mathfrak{L},\mathfrak{M}) is a matched pair of Leibniz algebroids (\mathfrak{L},[\cdot ,\cdot {]}_{\mathfrak{L}},{\rho }_{\mathfrak{L}}) and (\mathfrak{M},[\cdot ,\cdot {]}_{\mathfrak{M}},{\rho }_{\mathfrak{M}}), then there are Leibniz algebra representation ( {\rho }_1^L,{\rho }_1^R) of Leibniz algebroid \mathfrak{L} on \mathfrak{M} and Leibniz algebra representation ( {\rho }_2^L,{\rho }_2^R) of Leibniz algebroid \mathfrak{M} on \mathfrak{L}, for all g,h\in \mathrm{\Gamma }( \mathfrak{L}),x ,y\in \mathrm{\Gamma }(\mathfrak{M} ), they satisfy the following compatibility conditions:

(i) {\rho }_1({\rho }_2^R(y) g)+{\rho }_2({\rho }_1^L (g)y)= [{\rho }_1(g) ,{\rho }_2(y)];

(ii) {\rho }_2({\rho }_1^R(h) x)+{\rho }_1({\rho }_2^L (x)h)= [{\rho }_2(x) ,{\rho }_1(h)];

(iii) {\rho }_1^L (g)[x, y{]}_{\mathfrak{M}}= [{\rho }_1^L(g) x,y{]}_{\mathfrak{M}}+[x, {\rho }_1^L(g)y{]}_{\mathfrak{M}}+{\rho }_1^L({\rho }_2^R (x)g)y +{\rho }_1^R({\rho }_2^R(y) g)x;

(iv) {\rho }_1^R (g)[x, y{]}_{\mathfrak{M}}= [x,{\rho }_1^R(g)y{]}_{\mathfrak{M}}-[y,{\rho }_1^R(g)x{]}_{\mathfrak{M}}+{\rho }_1^R({\rho }_2^L (y)g)x -{\rho }_1^R({\rho }_2^L(x) g)y;

(v) [{\rho }_1^L(g) x,y{]}_{\mathfrak{M}}+{\rho }_1^L({\rho }_2^R (x)g)y +[{\rho }_1^R(g) x,y{]}_{\mathfrak{M}}+{\rho }_1^L({\rho }_2^L (x)g)y =0;

(vi) {\rho }_2^L (x)[g, h{]}_{\mathfrak{L}}= [{\rho }_2^L(x) g,h{]}_{\mathfrak{L}}+[g, {\rho }_2^L(x)h{]}_{\mathfrak{L}}+{\rho }_2^L({\rho }_1^R (g)x)h +{\rho }_2^R({\rho }_1^R(h) x)g;

(vii) {\rho }_2^R (x)[g, h{]}_{\mathfrak{L}}= [g,{\rho }_2^R(x)h{]}_{\mathfrak{L}}-[h,{\rho }_2^R(x)g{]}_{\mathfrak{L}}+{\rho }_2^R({\rho }_1^L (h)x)g -{\rho }_2^R({\rho }_1^L(g) x)h;

(viii) [{\rho }_2^L(x) g,h{]}_{\mathfrak{L}}+{\rho }_2^L({\rho }_1^R (g)x)h +[{\rho }_2^R(x) g,h{]}_{\mathfrak{L}}+{\rho }_2^L({\rho }_1^L (g)x)h =0.

( \mathfrak{L},\mathfrak{M}) is a matched pair of Leibniz algebroids, there is a Leibniz algebroid structure \mathfrak{L}\bowtie \mathfrak{M} on the direct sum vector bundle \mathfrak{L}⨁\mathfrak{M}, the Leibniz bracket is given by:

[g+x ,h+y{]}_{\mathfrak{L}\bowtie \mathfrak{M}}= [g,h {]}_{\mathfrak{L}}+{\rho }_2^R(y) g+{\rho }_2^L(x) h+[x ,y{]}_{\mathfrak{M}}+ {\rho }_1^L(g)y+{\rho }_1^R (h)x.

Through a series of calculations, we get the following proposition.

Proposition 3.4　 A nonabelian omni-Lie algebroid ( \mathfrak{D}E⨁\mathfrak{J}E, \{\cdot , \cdot {\}}_{\pi } ,{\rho }_{\pi }) constructs a matched pair of Leibniz algebroids (\mathfrak{D}E, [\cdot ,\cdot {]}_{\mathfrak{D}},\alpha ) and (\mathfrak{J}E, [\cdot ,\cdot {]}_{\pi },\alpha \circ {\pi }^{\mathrm{♯}}).

Proof　By Theorem 3.1, \mathfrak{D}E⨁ \mathfrak{J}E is a Leibniz algebroids, and \mathfrak{D}E, \mathfrak{J} E are Leibniz subalgebroids. According to the definition of a matched pair, this proposition can be proved.□

4 Conclusion

In this paper, we study a nonabelian omni-Lie algebroid on the basis of previous researches. We have given the definition of nonabelian omni-Lie algebroid with Dorfman bracket \{\cdot , \cdot {\}}_{\pi }, and show its related properties. It is proved that a nonabelian omni-Lie algebroid \mathfrak{D}E⨁\mathfrak{J}E can be regarded as a matched pair of Leibniz algebroids \mathfrak{D}E and \mathfrak{J} E. In the future, we can continue our research on the algebraic and geometric structures of higher nonabelian omni-Lie algebroids.

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