Left-symmetric bialgebroids and their corresponding Manin triples

TL;DR

This paper introduces the concept of left-symmetric bialgebroids, explores their relation to Manin triples, and constructs examples from pseudo-Hessian manifolds, expanding the geometric understanding of these algebraic structures.

Contribution

It defines left-symmetric bialgebroids, establishes their equivalence with Manin triples, and links them to pre-symplectic algebroids and Maurer-Cartan equations.

Findings

Defined left-symmetric bialgebroids and their properties.

Established the equivalence between Manin triples and bialgebroids.

Connected the structures to Maurer-Cartan equations and Dirac structures.

Abstract

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.

Equations

$$\text{mbox}quaLSalg\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$[x,fy{]}_{\mathcal{A}}=f[x,y{]}_{\mathcal{A}}+a_{\mathcal{A}}(x)(f)y,\forall f\in C^{\infty }(M).$$

$$a_{\mathcal{B}}\circ \sigma =a_{\mathcal{A}},\sigma [x,y{]}_{\mathcal{A}}=[\sigma (x),\sigma (y){]}_{\mathcal{B}}.$$

$$⟨{\rho }^{\ast }(x)(\xi ),u⟩=a_{\mathcal{A}}(x)⟨\xi ,u⟩-⟨\xi ,\rho (x)(u)⟩,\forall x\in \Gamma (\mathcal{A}),\xi \in \Gamma (E^{\ast }),u\in \Gamma (E).$$

$$⟨{\mathcal{L}}_x\xi ,y⟩=a_{\mathcal{A}}(x)⟨\xi ,y⟩-⟨\xi ,[x,y{]}_{\mathcal{A}}⟩,\forall y\in \Gamma (\mathcal{A}),\xi \in \Gamma ({\mathcal{A}}^{\ast }).$$

$$\mathrm{d}\varpi (x_1,\cdots ,x_{k+1})$$

$$a_{\mathcal{A}}(x)\varpi (y,z)-a_{\mathcal{A}}(y)\varpi (x,z)+a_{\mathcal{A}}(z)\varpi (x,y)-\varpi ([x,y{]}_{\mathcal{A}},z)+\varpi ([x,z{]}_{\mathcal{A}},y)-\varpi ([y,z{]}_{\mathcal{A}},x)=0.$$

$$(x,y,z)\triangleq x{\cdot }_{\mathfrak{g}}(y{\cdot }_{\mathfrak{g}}z)-(x{\cdot }_{\mathfrak{g}}y){\cdot }_{\mathfrak{g}}z$$

$$(x,y,z)=(y,x,z),\mathrm{or}\mathrm{equivalently},x{\cdot }_{\mathfrak{g}}(y{\cdot }_{\mathfrak{g}}z)-(x{\cdot }_{\mathfrak{g}}y){\cdot }_{\mathfrak{g}}z=y{\cdot }_{\mathfrak{g}}(x{\cdot }_{\mathfrak{g}}z)-(y{\cdot }_{\mathfrak{g}}x){\cdot }_{\mathfrak{g}}z.$$

$$L_{x}y=x{\cdot }_{A}y,R_{x}y=y{\cdot }_{A}x.$$

$$[x,y{]}_A=x{\cdot }_{A}y-y{\cdot }_{A}x,\forall x,y\in \Gamma (A).$$

$$[x+\xi ,y+\eta {]}_S=[x,y{]}_A+L_x^{\ast }\eta -L_y^{\ast }\xi ,$$

$$\omega (x+\xi ,y+\eta )=⟨\xi ,y⟩-⟨\eta ,x⟩,\forall x,y\in \Gamma (A),\xi ,\eta \in \Gamma (A^{\ast }).$$

$$\rho (x)\mu (y)e-\mu (y)\rho (x)e=\mu (x{\cdot }_{A}y)e-\mu (y)\mu (x)e.$$

$$C^{n+1}(A)=\Gamma ({\wedge }^n{A}^{\ast }\otimes A^{\ast }),n\ge 0.$$

$$\delta \phi (x_1,\cdots ,x_{n+1})$$

$$T(e_1,e_2,e_3)=(e_1\star e_2,e_3{)}_{-}+(e_1,e_2\star e_3{)}_{-}-(e_2\star e_1,e_3{)}_{-}-(e_2,e_1\star e_3{)}_{-},$$

$$(Df,e{)}_{-}=\rho (e)(f),$$

$$[e_1,e_2{]}_E=e_1\star e_2-e_2\star e_1.$$

$$e_1\star e_2={{\omega }^{♯}}^{-1}({\mathcal{L}}_{e_1}{\omega }^{♯}(e_2)+\frac{1}{2}\mathrm{d}(\omega (e_1,e_2)))\forall e_1,e_2\in \Gamma (E).$$

$$[e_1,e_2{]}_E=e_1\star e_2-e_2\star e_1,\forall e_1,e_2\in \Gamma (E),$$

$$(x+\xi )\star (y+\eta )=x{\cdot }_{A}y+{\mathcal{L}}_x\eta -R_y\xi -\frac{1}{2}\mathrm{d}(x+\xi ,y+\eta {)}_{+},$$

$$(x+\xi ,y+\eta {)}_{+}=⟨\xi ,y⟩+⟨\eta ,x⟩.$$

$${\mathfrak{L}}_x(y_1\wedge \cdots \wedge y_n\otimes y_{n+1})=\sum _{i=1}^n{y}_1\wedge \cdots \wedge x{\cdot }_A{y}_i\wedge \cdots \wedge y_n\otimes y_{n+1}+y_1\wedge \cdots \wedge y_n\otimes [x,y_{n+1}{]}_A,$$

$$R_x(y_1\wedge \cdots \wedge y_n\otimes y_{n+1})=-(\sum _{i=1}^n{y}_1\wedge \cdots y_i{\cdot }_{A}x\cdots \wedge y_n\otimes y_{n+1})+y_1\wedge \cdots \wedge y_n\otimes y_{n+1}{\cdot }_{A}x.$$

$${(}_{\xi }┘\phi )({\eta }_1,{\eta }_2,\cdots ,{\eta }_n)$$

$$(\phi {└}_{\xi })({\eta }_1,{\eta }_2,\cdots ,{\eta }_n)$$

$${\mathfrak{L}}_{[x,y{]}_A}$$

$${\mathfrak{L}}_{x}fX$$

$${\mathfrak{L}}_{fx}X$$

$${\mathfrak{L}}_{fx}{y}_1\wedge \cdots \wedge y_n\otimes y_{n+1}$$

$$⟨{\mathfrak{L}}_x\phi ,X⟩$$

Full text

1 Introduction

Left-symmetric algebras (or pre-Lie algebras) arose from the study of convex homogeneous cones [27], affine manifolds and affine structures on Lie groups [10], deformation and cohomology theory of associative algebras [6] and then appear in many fields in mathematics and mathematical physics. See the survey article [2] and the references therein. In particular, there are close relations between left-symmetric algebras and certain important left-invariant structures on Lie groups like aforementioned affine, symplectic, Kähler, and metric structures [8, 11, 17, 19, 20]. A quadratic left-symmetric algebra is a left-symmetric algebra together with a nondegenerate invariant skew-symmetric bilinear form [3]. A symplectic (Frobenius) Lie algebra is a Lie algebra $\mathfrak{g}$ equipped with a nondegenerate 2-cocycle $\omega\in\wedge^{2}\mathfrak{g}^{*}$. There is a one-to-one correspondence between symplectic (Frobenius) Lie algebras and quadratic left-symmetric algebras.

A left-symmetric algebroid, also called a Koszul-Vinberg algebroid, is a geometric generalization of a left-symmetric algebra. See [12, 22, 23] for more details and applications. In [13], we introduced the notion of a pre-symplectic algebroid, which is a geometric generalization of a quadratic left-symmetric algebra. Generalizing the relation between symplectic  (Frobenius)  Lie  algebras and quadratic left-symmetric algebras, we showed that there is a one-to-one correspondence between symplectic Lie algebroids and pre-symplectic algebroids. See [4, 7, 18, 21] for more details about symplectic Lie algebroids and their applications.

The purpose of this paper is studying the bialgebra theory for left-symmetric algebroids and the corresponding Manin triple theory. Motivated by [14, 16], we introduce the notion of a left-symmetric bialgebroid, which is a geometric generalization of a left-symmetric bialgebra [1]. The double of a left-symmetric bialgebroid is not a left-symmetric algebroid anymore, but a pre-symplectic algebroid. This result is parallel to the fact that the double of a Lie bialgebroid111The notion of a Lie bialgebroid was first introduced by Mackenzie and Xu in as the infinitesimal object of a Poisson groupoid [16]., is not a Lie algebroid, but a Courant algebroid [14]. Furthermore, if we consider the commutator of a left-symmetric bialgebroid, we can obtain a matched pair of Lie algebroids, whose double is the symplectic Lie algebroid associated to the pre-symplectic algebroid. The above results can be summarized into the following commutative diagram:

[TABLE]

[TABLE]

In the above diagram, qua is short for quadratic, LS is short for left-symmetric, alg is short for algebra, sym is short for symplectic, MP is short for matched pair and algd is short for algebroid. We establish a relation between left-symmetric bialgebroids and pseudo-Hessian manifolds. See [24, 25, 26] for more information about pseudo-Hessian Lie algebras and Hessian geometry. A flat manifold $(M,\nabla)$ gives rise to a left-symmetric algebroid $T_{\nabla}M$. We show that a pseudo-Riemannian metric $g$ on $(M,\nabla)$ is a pseudo-Hessian metric if $\delta g=0$, where $\delta$ is the cohomology operator of the left-symmetric algebroid $T_{\nabla}M$ (Proposition 4.12). Given a pseudo-Hessian manifold $(M,\nabla,g)$, $(T_{\nabla}M,T^{*}_{H}M)$ is a left-symmetric bialgebroid (Proposition 4.13), where $H$ is the inverse of $g$. This result is parallel to that $(TM,T^{*}_{\pi}M)$ is a Lie bialgebroid for any Poisson manifold $(M,\pi)$ [9, 16]. It seems that our theory is a symmetric analogue of Poisson geometry.

The paper is organized as follows. In Section $2$, we give a review on Lie algebroids, left-symmetric algebroids and pre-symplectic algebroids. In Section $3$, we develop the differential calculus on a left-symmetric algebroid which is the main tool in our later study. In Section $4$, we introduce the notion of a left-symmetric bialgebroid and study its properties. In Section 5, we introduce the notion of a Manin triple for left-symmetric algebroids and show the equivalence between left-symmetric bialgebroids and Manin triples for left-symmetric algebroids.

Throughout this paper, all the vector bundles are over the same manifold $M$. For two vector bundles $A$ and $B$, a bundle map from $A$ to $B$ is a base-preserving map and $C^{\infty}(M)$-linear.

Acknowledgement: We give our warmest thanks to Zhangju Liu and Jianghua Lu for very useful comments and discussions.

2 Preliminaries

We briefly recall Lie algebroids, left-symmetric algebroids and pre-symplectic algebroids.

Lie algebroids

The notion of a Lie algebroid was introduced by Pradines in 1967, which is a generalization of Lie algebras and tangent bundles. See [15] for general theory about Lie algebroids. They play important roles in various parts of mathematics.

Definition 2.1**.**

A Lie algebroid structure on a vector bundle $\mathcal{A}\longrightarrow M$ is a pair that consists of a Lie algebra structure $[\cdot,\cdot]_{\mathcal{A}}$ on the section space $\Gamma(\mathcal{A})$ and a bundle map $a_{\mathcal{A}}:\mathcal{A}\longrightarrow TM$, called the anchor, such that the following relation is satisfied:

[TABLE]

For a vector bundle $E\longrightarrow M$, we denote by $\mathfrak{D}(E)$ the gauge Lie algebroid of the frame bundle $\mathcal{F}(E)$, which is also called the covariant differential operator bundle of $E$.

Let $(\mathcal{A},[\cdot,\cdot]_{\mathcal{A}},a_{\mathcal{A}})$ and $(\mathcal{B},[\cdot,\cdot]_{\mathcal{B}},a_{\mathcal{B}})$ be two Lie algebroids (with the same base), a base-preserving morphism from $\mathcal{A}$ to $\mathcal{B}$ is a bundle map $\sigma:\mathcal{A}\longrightarrow\mathcal{B}$ such that

[TABLE]

A representation of a Lie algebroid $\mathcal{A}$ on a vector bundle $E$ is a base-preserving morphism $\rho$ form $\mathcal{A}$ to the Lie algebroid $\mathfrak{D}(E)$. Denote a representation by $(E;\rho).$ The dual representation of a Lie algebroid $\mathcal{A}$ on $E^{*}$ is the bundle map $\rho^{*}:\mathcal{A}\longrightarrow\mathfrak{D}(E^{*})$ given by

[TABLE]

As a generalization of a matched pair of Lie algebras, a matched pair of Lie algebroids is a pair of Lie algebroids $(\mathcal{A},\mathcal{B})$ together with two representations $\rho_{\mathcal{A}}:\mathcal{A}\longrightarrow\mathfrak{D}(\mathcal{B})$ and $\rho_{\mathcal{B}}:\mathcal{B}\longrightarrow\mathfrak{D}(\mathcal{A})$ such that some compatibility conditions are satisfied.

For all $x\in\Gamma(\mathcal{A})$, the Lie derivation $\mathcal{L}_{x}:\Gamma(\mathcal{A}^{*})\longrightarrow\Gamma(\mathcal{A}^{*})$ of the Lie algebroid $\mathcal{A}$ is given by

[TABLE]

A Lie algebroid $(\mathcal{A},[\cdot,\cdot]_{\mathcal{A}},a_{\mathcal{A}})$ naturally represents on the trivial line bundle $E=M\times\mathbb{R}$ via the anchor map $a_{\mathcal{A}}:\mathcal{A}\longrightarrow TM$. The corresponding coboundary operator $\mathrm{d}:\Gamma(\wedge^{k}\mathcal{A}^{*})\longrightarrow\Gamma(\wedge^{k+1}\mathcal{A}^{*})$ is given by

[TABLE]

In particular, a $2$-form $\varpi\in\Gamma(\wedge^{2}\mathcal{A}^{*})$ is a 2-cocycle if $\mathrm{d}\varpi=0$, i.e.

[TABLE]

A symplectic Lie algebroid is a Lie algebroid together with a nondegenerate closed $2$-form. A subalgebroid of a symplectic Lie algebroid $(\mathcal{A},[\cdot,\cdot]_{\mathcal{A}},a_{\mathcal{A}},\varpi)$ is called Lagrangian if it is maximal isotropic with respect to the skew-symmetric bilinear form $\varpi$.

Left-symmetric algebroids

Definition 2.2**.**

A left-symmetric algebra is a pair $(\mathfrak{g},\cdot_{\mathfrak{g}})$, where $\mathfrak{g}$ is a vector space, and $\cdot_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ is a bilinear multiplication satisfying that for all $x,y,z\in\mathfrak{g}$, the associator

[TABLE]

is symmetric in $x,y$, i.e.

[TABLE]

A left-symmetric algebroid is also called a Koszul-Vinberg algebroid in [22].

Definition 2.3**.**

[12, 22]* A left-symmetric algebroid structure on a vector bundle $A\longrightarrow M$ is a pair that consists of a left-symmetric algebra structure $\cdot_{A}$ on the section space $\Gamma(A)$ and a vector bundle morphism $a_{A}:A\longrightarrow TM$, called the anchor, such that for all $f\in C^{\infty}(M)$ and $x,y\in\Gamma(A)$, the following conditions are satisfied:*

• (i)

$~{}x\cdot_{A}(fy)=f(x\cdot_{A}y)+a_{A}(x)(f)y,$**

• (ii)

$(fx)\cdot_{A}y=f(x\cdot_{A}y).$**

We usually denote a left-symmetric algebroid by $(A,\cdot_{A},a_{A})$. Any left-symmetric algebra is a left-symmetric algebroid over a point.

Example 2.4**.**

Let $M$ be a differential manifold with a flat torsion free connection $\nabla$. Then $(TM,\nabla,\rm{id})$ is a left-symmetric algebroid whose sub-adjacent Lie algebroid is exactly the tangent Lie algebroid. We denote this left-symmetric algebroid by $T_{\nabla}M$, which will be frequently used below. **

For any $x\in\Gamma(A)$, we define $L_{x}:\Gamma(A)\longrightarrow\Gamma(A)$ and $R_{x}:\Gamma(A)\longrightarrow\Gamma(A)$ by

[TABLE]

Condition (i) in the above definition means that $L_{x}\in\mathfrak{D}(A)$. Condition (ii) means that the map $x\longmapsto L_{x}$ is $C^{\infty}(M)$-linear. Thus, $L:A\longrightarrow\mathfrak{D}(A)$ is a bundle map.

Proposition 2.5**.**

[12]* Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid. Define a skew-symmetric bilinear bracket operation $[\cdot,\cdot]_{A}$ on $\Gamma(A)$ by*

[TABLE]

Then, $(A,[\cdot,\cdot]_{A},a_{A})$ is a Lie algebroid, and denoted by $A^{c}$, called the sub-adjacent Lie algebroid of $(A,\cdot_{A},a_{A})$. Furthermore, $L:A\longrightarrow\mathfrak{D}(A)$ gives a representation of the Lie algebroid $A^{c}$.

Theorem 2.6**.**

[12]* Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid. Then $(A^{c}\ltimes_{L^{*}}A^{*},[\cdot,\cdot]_{S},\rho,\omega)$ is a symplectic Lie algebroid, where $A^{c}\ltimes_{L^{*}}A^{*}$ is the semidirect product of $A^{c}$ and $A^{*}$ in which $L^{*}$ is the dual representation of $L$. More precisely, the Lie bracket $[\cdot,\cdot]_{S}$ and the anchor $\rho$ are given by*

[TABLE]

and $\rho(x+\xi)=a_{A}(x)$ respectively. Furthermore, the symplectic form $\omega$ is given by

[TABLE]

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid and $E$ a vector bundle. A representation of $A$ on $E$ consists of a pair $(\rho,\mu)$, where $\rho:A\longrightarrow\mathfrak{D}(E)$ is a representation of $A^{c}$ on $E$ and $\mu:A\longrightarrow\mathrm{End}(E)$ is a bundle map, such that for all $x,y\in\Gamma(A),\ e\in\Gamma(E)$, we have

[TABLE]

Denote a representation by $(E;\rho,\mu)$.

Let us recall the cohomology complex with the coefficients in the trivial representation, i.e. $\rho=a_{A}$ and $\mu=0$. See [5, 12] for general theory of cohomologies of right-symmetric algebras and left-symmetric algebroids respectively. The set of $(n+1)$-cochains is given by

[TABLE]

For all $\varphi\in C^{n}(A)$ and $x_{i}\in\Gamma(A),~{}i=1,\cdots,n+1$, the corresponding coboundary operator $\delta$ is given by

[TABLE]

Pre-symplectic algebroids

Here we recall the notion of pre-symplectic algebroids and the relation with symplectic Lie algebroids. See [13] for more details.

Definition 2.7**.**

A pre-symplectic algebroid is a vector bundle $E\rightarrow M$ equipped with a nondegenerate skew-symmetric bilinear form $(\cdot,\cdot)_{-}$, a multiplication $\star:\Gamma(E)\times\Gamma(E)\longrightarrow\Gamma(E)$, and a bundle map $\rho:E\rightarrow TM$, such that for all $e,e_{1},e_{2},e_{3}\in\Gamma(E),~{}~{}f\in C^{\infty}(M)$, the following conditions are satisfied:

• $\rm(i)$

$(e_{1},e_{2},e_{3})-(e_{2},e_{1},e_{3})=\frac{1}{6}DT(e_{1},e_{2},e_{3})$;

• $\rm(ii)$

$\rho(e_{1})(e_{2},e_{3})_{-}=(e_{1}{\star}e_{2}-\frac{1}{2}D(e_{1},e_{2})_{-},e_{3})_{-}+(e_{2},[e_{1},e_{3}]_{E})_{-},$**

where $(e_{1},e_{2},e_{3})$ is the associator for the multiplication $\star$ given by (3), $T:\Gamma(E)\times\Gamma(E)\times\Gamma(E)\longrightarrow C^{\infty}(M)$ is defined by

[TABLE]

$D:C^{\infty}(M)\longrightarrow\Gamma(E)$* is defined by*

[TABLE]

and the bracket $[\cdot,\cdot]_{E}:\wedge^{2}\Gamma(E)\longrightarrow\Gamma(E)$ is defined by

[TABLE]

We denote a pre-symplectic algebroid by $(E,\star,\rho,(\cdot,\cdot)_{-})$.

Theorem 2.8**.**

Let $(E,\star,\rho,(\cdot,\cdot)_{-})$ be a pre-symplectic algebroid. Then $(E,[\cdot,\cdot]_{E},\rho,\omega=(\cdot,\cdot)_{-})$ is a symplectic Lie algebroid.

Given a symplectic Lie algebroid $(E,[\cdot,\cdot]_{E},\rho,\omega)$, define a multiplication $\star:\Gamma(E)\times\Gamma(E)\longrightarrow\Gamma(E)$ by

[TABLE]

Theorem 2.9**.**

Let $(E,[\cdot,\cdot]_{E},\rho,\omega)$ be a symplectic Lie algebroid. Then $(E,\star,\rho,(\cdot,\cdot)_{-}=\omega)$ is a pre-symplectic algebroid, and satisfies

[TABLE]

where the multiplication $\star$ is given by $(\ref{LSCA bracket})$.

Example 2.10**.**

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid and $(A^{c}\ltimes_{L^{*}}A^{*},\omega)$ the corresponding symplectic Lie algebroid, where $\omega$ is given by (5). Then the corresponding pre-symplectic algebroid structure is given by

[TABLE]

where $\mathcal{L}$ is given by (1), $R$ is given by (23), and $(\cdot,\cdot)_{+}$ is the nondegenerate symmetric bilinear form on $A\oplus A^{*}$ given by

[TABLE]

Definition 2.11**.**

Let $(E,\star,\rho,(\cdot,\cdot)_{-})$ be a pre-symplectic algebroid. A subbundle $F$ of $E$ is called isotropic if it is isotropic under the skew-symmetric bilinear form $(\cdot,\cdot)_{-}$. It is called integrable if $\Gamma(F)$ is closed under the operation $\star$. A Dirac structure is a subbundle $F$ which is maximal isotropic and integrable.

The following proposition is obvious.

Proposition 2.12**.**

Let $F$ be a Dirac structure of a pre-symplectic algebroid $(E,\star,\rho,(\cdot,\cdot)_{-})$. Then $(F,\star|_{F},\rho|_{F})$ is a left-symmetric algebroid.

3 Differential calculus on left-symmetric algebroids

In this section, we develop the differential calculus on left-symmetric algebroids, which is the fundamental tool in the following study. Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid. For all $x\in\Gamma(A)$, define the Lie derivative $\mathfrak{L}_{x}:\Gamma(\wedge^{n}A\otimes A)\longrightarrow\Gamma(\wedge^{n}A\otimes A)$ by

[TABLE]

where $y_{1},\cdots,y_{n+1}\in\Gamma(A)$. Define the right multiplication $R_{x}:\Gamma(\wedge^{n}A\otimes A)\longrightarrow\Gamma(\wedge^{n}A\otimes A)$ by

[TABLE]

Remark 3.1**.**

For all $x,y\in\Gamma(A)$, we have $\mathfrak{L}_{x}y=[x,y]_{A}$. Thus, $\mathfrak{L}_{x}$ is not a straightforward generalization of the left multiplication $L_{x}:\Gamma(A)\longrightarrow\Gamma(A)$ given by (4). This is why we use different notations. However, $R_{x}:\Gamma(\wedge^{n}A\otimes A)\longrightarrow\Gamma(\wedge^{n}A\otimes A)$ is a straightforward generalization of the right multiplication $R_{x}:\Gamma(A)\longrightarrow\Gamma(A)$ given by (4). Therefore, we use the same notations which will not cause confusion.

For all $\xi\in\Gamma(A^{*})$, the left contraction and right contraction, which we denote by ${}_{\xi}\lrcorner:\Gamma(\wedge^{n}A\otimes A)\longrightarrow\Gamma(\wedge^{n-1}A\otimes A)$ and $\llcorner_{\xi}:\Gamma(\wedge^{n}A\otimes A)\longrightarrow\Gamma(\wedge^{n}A)$ respectively, are defined by

[TABLE]

where $\varphi\in\Gamma(\wedge^{n}A\otimes A),\quad n\geq 1$ and $\eta_{1},\eta_{2},\cdots,\eta_{n}\in\Gamma(A^{*})$.

The Lie derivative $\mathfrak{L}_{x}$ has the following properties.

Proposition 3.2**.**

For all $x,y\in\Gamma(A),~{}~{}f\in C^{\infty}(M),~{}~{}X\in\Gamma(\wedge^{n}A\otimes A),n\geq 0$, we have

[TABLE]

Proof. We only prove (21). Others can be proved similarly. For all $x,y_{1},y_{2},\cdots,y_{n+1}\in\Gamma(A),~{}~{}f\in C^{\infty}(M)$, without loss of generality we can assume that $X=y_{1}\wedge\cdots\wedge y_{n}\otimes y_{n+1}$, then we have

[TABLE]

The proof is finished.

For all $x\in\Gamma(A)$, the Lie derivative $\mathfrak{L}_{x}:\Gamma(\wedge^{n}A^{*}\otimes A^{*})\longrightarrow\Gamma(\wedge^{n}A^{*}\otimes A^{*})$ and the right multiplication $R_{x}:\Gamma(\wedge^{n}A^{*}\otimes A^{*})\longrightarrow\Gamma(\wedge^{n}A^{*}\otimes A^{*})$ are defined respectively by222Here we use the same notations as before and this will not bring confusion since it depends on what it acts.

[TABLE]

where $\varphi\in\Gamma(\wedge^{n}A^{*}\otimes A^{*}),\quad n\geq 1$ and $X\in\Gamma(\wedge^{n}A\otimes A)$.

These operators satisfy the following equalities which are repeatedly used below.

Proposition 3.3**.**

For all $\varphi\in\Gamma(\wedge^{n}A^{*}\otimes A^{*}),n\geq 1,~{}~{}x,y\in\Gamma(A),\xi\in\Gamma(A^{*})$, we have

[TABLE]

Proof. We only give the proof of (26) and (27). Others can be proved similarly. For all $\varphi\in\Gamma(\wedge^{n}A^{*}\otimes A^{*}),\quad n\geq 1$ and $x,y,y_{1},y_{2},\cdots,y_{n}\in\Gamma(A)$, we have

[TABLE]

and

[TABLE]

Therefore, we have

[TABLE]

which implies that (26) holds.

Next we prove that (27) holds. On one hand, for all $\varphi\in\Gamma(\wedge^{n}A^{*}\otimes A^{*}),~{}n\geq 1$ and $x,y_{1},\cdots,y_{n+1}\in\Gamma(A)$, we have

[TABLE]

and

[TABLE]

On the other hand, by (22) and (23), we have

[TABLE]

Thus, (27) holds.

4 Left-symmetric bialgebroids

In this section, we introduce the concept of a left-symmetric bialgebroid and study its properties. We construct a left-symmetric algebroid using a symmetric tensor satisfying a condition, which can be viewed as a generalization of the S-equation introduced in [1] for left-symmetric bialgebras. In particular, we construct a left-symmetric bialgebroid using a pseudo-Hessian manifold.

Definition 4.1**.**

Let $(A,\cdot_{A},a_{A})$ and $(A^{*},\cdot_{A^{*}},a_{A^{*}})$ be two left-symmetric algebroids. Then $(A,A^{*})$ is a left-symmetric bialgebroid if for all $x,y\in\Gamma(A),~{}~{}\xi,\eta\in\Gamma(A^{*})$, the following equalities hold:

[TABLE]

where $\delta$ and $\delta_{*}$ are coboundary operators of left-symmetric algebroids $A$ and $A^{*}$ respectively and $\mathfrak{L}$ is the Lie derivative associated to a left-symmetric algebroid given by (15).

Remark 4.2**.**

By (15), it is not hard to see that a left-symmetric bialgebroid reduces to a left-symmetric bialgebra when the base manifold is a point. Thus, a left-symmetric bialgebroid can be viewed as a geometric generalization of a left-symmetric bialgebra. See [1] for more details about left-symmetric bialgebras.

Lemma 4.3**.**

Let $(A,A^{*})$ be a left-symmetric bialgebroid. For all $x\in\Gamma(A),~{}~{}\xi\in\Gamma(A^{*}),~{}~{}f\in C^{\infty}(M)$, we have

[TABLE]

Proof. By (20), $(\ref{pro 3 of lie der})$ and $(\ref{cond2})$, we have

[TABLE]

On the other hand, we have

[TABLE]

Thus, we have

[TABLE]

which implies that (34) holds. (35) can be proved similarly.

Recall that $R_{x}:A^{*}\longrightarrow A^{*}$ and $R_{\xi}:A\longrightarrow A$ are defined by

[TABLE]

Corollary 4.4**.**

Let $(A,A^{*})$ be a left-symmetric bialgebroid. For all $x\in\Gamma(A),~{}~{}\xi\in\Gamma(A^{*}),~{}~{}f\in C^{\infty}(M)$, we have

[TABLE]

Proof. By (34), we have

[TABLE]

which implies that (36) holds. (37) can be proved similarly.

Corollary 4.5**.**

Let $(A,A^{*})$ be a left-symmetric bialgebroid. For all $x\in\Gamma(A),~{}~{}\xi\in\Gamma(A^{*})$, we have

[TABLE]

Proof. For all $f\in C^{\infty}(M)$, by (36), we have

[TABLE]

which implies that (38) holds.

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid. Define

[TABLE]

For any $H\in\mathsf{Sym}^{2}(A)$, the bundle map $H^{\sharp}:A^{*}\longrightarrow A$ is given by $H^{\sharp}(\xi)(\eta)=H(\xi,\eta)$. We introduce $\llbracket H,H\rrbracket\in\wedge^{2}A\otimes A$ as follows:

[TABLE]

Suppose that $H^{\sharp}:A^{*}\longrightarrow A$ is nondegenerate. Then $(H^{\sharp})^{-1}:A\longrightarrow A^{*}$ is also a symmetric bundle map, which gives rise to an element, denoted by $H^{-1}$, in $\mathsf{Sym}^{2}(A^{*})$.

Proposition 4.6**.**

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid and $H\in\mathsf{Sym}^{2}(A)$. If $H$ is nondegenerate, then the following two statements are equivalent:

• $\rm(1)$

$\llbracket H,H\rrbracket=0$;

• $\rm(2)$

$\delta(H^{-1})=0.$**

Proof. By direct calculation, we have the following formula

[TABLE]

Thus, the conclusion follows immediately.

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid, and $H\in\mathsf{Sym}^{2}(A)$. Define

[TABLE]

Proposition 4.7**.**

With the above notations, for all $\xi,\eta\in\Gamma(A^{*})$, we have

[TABLE]

Proof. First, for all $\xi,\eta,\zeta\in\Gamma(A^{*})$, we have

[TABLE]

Thus, by (39), we have

[TABLE]

which finishes the proof.

By direct calculation, we have

Corollary 4.8**.**

For all $\xi,\eta\in\Gamma(A^{*})$, we have

[TABLE]

where ${[\cdot,\cdot]}_{H}$ is the commutator bracket of $\cdot_{H}$.

Theorem 4.9**.**

With the above notations, if $\llbracket H,H\rrbracket=0$, then $(A^{*},\cdot_{H},a_{A^{*}}=a_{A}\circ H^{\sharp})$ is a left-symmetric algebroid, and $H^{\sharp}$ is a left-symmetric algebroid homomorphism from $(A^{*},\cdot_{H},a_{A^{*}})$ to $(A,\cdot_{A},a_{A})$. Furthermore, $(A,A^{*})$ is a left-symmetric bialgebroid.

Proof. By direct calculation, we can show that $(\Gamma(A^{*}),\cdot_{H})$ is a left-symmetric algebra. Moreover, we have

[TABLE]

Thus, $(A^{*},\cdot_{H},a_{A}\circ H)$ is a left-symmetric algebroid. By $(\ref{homo})$, $H^{\sharp}$ is a left-symmetric algebroid homomorphism.

To obtain that $(A,A^{*})$ is a left-symmetric bialgebroid, we need to prove that $(\ref{cond1})$ and $(\ref{cond2})$ hold. By tedious calculation, $(\ref{cond1})$ can be obtained directly and $(\ref{cond2})$ is equivalent to the following equation:

[TABLE]

By direct calculation, we have

[TABLE]

which implies that $(\ref{homo2})$ holds.

Remark 4.10**.**

When $M$ is a point, $\llbracket H,H\rrbracket=0$ is exactly the S-equation introduced in [1]. Thus, the equation $\llbracket H,H\rrbracket=0$ can be viewed as a geometric generalization of the S-equation.

At the end of this section, we give some applications in Hessian geometry and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold.

Recall that a pseudo-Hessian metric $g$ is a pseudo-Riemannian metric $g$ on a flat manifold $(M,\nabla)$ such that $g$ can be locally expressed by $g_{ij}=\frac{\partial^{2}\varphi}{\partial x^{i}\partial x^{j}},$ where $\varphi\in C^{\infty}(M)$ and $\{x^{1},\cdots,x^{n}\}$ is an affine coordinate system with respect to $\nabla$, i.e.

[TABLE]

Then the pair $(\nabla,g)$ is called a pseudo-Hessian structure on $M$. A manifold $M$ with a pseudo-Hessian structure $(\nabla,g)$ is called a pseudo-Hessian manifold. See [26] for more details about pseudo-Hessian manifolds.

Proposition 4.11**.**

[26]* Let $(M,\nabla)$ be a flat manifold and $g$ a pseudo-Riemannian metric on $M$. Then the following conditions are equivalent:*

• $\rm(1)$

$\mathfrak{g}$* is a pseudo-Hessian metric;*

• $\rm(2)$

for all $x,y,z\in\Gamma(TM),$ there holds $\nabla_{x}g(y,z)=\nabla_{y}g(x,z),$ where $\nabla_{x}g(y,z)$ is given by

[TABLE]

Proposition 4.12**.**

Let $(M,\nabla)$ be a flat manifold and $g$ a pseudo-Riemannian metric on $M$. Then $(M,\nabla,g)$ is a pseudo-Hessian manifold if and only if $\delta g(x,y,z)=0$, where $\delta$ is the coboundary operator given by (7) associated to the left-symmetric algebroid $T_{\nabla}M$ given in Example 2.4.

Proof. By Proposition 4.11 and the following formula

[TABLE]

we can obtain the conclusion immediately.

By Proposition 4.12, Proposition 4.6 and Theorem 4.9, we can construct a left-symmetric bialgebroid from a pseudo-Hessian manifold.

Proposition 4.13**.**

Let $(M,\nabla,g)$ be a pseudo-Hessian manifold. Define $H\in\mathsf{Sym}^{2}(TM)$ by

[TABLE]

Then $(T^{*}M,\cdot_{H},H^{\sharp})$ is a left-symmetric algebroid, which we denote by $T^{*}_{H}M$, where $\cdot_{H}$ is given by (40). Furthermore, $H^{\sharp}$ is a left-symmetric algebroid homomorphism from $T^{*}_{H}M$ to $T_{\nabla}M$ and $(T_{\nabla}M,T^{*}_{H}M)$ is a left-symmetric bialgebroid.

Remark 4.14**.**

The above result is parallel to that $(TM,T^{*}_{\pi}M)$ is a Lie bialgebroid for any Poisson manifold $(M,\pi)$. See [9] for more details.

5 Equivalence between Manin triples for left-symmetric algebroids and left-symmetric bialgebroids

In this section, we introduce the notion of a Manin triple for left-symmetric algebroids as a geometric generalization of a Manin triple for left-symmetric algebras. We would like to point out that unlike the latter, the double structure of the former is a pre-symplectic algebroid rather than a left-symmetric algebroid. We show that Manin triples for left-symmetric algebroids are equivalent to left-symmetric bialgebroids. At the end of this section, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.

Definition 5.1**.**

A Manin triple for left-symmetric algebroids is a triple $(E;L_{1},L_{2})$, where $E$ is a pre-symplectic algebroid, $L_{1}$ and $L_{2}$ are transversal Dirac structures.

Example 5.2**.**

Let $(A,\cdot_{A},a_{A})$ be a left-symmetric algebroid and $(A\oplus A^{*},\star,\rho,(\cdot,\cdot)_{-})$ the corresponding pre-symplectic algebroid given in Example 2.10. Then $(A\oplus A^{*};A,A^{*})$ is a Manin triple for left-symmetric algebroids. **

More generally, we have

Theorem 5.3**.**

There is a one-to-one correspondence between Manin triples for left-symmetric algebroids and left-symmetric bialgebroids.

Proof. Follows form the following Proposition 5.4 and Proposition 5.5.

Suppose that both $(A,\cdot_{A},a_{A})$ and $(A^{*},\cdot_{A^{*}},a_{A^{*}})$ are left-symmetric algebroids. Let $E=A\oplus A^{*}$. We introduce a multiplication $\star:\Gamma(E)\times\Gamma(E)\longrightarrow\Gamma(E)$ by

[TABLE]

where $e_{1}=x_{1}+\xi_{1},e_{2}=x_{2}+\xi_{2}$, $\mathcal{L}$ is given by (1), $R$ is given by (23) and $(\cdot,\cdot)_{+}$ is given by (14). Let $\rho:E\rightarrow TM$ be the bundle map defined by $\rho=a_{A}+a_{A^{*}}$. That is,

[TABLE]

It is obvious that in this case the operator $D$ (see (9)) is given by

[TABLE]

where $\mathrm{d}:C^{\infty}(M)\rightarrow\Gamma({A^{*}})$ and $\mathrm{d}_{*}:C^{\infty}(M)\rightarrow\Gamma(A)$ are the usual differential operators associated to the sub-adjacent Lie algebroids $A^{c}$ and ${A^{*}}^{c}$ respectively.

Proposition 5.4**.**

With the above notations, let $(A,A^{*})$ be a left-symmetric bialgebroid. Then $(A\oplus A^{*},\star,\rho,(\cdot,\cdot)_{-})$ is a pre-symplectic algebroid, where the multiplication $\star$ is given by $(\ref{operation})$, $\rho=a_{A}+a_{A^{*}},$ and $(\cdot,\cdot)_{-}$ is given by (5).

Conversely, we have

Proposition 5.5**.**

Let $(E,\star,\rho,(\cdot,\cdot)_{-})$ be a pre-symplectic algebroid. Suppose that $L_{1}$ and $L_{2}$ are Dirac subbundles transversal to each other. Then $(L_{1},L_{2})$ is a left-symmetric bialgebroid, where $L_{2}$ is considered as the dual bundle of $L_{1}$ under the nondegenerate bilinear form $(\cdot,\cdot)_{-}$.

We prove some lemmas first.

Lemma 5.6**.**

Let $(A,\cdot_{A},a_{A})$ and $(A^{*},\cdot_{A^{*}},a_{A^{*}})$ be two left-symmetric algebroids. For all $e_{i}=x_{i}+\xi_{i}\in\Gamma(E),~{}i=1,2,3,4$, we have

[TABLE]

where $I_{1},I_{2},I_{3}$ is defined by

[TABLE]

Proof. First, we have

[TABLE]

By direct calculation, we have

[TABLE]

Therefore, we obtain

[TABLE]

It is easy to see that

[TABLE]

The proof is finished.

Lemma 5.7**.**

With the above notations, we have

[TABLE]

where $J_{1},J_{2},J_{3}$ is defined by

[TABLE]

Proof. First we have

[TABLE]

Similarly, by direct calculation, we have

[TABLE]

Therefore, we obtain

[TABLE]

It is easy to see that

[TABLE]

The proof is finished.

Lemma 5.8**.**

With the above notations, we have

[TABLE]

where $J_{1},J_{2},J_{3}$ is defined by

[TABLE]

Proof. By Lemma 5.7 and $T(e_{1},e_{2},e_{3})=-T(e_{2},e_{1},e_{3})$. The lemma follows immediately.

**Proof of Proposition 5.4 ** To prove Proposition 5.4, it is sufficient to verify that conditions (i) and (ii) in Definition 2.7 hold. First, condition (i) in Definition 2.7 follows directly from Lemma $\ref{associator 1}-\ref{associator 3}$ and properties of left-symmetric bialgebroids. Below, we show that condition (ii) in Definition 2.7 holds. On one hand, we have

[TABLE]

On the other hand, we have

[TABLE]

and

[TABLE]

which implies that condition (ii) in Definition 2.7 holds.

Proof of Proposition 5.5. Since the pairing $(\cdot,\cdot)_{-}$ is nondegenerate, $L_{2}$ is isomorphic to $L_{1}^{*}$, the dual bundle of $L_{1}$, via $\langle\xi,y\rangle=(\xi,y)_{-}$ for all $y\in\Gamma(L_{1}),~{}\xi\in\Gamma(L_{2})$. Under this isomorphism, the skew-symmetric bilinear form $(\cdot,\cdot)_{-}$ on $E$ is given by

[TABLE]

By Proposition 2.12, both $L_{1}$ and $L_{2}$ are left-symmetric algebroids, their anchors are given by $a=\rho\mid_{L_{1}}$ and $a_{*}=\rho\mid_{L_{2}}$ respectively. We shall use $\delta,~{}~{}\mathrm{d}$ and $\delta_{*},~{}~{}\mathrm{d}_{*}$ to denote their differential of left-symmetric algebroids and corresponding sub-adjacent Lie algebroids respectively.

By condition (ii) in Definition 2.7, we deduce that the bracket between $x\in\Gamma(L_{1})$ and $\xi\in\Gamma(L_{2})$ is given by

[TABLE]

Thus, the multiplication $\star$ is given by $(\ref{operation})$.

It follows from Lemma 5.6 that $I_{1}+I_{2}-I_{3}=0$. Since the anchor $\rho$ is a Lie algebroid homomorphism, we have $I_{2}=I_{3}=0$. Thus, $I_{1}=0$, i.e.

[TABLE]

Similarly, we have

[TABLE]

Thus, $(L_{1},L_{2})$ is a left-symmetric bialgebroid.

By Theorem 5.3, we have

Corollary 5.9**.**

Let $(A,A^{*})$ be a left-symmetric bialgebroid. Then $(A^{c},{A^{*}}^{c})$ is a matched pair of Lie algebroids and the bracket on ${A}^{c}\oplus{A^{*}}^{c}$ is defined by

[TABLE]

Furthermore, $({A}^{c}\oplus{A^{*}}^{c},\omega)$ is a symplectic Lie algebroid, where $\omega$ is given by (5).

By Proposition 4.13 and Proposition 5.4, we obtain

Proposition 5.10**.**

Let $(M,\nabla,g)$ be a pseudo-Hessian manifold. Then $(TM\oplus T^{*}M,\star,{\rm{id}}+H^{\sharp},(\cdot,\cdot)_{-})$ is a pre-symplectic algebroid, where for all $e_{1}=x_{1}+\xi_{1},e_{2}=x_{2}+\xi_{2}\in\Gamma(TM\oplus T^{*}M)$, the multiplication $\star$ is given by

[TABLE]

Here $\mathcal{L}$ is given by (1), $R$ is given by (23) and $(\cdot,\cdot)_{+}$ is given by (14).

Assume that $(A,A^{*})$ is a left-symmetric bialgebroid and $H\in\Gamma(A\otimes A)$. We denote by $G_{H}$ the graph of $H^{\sharp}$, i.e. $G_{H}=\{H^{\sharp}(\xi)+\xi|~{}~{}\forall\xi\in A^{*}\}$.

Theorem 5.11**.**

With the above notations, $G_{H}$ is a Dirac structure of the pre-symplectic algebroid $(A\oplus A^{*},\star,\rho,(\cdot,\cdot)_{-})$ given by Proposition 5.4 if and only if $H\in\mathsf{Sym}^{2}(A)$ and the following Maurer-Cartan type equation is satisfied:

[TABLE]

where $\llbracket H,H\rrbracket$ is given by (39).

Proof. First it is easy to see that $G_{H}$ is isotropic if and only if $H\in\mathsf{Sym}^{2}(A)$. By (44), we have

[TABLE]

which implies that

[TABLE]

Then by $(\ref{homo})$, we have

[TABLE]

Thus, $G_{H}$ is integrable if and only if for all $\xi,\eta\in\Gamma(A^{*})$,

[TABLE]

On the other hand, we have

[TABLE]

By $(\ref{integ})$ and (49), $G_{H}$ is a Dirac structure if and only if

[TABLE]

The proof is finished.