A Herglotz-based integrator for nonholonomic mechanical systems

1.   Introduction

In this paper, we study the categorification of $\mathsf{VB}$-Lie algebroids and $\mathsf{VB}$-Courant algebroids, and establish the relations between these higher structures and super representations of Lie 2-algebroids, tangent prolongations of Lie 2-algebroids, N-manifolds of degree 3, tangent prolongations of $\mathsf{CLWX}$ 2-algebroids and higher analogues of the string Lie 2-algebra.

1.1.   Lie $n$-algebroids, Courant algebroids and $\mathsf{CLWX}$ 2-algebroids

An NQ-manifold is an N-manifold $\mathcal{M}$ together with a degree $1$ vector field $Q$ satisfying $[Q, Q] = 0$. It is well known that a degree 1 NQ manifold corresponds to a Lie algebroid. Thus, people usually think that

Some work in this direction appeared in ^[54]. Strictly speaking, a Lie $n$-algebroid gives arise to an NQ-manifold only after a degree 1 shift, just as a Lie algebroid $A$ corresponds to a degree 1 NQ manifold $A[1]$. To make the shifting manifest, and to present a Lie $n$-algebroid in a way more used to differential geometers, that is, to use the language of vector bundles, the authors introduced the notion of a split Lie $n$-algebroid in ^[52] to study the integration of a Courant algebroid. The equivalence between the category of split NQ manifolds and the category of split Lie $n$-Lie algebroids was proved in ^[5]. The language of split Lie $n$-algebroids has slowly become a useful tool for differential geometers to study problems related to NQ-manifolds (^[14,24,25]). Since Lie 2-algebras are the categorification of Lie algebras (^[4]), we will view Lie 2-algebroids as the categorification of Lie algebroids.

To study the double of a Lie bialgebroid (^[42]), Liu, Weinstein and Xu introduced the notion of a Courant algebroid in ^[35]. See ^[44] for an alternative definition. There are many important applications of Courant algebroids, e.g. in generalized complex geometry (^[8,17,22]), Poisson geometry (^[33]), moment maps (^[9]), Poisson-Lie T-duality (^[47,48]) and topological field theory (^[46]). In ^[34], the authors introduced the notion of a $\mathsf{CLWX}$ 2-algebroid (named after Courant-Liu-Weinstein-Xu), which can be viewed as the categorification of a Courant algebroid. Furthermore, $\mathsf{CLWX}$ 2-algebroids are in one-to-one correspondence with QP-manifolds (symplectic NQ-manifolds) of degree 3, and have applications in the fields theory. See ^[23] for more details. The underlying algebraic structure of a $\mathsf{CLWX}$ 2-algebroid is a Leibniz 2-algebra, or a Lie 3-algebra. There is also a close relationship between $\mathsf{CLWX}$ 2-algebroids and the first Pontryagin classes of quadratic Lie 2-algebroids, which are represented by closed 5-forms. More precisely, as the higher analogue of the results given in ^[6,13], it was proved in ^[49] that the first Pontryagin class of a quadratic Lie algebroid is the obstruction of the existence of a $\mathsf{CLWX}$-extension.

1.2.   $\mathsf{VB}$-Lie algebroids and $\mathsf{VB}$-Courant algebroids

Double structures in geometry can be traced back to the work of Ehresmann on connection theory, and have been found many applications in Poisson geometry. See ^[40] for more details. We use the word "doublization" to indicate putting geometric structures on double vector bundles in the sequel. In ^[19], Gracia-Saz and Mehta introduced the notion of a $\mathsf{VB}$-Lie algebroid, which is equivalent to Mackenzie's $\mathcal{L} \mathcal{A}$-vector bundle (^[38]). A $\mathsf{VB}$-Lie algebroid is a Lie algebroid object in the category of vector bundles and one important property is that it is closely related to superconnection (also called representation up to homotopy ^[1,2]) of a Lie algebroid on a 2-term complex of vector bundles. Recently, the relation between $\mathsf{VB}$-algebroid morphisms and representations up to homotopy were studied in ^[15].

In his PhD thesis ^[32], Li-Bland introduced the notion of a $\mathsf{VB}$-Courant algebroid which is the doublization of a Courant algebroid ^[35], and established abstract correspondence between NQ-manifolds of degree 2 and $\mathsf{VB}$-Courant algebroids. Then in ^[24], Jotz Lean provided a more concrete description of the equivalence between the category of split Lie 2-algebroids and the category of decomposed $\mathsf{VB}$-Courant algebroids.

Double structures, such as double principle (vector) bundles (^{[12,16,26,30]}), double Lie algebroids (^{[18,37,38,39,41,55]}), double Lie groupoids (^[43]), $\mathsf{VB}$-Lie algebroids (^[7,19]) and $\mathsf{VB}$-Lie groupoids (^[7,20]) became more and more important recently and are widely studied. In particular, the Lie theory relating $\mathsf{VB}$-Lie algebroids and $\mathsf{VB}$-Lie groupoids, i.e. their relation via differentiation and integration, is established in ^[7].

1.3.   Summary of the results and outline of the paper

In this paper, we combine the aforementioned higher structures and double structures. First we introduce the notion of a $\mathsf{VB}$-Lie 2-algebroid, which can be viewed as the categorification of a $\mathsf{VB}$-Lie algebroid, or doublization of a Lie 2-algebroid:

We show that the tangent prolongation of a Lie 2-algebroid is a $\mathsf{VB}$-Lie 2-algebroid and the graded fat bundle associated to a $\mathsf{VB}$-Lie 2-algebroid is Lie 2-algebroid. Consequently, the graded jet bundle of a Lie 2-algebroid is also a Lie 2-algebroid. In ^[19], the authors showed that a $\mathsf{VB}$-Lie algebroid is equivalent to a flat superconnection (representation up to homotopy (^[1])) of a Lie algebroid on a 2-term complex of vector bundles after choosing a splitting. Now for a $\mathsf{VB}$-Lie 2-algebroid, we establish a higher analogous result, namely, we show that after choosing a splitting, it is equivalent to a flat superconnection of a Lie 2-algebroid on a 3-term complex of vector bundles.

Then we introduce the notion of a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid, which can be viewed as both the doublization of a $\mathsf{CLWX}$ 2-algebroid and the categorification of a $\mathsf{VB}$-Courant algebroid. More importantly, we show that after choosing a splitting, there is a one-to-one correspondence between $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroids and split Lie 3-algebroids (NQ-manifolds of degree 3). The tangent prolongation of a $\mathsf{CLWX}$ 2-algebroid is a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid naturally. We go on defining $E$-$\mathsf{CLWX}$ 2-algebroid, which can be viewed as the categorification of an $E$-Courant algebroid introduced in ^[11]. As a higher analogue of the result that associated to a $\mathsf{VB}$-Courant algebroid, there is an $E$-Courant algebroid ^[24,31], we show that on the graded fat bundle associated to a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid, there is an $E$-$\mathsf{CLWX}$ 2-algebroid structure naturally. Similar to the case of a $\mathsf{CLWX}$ 2-algebroid, an $E$-$\mathsf{CLWX}$ 2-algebroid also gives rise to a Lie 3-algebra naturally. Thus through the following procedure:

we can construct a Lie 3-algebra from a Lie 3-algebra. We obtain new interesting examples, including the higher analogue of the string Lie 2-algebra.

The paper is organized as follows. In Section 2, we recall double vector bundles, $\mathsf{VB}$-Lie algebroids and $\mathsf{VB}$-Courant algebroids. In Section 3, we introduce the notion of a $\mathsf{VB}$-Lie 2-algebroid, and show that both the graded side bundle and the graded fat bundle are Lie 2-algebroids. The tangent prolongation of a Lie 2-algebroid is a $\mathsf{VB}$-Lie 2-algebroid naturally. In Section 4, first we construct a strict Lie 3-algebroid $\mathrm{End}(\mathcal{E}) = (\mathrm{End}^{-2}(\mathcal{E}), \mathrm{End}^{-1}(\mathcal{E}), \mathfrak{D}(\mathcal{E}), \mathfrak p, \mathrm{d}, [\cdot, \cdot]_C)$ from a 3-term complex of vector bundles $\mathcal{E}:E_{-2}\stackrel{\pi}{\longrightarrow}E_{-1}\stackrel{\pi}{\longrightarrow}E_{0}$ and then we define a flat superconnection of a Lie 2-algebroid $\mathcal{A} = (A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ on this 3-term complex of vector bundles to be a morphism from $\mathcal{A}$ to $\mathrm{End}(\mathcal{E})$. We show that after choosing a splitting, $\mathsf{VB}$-Lie 2-algebroids one-to-one correspond to flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. In Section 5, we introduce the notion of a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid and show that after choosing a splitting, there is a one-to-one correspondence between $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroids and Lie 3-algebroids. In Section 6, we introduce the notion of an $E$-$\mathsf{CLWX}$ 2-algebroid and show that the graded fat bundle associated to a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid is an $E$-$\mathsf{CLWX}$ 2-algebroid naturally. In particular, the graded jet bundle of a $\mathsf{CLWX}$ 2-algebroid, which is the graded fat bundle of the tangent prolongation of this $\mathsf{CLWX}$ 2-algebroid, is a $T^*M$-$\mathsf{CLWX}$ 2-algebroid. We can also obtain a Lie 3-algebra from an $E$-$\mathsf{CLWX}$ 2-algebroid. In Section 7, we construct a Lie 3-algebra from a given Lie 3-algebra using the theories established in Section 5 and Section 6, and give interesting examples. In particular, we show that associated to a quadratic Lie 2-algebra, we can obtain a Lie 3-algebra, which can be viewed as the higher analogue of the string Lie 2-algebra.

2.   Preliminaries

See [40,Definition 9.1.1] for the precise definition of a double vector bundle. We denote a double vector bundle

with core $C$ by $(D; A, B; M)$. We use $D^B$ and $D^A$ to denote vector bundles $D\longrightarrow B$ and $D\longrightarrow A$ respectively. For a vector bundle $A$, both the tangent bundle $TA$ and the cotangent bundle $T^*A$ are double vector bundles:

A morphism of double vector bundles

consists of maps $\varphi$: $D\rightarrow D'$, $f_A:A\rightarrow A'$, $f_B:B\rightarrow B'$, $f_M:M\rightarrow M'$, such that each of $(\varphi, f_B)$, $(\varphi, f_A)$, $(f_A, f_M)$ and $(f_B, f_M)$ is a morphism of the relevant vector bundles.

The space of sections $\Gamma_B(D)$ of the vector bundle $D^B$ is generated as a $C^\infty(B)$-module by core sections $\Gamma_B^c(D)$ and linear sections $\Gamma_B^l(D)$. See ^[41] for more details. For a section $c: M\rightarrow C$, the corresponding core section $c^\dagger: B\rightarrow D$ is defined as

where $\bar{\cdot}$ means the inclusion $C\hookrightarrow D$. A section $\xi: B\rightarrow D$ is called linear if it is a bundle morphism from $B\rightarrow M$ to $D\rightarrow A$ over a section $X\in \Gamma(A)$. We will view $B^*\otimes C$ both as $\mathrm{Hom}(B, C)$ and $\mathrm{Hom}(C^*, B^*)$ depending on what it acts. Given $\psi\in \Gamma(B^*\otimes C)$, there is a linear section $\tilde{\psi}: B\rightarrow D$ over the zero section $0^A: M\rightarrow A$ given by

Note that $\Gamma_B^l(D)$ is locally free as a $C^\infty (M)$-module. Therefore, $\Gamma_B^l(D)$ is equal to $\Gamma(\hat{A})$ for some vector bundle $\hat{A}\rightarrow M$. The vector bundle $\hat{A}$ is called the fat bundle of the double vector bundle $(D; A, B; M)$. Moreover, we have the following short exact sequence of vector bundles over $M$

Definition 2.1. ([19,Definition 3.4]) A $\mathsf{VB}$-Lie algebroid is a double vector bundle $(D; A, B; M)$ equipped with a Lie algebroid structure $(D^B, a, [\cdot, \cdot]_D)$ such that the anchor $a:D\longrightarrow TB$ is linear, i.e. $a: (D; A, B; M)\longrightarrow (TB; TM, B; M)$ is a morphism of double vector bundles, and the Lie bracket $[\cdot, \cdot]_D$ is linear:

The vector bundle $A\longrightarrow M$ is then also a Lie algebroid, with the anchor $\mathfrak a$ and the bracket $[\cdot, \cdot]_A$ defined as follows: if $\xi_1, \xi_2$ are linear over $X_1, X_2\in\Gamma(A)$, then the bracket $[\xi_1, \xi_2]_D$ is linear over $[X_1, X_2]_A$.

Definition 2.2. ([32,Definition 3.1.1]) A $\mathsf{VB}$-Courant algebroid is a metric double vector bundle $(D; A, B; M)$ such that $({{D}^{B}},S,\text{ }\left[\!\left[ \cdot ,\cdot \right]\!\right],\rho )$ is a Courant algebroid and the following conditions are satisfied:

${\rm{(i)}}$ The anchor map $\rho:D\rightarrow TB$ is linear;

${\rm{(ii) }}$ The Courant bracket is linear. That is

Theorem 2.3. ([32,Proposition 3.2.1]) There is a one-to-one correspondence between Lie $2$-algebroids and $\mathsf{VB}$-Courant algebroids.

3.   $\mathsf{VB}$-Lie 2-algebroids

In this section, we introduce the notion of a $\mathsf{VB}$-Lie 2-algebroid, which can be viewed as the categorification of a $\mathsf{VB}$-Lie algebroid introduced in ^[19]. First we recall the notion of a Lie $n$-algebroid. See ^[28,29] for more information of $L_\infty$-algebras.

Definition 3.1. ([52,Definition 2.1]) A split Lie $n$-algebroid is a non-positively graded vector bundle $\mathcal{A} = A_0\oplus A_{-1}\oplus\cdots\oplus A_{-n+1}$ over a manifold $M$ equipped with a bundle map $a:A_0\longrightarrow TM$ (called the anchor), and $n+1$ many brackets $l_i:\Gamma(\wedge^i \mathcal{A})\longrightarrow \Gamma(\mathcal{A})$ with degree $2-i$ for $1\le i \le n+1$, such that

$1.$ $\Gamma(\mathcal{A})$ is an $n$-term $L_\infty$-algebra:

where the summation is taken over all $(i, k-i)$-unshuffles $Sh^{-1}_{i, k-i}$ with $i\geq1$ and "$\mathrm{Ksgn}(\sigma)$" is the Koszul sign for a permutation $\sigma\in S_k$, i.e.

$2.$ $l_2$ satisfies the Leibniz rule with respect to the anchor $a$:

$3.$ For $i\neq 2$, $l_i$'s are $C^\infty(M)$-linear.

Denote a split Lie $n$-algebroid by $(A_{-n+1}, \cdots, A_0, a, l_1, \cdots, l_{n+1})$, or simply by $\mathcal{A}$. We will only use a split Lie 2-algebroid $(A_{-1}, A_0, a, l_1, l_2, l_3)$ and a split Lie 3-algebroid $(A_{-2}, A_{-1}, A_0, a, l_1, l_2, l_3, l_4)$. For a split Lie $n$-algebroid, we have a generalized Chevalley-Eilenberg complex $(\Gamma(\mathsf{Symm}(\mathcal{A}[1])^*), \delta)$. See ^[5,52] for more details. Then $\mathcal{A}[1]$ is an NQ-manifold of degree $n$. A split Lie $n$-algebroid morphism $\mathcal{A} \to \mathcal{A}'$ can be defined to be a graded vector bundle morphism $f: \mathsf{Symm}(\mathcal{A}[1]) \to \mathsf{Symm}(\mathcal{A}'[1])$ such that the induced pull-back map $f^*: C(\mathcal{A}'[1]) \to C(\mathcal{A}[1])$ between functions is a morphism of NQ manifolds. However it is rather complicated to write down a morphism between split Lie $n$-algebroids in terms of vector bundles, anchors and brackets, please see [5,Section 4.1] for such details. We only give explicit formulas of a morphism from a split Lie 2-algebroid to a strict split Lie $3$-algebroid $(l_3 = 0, l_4 = 0)$ and this is what we will use in this paper to define flat superconnections.

Definition 3.2. Let $\mathcal{A} = (A_{-1}, A_0, a, l_1, l_{2}, l_3)$ be a split Lie $2$-algebroid and $\mathcal{A}' = (A_{-2}', A_{-1}', A_0', a', l_1', l_{2}')$ a strict split Lie $3$-algebroid. A morphism $F$ from $\mathcal{A}$ to $\mathcal{A}'$ consists of:

$\bullet$ a bundle map $F^{0}:A_{0}\longrightarrow A_{0}'$,

$\bullet$ a bundle map $F^{1}:A_{-1}\longrightarrow A_{-1}',$

$\bullet$ a bundle map $F^{2}_0:\wedge^2 A_{0} \longrightarrow A_{-1}'$,

$\bullet$ a bundle map $F^{2}_1:A_{0}\wedge A_{-1}\longrightarrow A_{-2}',$

$\bullet$ a bundle map $F^{3}: \wedge^3 A_0\longrightarrow A_{-2}',$

such that for all $X^0, Y^0, Z^0, X^0_i\in \Gamma(A_{0})$, $i = 1, 2, 3, 4$, $\; X^1, Y^1\in \Gamma(A_{-1}),$ we have

and

where $k < l$ and $\{k, l\}\cap \{i, j\} = \emptyset.$

Let $(A_{-1}, A_0, a, l_1, l_2, l_3)$ be a split Lie 2-algebroid. Then for all $X^0, Y^0\in\Gamma(A_0)$ and $X^1\in\Gamma(A_{-1})$, Lie derivatives $L^0_{X^0}:\Gamma(A^*_{-i})\longrightarrow \Gamma(A^*_{-i})$, $i = 0, 1$, $L^1_{X^1}:\Gamma(A_{-1}^*)\longrightarrow \Gamma(A^*_0)$ and $L^3_{X^0, Y^0}:\Gamma(A_{-1}^*)\longrightarrow \Gamma(A^*_0)$ are defined by

for all $\alpha^0\in\Gamma(A^*_0), \; \alpha^1\in\Gamma(A_{-1}^*), \; Y^1\in\Gamma(A_{-1}), \; Z^0\in\Gamma(A_0)$. If $(\mathcal{A}^*[1], \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ is also a split Lie 2-algebroid, we denote by $\mathcal{L}^0, \mathcal{L}^1, \mathcal{L}^3, \delta_*$ the corresponding operations.

A graded double vector bundle consists of a double vector bundle of degree $-1$ and a double vector bundle of degree $0$:

We denote a graded double vector bundle by $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B_{-1};&M_{-1}\\ D_{0};&A_{0}, B_{0};&M_0 \end{array}\right)$. Morphisms between graded double vector bundles can be defined in an obvious way. We will denote by $\mathcal{D}$ and $\mathcal{A}$ the graded vector bundles $D_0^B\oplus D_{-1}^B$ and $A_0\oplus A_{-1}$ respectively. Now we are ready to introduce the main object in this section.

Definition 3.3. A $\mathsf{VB}$-Lie $2$-algebroid is a graded double vector bundle

equipped with a Lie $2$-algebroid structure $(D_{-1}^B, D_0^B, a, l_1, l_2, l_3)$ on $\mathcal{D}$ such that

${\rm(i)}$ The anchor $a:D_0\longrightarrow TB$ is linear, i.e. we have a bundle map $\mathfrak a:A_0\longrightarrow TM$ such that $(a; \mathfrak a, {\rm{id}_B}; {\rm{id}_M})$ is a double vector bundle morphism (see Diagram ${{\rm(i)}}$);

${\rm(ii)} \ l_1$ is linear, i.e. we have a bundle map $\mathfrak l_1:A_{-1}\longrightarrow A_0$ such that $(l_1; \mathfrak l_1, {\rm{id}_B}; {\rm{id}_M})$ is a double vector bundle morphism (see Diagram ${{\rm(ii)}}$);

${\rm(iii)}$ $l_2$ is linear, i.e.

${\rm(iv)}$ $l_3$ is linear, i.e.

Since Lie 2-algebroids are the categorification of Lie algebroids, $\mathsf{VB}$-Lie 2-algebroids can be viewed as the categorification of $\mathsf{VB}$-Lie algebroids.

Recall that if $(D; A, B; M)$ is a $\mathsf{VB}$-Lie algebroid, then $A$ is a Lie algebroid. The following result is its higher analogue.

Theorem 3.4. Let $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$ be a $\mathsf{VB}$-Lie $2$-algebroid. Then

is a split Lie $2$-algebroid, where $\mathfrak l_2$ is defined by the property that if $\xi^0_1, \; \xi^0_2, \; \xi^0 \in\Gamma^l_B(D_0)$ are linear sections over $X^0_1, \; X^0_2, \; X^0\in\Gamma(A_0)$, and $\xi^1\in\Gamma^l_B(D_{-1})$ is a linear section over $X^1\in\Gamma(A_{-1})$, then $l_2(\xi^0_1, \xi^0_2)\in\Gamma^l_B(D_0)$ is a linear section over $\mathfrak l_2(X^0_1, X^0_2)\in\Gamma(A_0)$ and $l_2(\xi^0, \xi^1)\in \Gamma^l_B(D_{-1})$ is a linear section over $\mathfrak l_2(X^0, X^1)\in\Gamma(A_{-1})$. Similarly, $\mathfrak l_3$ is defined by the property that if $\xi^0_1, \; \xi^0_2, \; \xi^0_3\in\Gamma^l_B(D_0)$ are linear sections over $X^0_1, \; X^0_2, \; X^0_3\in\Gamma(A_0)$, then $l_3(\xi^0_1, \xi^0_2, \xi^0_3)\in\Gamma^l_B(D_{-1})$ is a linear section over $\mathfrak l_3(X^0_1, X^0_2, X^0_3)\in\Gamma(A_{-1})$.

Proof. Since $l_2$ is linear, for any $\xi^i\in\Gamma^l_B(D_{-i})$ satisfying $\pi^{A_{-i}}(\xi^i) = 0$, we have

This implies that $\mathfrak l_2$ is well-defined. Similarly, $\mathfrak l_3$ is also well-defined.

By the fact that $l_1:D_{-1}\longrightarrow D_0$ is a double vector bundle morphism over $\mathfrak l_1:A_{-1}\longrightarrow A_0$, we can deduce that $(\Gamma(A_{-1}), \Gamma(A_0), \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ is a Lie 2-algebra. We only give a proof of the property

The other conditions in the definition of a Lie 2-algebra can be proved similarly. In fact, let $\xi^0\in\Gamma^l_B(D_0), \; \xi^1\in\Gamma^l_B(D_{-1})$ be linear sections over $X^0, X^1$ respectively, then by the equality $l_1(l_2(\xi^0, \xi^1)) = l_2(\xi^0, l_1(\xi^1))$, we have

Since $l_1:D_{-1}\longrightarrow D_0$ is a double vector bundle morphism over $\mathfrak l_1:A_{-1}\longrightarrow A_0$, the left hand side is equal to

and the right hand side is equal to

Thus, we deduce that (3.2) holds.

Finally, for all $X^0\in\Gamma(A_0)$, $Y^i\in\Gamma(A_{-i})$ and $f\in C^{\infty}(M)$, let $\xi^0\in\Gamma^l_B(D_0)$ and $\eta^i\in\Gamma^l_B(D_{-i}), \; i = 0, 1$ be linear sections over $X^0$ and $Y^i$. Then $q_B^*(f)\eta^i$ is a linear section over $fY^i$. By the fact that $a$ is a double vector bundle morphism over $\mathfrak a$, we have

Therefore, $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ is a Lie $2$-algebroid.

Remark 1. By the above theorem, we can view a $\mathsf{VB}$-Lie $2$-algebroid as a Lie $2$-algebroid object in the category of double vector bundles.

Consider the associated graded fat bundle $\hat{A}_{-1}\oplus \hat{A}_0$, obviously we have

Proposition 1. Let $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$ be a $\mathsf{VB}$-Lie $2$-algebroid. Then $(\hat{A}_{-1}, \hat{A}_0, \hat{a}, \hat{l}_1, \hat{l}_2, \hat{l}_3)$ is a split Lie $2$-algebroid, where $\hat{a} = \mathfrak a\circ \mathrm{pr}$ and $\hat{l}_1, \; \hat{l}_2, \; \hat{l}_3$ are the restriction of $l_1, \; l_2, \; l_3$ on linear sections respectively.

Consequently, we have the following exact sequences of split Lie $2$-algebroids:

It is helpful to give the split Lie 2-algebroid structure on $B^*\otimes C_{-1}\oplus B^*\otimes C_0$. Since $l_1$ is linear, it induces a bundle map $l_1^C:C_{-1}\longrightarrow C_0$. The restriction of $\hat{l}_1$ on $B^*\otimes C_{-1}$ is given by

Since the anchor $a:D_0\longrightarrow TB$ is a double vector bundle morphism, it induces a bundle map $\varrho:C_0\longrightarrow B$ via

Then by the Leibniz rule, we deduce that the restriction of $\hat{l}_2$ on $\Gamma(B^*\otimes C_{-1}\oplus B^*\otimes C_0)$ is given by

for all $\phi^0, \psi^0\in\Gamma(B^*\otimes C_{0}) = \Gamma(\mathrm{Hom}(B, C_{0}))$, $\psi^1\in \Gamma(B^*\otimes C_{-1}) = \Gamma(\mathrm{Hom}(B, C_{-1}))$. Since $l_3$ is linear, the restriction of $l_3$ on $B^*\otimes C_{-1}\oplus B^*\otimes C_0$ vanishes. Obviously, the anchor is trivial. Thus, the split Lie 2-algebroid structure on $B^*\otimes C_{-1}\oplus B^*\otimes C_0$ is exactly given by (3.4), (3.6) and (3.7). Therefore, $B^*\otimes C_{-1}\oplus B^*\otimes C_0$ is a graded bundle of strict Lie 2-algebras.

An important example of $\mathsf{VB}$-Lie algebroids is the tangent prolongation of a Lie algebroid. Now we explore the tangent prolongation of a Lie 2-algebroid. Recall that for a Lie algebroid $A\stackrel{ }{\longrightarrow}M$, $TA$ is a Lie algebroid over $TM$. A section $\sigma:M\longrightarrow A$ gives rise to a linear section $\sigma_T\triangleq d\sigma:TM\longrightarrow TA$ and a core section $\sigma_C:TM\longrightarrow TA$ by contraction. Any section of $TA$ over $TM$ is generated by such sections. A function $f\in C^{\infty}(M)$ induces two types of functions on $TM$ by

where $q:TM\longrightarrow M$ is the projection. We have the following relations about the module structure:

In particular, for $A = TM$, we have

for all $X\in \mathfrak X(M).$ See [32,Example 2.5.4] and ^[40] for more details.

Now for split Lie 2-algebroids, we have

Proposition 2. Let $\mathcal{A} = (A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ be a split Lie $2$-algebroid. Then

is a split Lie $2$-algebroid over $TM$, where $a:TA_0\longrightarrow TTM$ is given by

$l_1:\Gamma_{TM}(TA_{-1})\longrightarrow \Gamma_{TM}(TA_0)$ is given by

$l_2:\Gamma_{TM}(TA_{-i})\times \Gamma_{TM}(TA_{-j})\longrightarrow \Gamma_{TM}(TA_{-(i+j)})$ is given by

and $l_3:\wedge^3\Gamma_{TM}(TA_{0})\longrightarrow \Gamma_{TM}(TA_{-1})$ is given by

and $l_3(\sigma^0_T, \tau^0_C, \varsigma^0_C) = 0,$ for all $\sigma^0, \tau^0, \varsigma^0\in\Gamma(A_0)$ and $\sigma^1, \tau^1\in\Gamma(A_{-1}).$ Moreover, we have the following $\mathsf{VB}$-Lie $2$-algebroid:

Proof. By the fact that $\mathcal{A} = (A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ is a split Lie $2$-algebroid, it is straightforward to deduce that $(TA_{-1}, TA_0, a, l_1, l_2, l_3)$ is a split Lie $2$-algebroid over $TM$. Moreover, $a, \; l_1, \; l_2, \; l_3$ are all linear, which implies that it is a $\mathsf{VB}$-Lie 2-algebroid.

The associated fat bundles of double vector bundles $(TA_{-1};A_{-1}, TM; M)$ and $(TA_{0};A_{0}, TM; M)$ are the jet bundles $\mathfrak{J} A_{-1}$ and $\mathfrak{J} A_0$ respectively. By Proposition 2 and Proposition 1, we obtain the following result, which is the higher analogue of the fact that the jet bundle of a Lie algebroid is a Lie algebroid.

Corollary 1. Let $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ be a split Lie $2$-algebroid. Then we obtain that $(\mathfrak{J} A_{-1}, \mathfrak{J} A_0, \hat{a}, \hat{l}_1, \hat{l}_2, \hat{l}_3)$ is a split Lie $2$-algebroid, where $\hat{a}, \; \hat{l}_1, \; \hat{l}_2, \; \hat{l}_3$ is given by

for all $\sigma^0, \; \tau^0, \; \zeta^0\in\Gamma(A_0)$ and $\tau^1\in\Gamma(A_{-1})$.

4.   Superconnections of a split Lie $2$-algebroid on a $3$-term complex of vector bundles

In the section, we introduce the notion of a superconnection of a split Lie $2$-algebroid on a $3$-term complex of vector bundles, which generalizes the notion of a superconnection of a Lie algebroid on a $2$-term complex of vector bundles studied in ^[19]. We show that a $\mathsf{VB}$-Lie $2$-algebroid structure on a split graded double vector bundle is equivalent to a flat superconnection of a split Lie $2$-algebroid on a $3$-term complex of vector bundles.

Denote a 3-term complex of vector bundles $E_{-2}\stackrel{\pi}{\longrightarrow}E_{-1}\stackrel{\pi}{\longrightarrow}E_{0}$ by $\mathcal{E}$. Sections of the covariant differential operator bundle $\mathfrak{D}(\mathcal{E})$ are of the form $\mathfrak d = (\mathfrak d_0, \mathfrak d_1, \mathfrak d_2)$, where $\mathfrak d_i:\Gamma(E_{-i})\longrightarrow\Gamma(E_{-i})$ are $\mathbb R$-linear maps such that there exists $X\in \mathfrak X(M)$ satisfying

Equivalently, $\mathfrak{D}(\mathcal{E}) = \mathfrak{D}(E_0)\times_{TM} \mathfrak{D}(E_{-1})\times_{TM} \mathfrak{D}(E_{-2})$. Define $\mathfrak p: \mathfrak{D}(\mathcal{E})\longrightarrow TM$ by

Then the covariant differential operator bundle $\mathfrak{D}(\mathcal{E})$ fits the following exact sequence:

Denote by $\mathrm{End}^{-1}(\mathcal{E}) = \mathrm{Hom}(E_0, E_{-1})\oplus \mathrm{Hom}(E_{-1}, E_{-2})$. Denote by $\mathrm{End}^{-2}(\mathcal{E}) = \mathrm{Hom}(E_0, E_{-2})$. Define $\mathrm{d}: \mathrm{End}^{-2}(\mathcal{E})\longrightarrow \mathrm{End}^{-1}(\mathcal{E})$ by

and define $\mathrm{d}: \mathrm{End}^{-1}(\mathcal{E})\longrightarrow \mathfrak{D}(\mathcal{E})$ by

Then we define a degree $0$ graded symmetric bracket operation $[\cdot, \cdot]_C$ on the section space of the graded bundle $\mathrm{End}^{-2}(\mathcal{E})\oplus \mathrm{End}^{-1}(\mathcal{E})\oplus \mathfrak{D}(\mathcal{E})$ by

Denote by $\mathfrak{D}_\pi(\mathcal{E})\subset \mathfrak{D}(\mathcal{E})$ the subbundle of $\mathfrak{D}(\mathcal{E})$ whose section $\mathfrak d\in\Gamma(\mathfrak{D}_\pi(\mathcal{E}))$ satisfying $\pi\circ \mathfrak d = \mathfrak d\circ\pi$, or in term of components,

It is obvious that $\Gamma(\mathfrak{D}_\pi(\mathcal{E}))$ is closed under the bracket operation $[\cdot, \cdot]_C$ and

Then it is straightforward to verify that

Theorem 4.1. Let $E_{-2}\stackrel{\pi}{\longrightarrow}E_{-1}\stackrel{\pi}{\longrightarrow}E_{0}$ be a $3$-term complex of vector bundles over $M$. Then $(\mathrm{End}^{-2}(\mathcal{E}), \mathrm{End}^{-1}(\mathcal{E}), \mathfrak{D}_\pi(\mathcal{E}), \mathfrak p, \mathrm{d}, [\cdot, \cdot]_C)$ is a strict split Lie $3$-algebroid.

With above preparations, we give the definition of a superconnection of a split Lie 2-algebroid on a $3$-term complex of vector bundles as follows.

Definition 4.2 A superconnection of a split Lie $2$-algebroid $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ on a $3$-term complex of vector bundles $E_{-2}\stackrel{\pi}{\longrightarrow}E_{-1}\stackrel{\pi}{\longrightarrow}E_{0}$ consists of:

$\bullet$ a bundle morphism $F^0:A_0\longrightarrow \mathfrak{D}_\pi(\mathcal{E})$,

$\bullet$ a bundle morphism $F^1:A_{-1}\longrightarrow \mathrm{End}^{-1}(\mathcal{E})$,

$\bullet$ a bundle morphism $F^2_0:\wedge^2A_{0}\longrightarrow \mathrm{End}^{-1}(\mathcal{E})$,

$\bullet$ a bundle morphism $F^2_1: A_{0}\wedge A_{-1}\longrightarrow \mathrm{End}^{-2}(\mathcal{E})$,

$\bullet$ a bundle morphism $F^3: \wedge^3 A_{0} \longrightarrow \mathrm{End}^{-2}(\mathcal{E})$.

A superconnection is called flat if $(F^0, F^1, F^2_0, F^2_1, F^3)$ is a Lie $n$-algebroid morphism from the split Lie $2$-algebroid $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ to the strict split Lie $3$-algebroid $(\mathrm{End}^{-2}(\mathcal{E}), \mathrm{End}^{-1}(\mathcal{E}), \mathfrak{D}_\pi(\mathcal{E}), \mathfrak p, \mathrm{d}, [\cdot, \cdot]_C).$

Remark 2. If the split Lie $2$-algebroid reduces to a Lie algebroid $A$ and the $3$-term complex reduces to a $2$-term complex $E_{-1}\stackrel{\pi}{\longrightarrow} E_0$, a superconnection will only consists of

$\bullet$ a bundle morphism $F^0 = (F^0_0, F^0_1):A\longrightarrow \mathfrak{D}_\pi(\mathcal{E})$,

$\bullet$ a bundle morphism $F^2_0:\wedge^2A_{0}\longrightarrow \mathrm{Hom}(E_0, E_{-1})$.

Thus, we recover the notion of a superconnection (also called representation up to homotopy if it is flat) of a Lie algebroid on a $2$-term complex of vector bundles. See ^[1,19] for more details.

Now we come back to $\mathsf{VB}$-Lie $2$-algebroids. Let $(D_{-1}^B, D_0^B, a, l_1, l_2, l_3)$ be a $\mathsf{VB}$-Lie $2$-algebroid structure on the graded double vector bundle $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$. Recall from Theorem 3.4 and Proposition 1 that both $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ and $(\hat{A}_{-1}, \hat{A}_0, \hat{a}, \hat{l}_1, \hat{l}_2, \hat{l}_3)$ are split Lie $2$-algebroids.

Choose a horizontal lift $s = (s_0, s_1):A_0\oplus A_{-1}\longrightarrow \hat{A}_0\oplus \hat{A}_{-1}$ of the short exact sequence of split Lie 2-algebroids (3.3). Define $\nabla^B:A_0\longrightarrow \mathfrak{D}(B)$ by

Since for all $\phi^0\in\Gamma(B^*\otimes C_0)$, we have $\hat{a}(\phi^0) = 0$, it follows that $\nabla^B$ is well-defined.

We define $\nabla^0:A_0\longrightarrow \mathfrak{D}(C_0)$ and $\nabla^1:A_0\longrightarrow \mathfrak{D}(C_{-1})$ by

for all $X^0\in\Gamma(A_0), \; c^0\in\Gamma(C_0), \; c^1\in\Gamma(C_{-1}).$ Define $\Upsilon^1:A_{-1}\longrightarrow \mathrm{Hom}(B, C_0)$ and $\Upsilon^2:A_{-1}\longrightarrow \mathrm{Hom}(C_0, C_{-1})$ by

for all $X^1\in\Gamma(A_{-1}), \; c^0\in\Gamma(C_0).$ Since $l_2$ is linear, $\nabla^0$, $\nabla^1$ and $\Upsilon$ are well-defined.

Define $R^0:\wedge^2\Gamma(A_0)\longrightarrow \Gamma(\mathrm{Hom}(B, C_0))$, $\Lambda:\wedge^2\Gamma(A_0)\longrightarrow \Gamma(\mathrm{Hom}(C_0, C_{-1}))$ and $R^1:\Gamma(A_0)\wedge\Gamma(A_{-1})\longrightarrow\Gamma(\mathrm{Hom}(B, C_{-1}))$ by

for all $X^0, \; Y^0\in\Gamma(A_0)$ and $Y^1\in \Gamma(A_{-1})$ Finally, define $\Xi:\wedge^3\Gamma(A_0)\longrightarrow \mathrm{Hom}(B, C_{-1})$ by

By the equality $l_1l_2(s_0(X^0), c^1) = l_2(s_0(X^0), l_1^C(c^1))$, we obtain

By the fact that $a:D_0\longrightarrow TB$ preserves the bracket operation, we obtain

which implies that

By (4.14) and (4.15), we deduce that $(\nabla^B_{X^0}, \nabla^0_{X^0}, \nabla^1_{X^0})\in \mathfrak{D} (\mathcal{E})$, where $\mathcal{E}$ is the 3-term complex of vector bundles $C_{-1}\stackrel{l_1^C}{\longrightarrow}C_{0}\stackrel{\varrho}{\longrightarrow}B$. Then we obtain a superconnection $(F^0, F^1, F^2_0, F^2_1, F^3)$ of the Lie 2-algebroid $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ on the $3$-term complex of vector bundles $C_{-1}\stackrel{l_1^C}{\longrightarrow}C_{0}\stackrel{\varrho}{\longrightarrow}B$, where

Theorem 4.3. There is a one-to-one correspondence between $\mathsf{VB}$-Lie $2$-algebroids $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$ and flat superconnections $(F^0, F^1, F^2_0, F^2_1, F^3)$ of the split Lie $2$-algebroid $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ on the $3$-term complex of vector bundles $C_{-1}\stackrel{l_1^C}{\longrightarrow}C_{0}\stackrel{\varrho}{\longrightarrow}B$ by choosing a horizontal lift $s = (s_0, s_1):A_0\oplus A_{-1}\longrightarrow \hat{A}_0\oplus \hat{A}_{-1}$.

Proof. First it is obvious that

Using equalities $\mathfrak a\circ \mathfrak l_1 = 0$ and $a\circ l_1 = 0$, we have

which implies that

For $\nabla^0$, we can obtain

For $\nabla^1$, we have

By (4.17), (4.18) and (4.19), we deduce that

By straightforward computation, we have

which implies that

Similarly, we have

which implies that

and

which implies that

By (4.21), (4.22) and (4.23), we obtain

By the equality

we obtain

Furthermore, we have

By (4.25) and (4.26), we deduce that

By straightforward computation, we have

By the equality

we deduce that

By the equality

we deduce that

By (4.29) and (4.30), we obtain

Then by the equality

we deduce that

Finally, by the equality

we deduce that

By (4.16), (4.20), (4.24), (4.27), (4.28), (4.31)-(4.33), we deduce that $(F^0, F^1, F^2_0, F^2_1, F^3)$ is a morphism from the split Lie 2-algebroid $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ to the strict split Lie 3-algebroid

Conversely, let $(A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3)$ be a split Lie 2-algebroid and $(F^0, F^1, F^2_0, F^2_1, F^3)$ a flat superconnection on the $3$-term complex $C_{-1}\stackrel{l_1^C}{\longrightarrow}C_{0}\stackrel{\varrho}{\longrightarrow}B$. Then we can obtain a $\mathsf{VB}$-Lie $2$-algebroid structure on the split graded double vector bundle $\left(\begin{array}{ccc} A_{-1}\oplus B\oplus C_{-1};&A_{-1}, B; &M\\ A_0\oplus B\oplus C_0;&A_{0}, B; &M \end{array}\right).$ We leave the details to readers. The proof is finished.

5.   $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroids

In this section, first we recall the notion of a $\mathsf{CLWX}$ 2-algebroid. Then we explore what is a metric graded double vector bundle, and introduce the notion of a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid, which can be viewed as the categorification of a $\mathsf{VB}$-Courant algebroid introduced in ^[32].

As a model for "Leibniz algebras that satisfy Jacobi identity up to all higher homotopies", the notion of a strongly homotopy Leibniz algebra, or a $Lod_\infty$-algebra was given in ^[36] by Livernet, which was further studied by Ammar and Poncin in ^[3]. In ^[50], the authors introduced the notion of a Leibniz 2-algebra, which is the categorification of a Leibniz algebra, and proved that the category of Leibniz 2-algebras and the category of 2-term $Lod_\infty$-algebras are equivalent. Due to this reason, a 2-term $Lod_\infty$-algebra will be called a Leibniz 2-algebra directly in the sequel.

Definition 5.1. (^[34]) A $\mathsf{CLWX}$ $2$-algebroid is a graded vector bundle $\mathcal{E} = E_{-1}\oplus E_0$ over $M$ equipped with a non-degenerate graded symmetric bilinear form $S$ on $\mathcal{E}$, a bilinear operation $\diamond:\Gamma(E_{-i})\times \Gamma(E_{-j})\longrightarrow \Gamma(E_{-(i+j)})$, $0\leq i+j\leq 1$, which is skewsymmetric on $\Gamma(E_0)\times \Gamma(E_0)$, an $E_{-1}$-valued $3$-form $\Omega$ on $E_0$, two bundle maps $\partial:E_{-1}\longrightarrow E_0$ and $\rho:E_0\longrightarrow TM$, such that $E_{-1}$ and $E_0$ are isotropic and the following axioms are satisfied:

${\rm(i)} \ (\Gamma(E_{-1}), \Gamma(E_0), \partial, \diamond, \Omega)$ is a Leibniz $2$-algebra;

${\rm(ii)}$ for all $e\in\Gamma(\mathcal{E})$, $e\diamond e = \frac{1}{2} \mathcal{D} S(e, e)$, where $\mathcal{D}: C^{\infty}(M)\longrightarrow \Gamma(E_{-1})$ is defined by

${\rm(iii)}$ for all $e^1_1, e^1_2\in\Gamma(E_{-1})$, $S(\partial(e^1_1), e^1_2) = S(e^1_1, \partial(e^1_2))$;

${\rm(iv)}$ for all $e_1, e_2, e_3\in\Gamma(\mathcal{E})$, $\rho(e_1)S(e_2, e_3) = S(e_1\diamond e_2, e_3)+S(e_2, e_1\diamond e_3)$;

${\rm(v)}$ for all $e^0_1, e^0_2, e^0_3, e^0_4\in\Gamma(E_0)$, $S(\Omega(e^0_1, e^0_2, e^0_3), e^0_4) = -S(e^0_3, \Omega(e^0_1, e^0_2, e^0_4))$.

Denote a $\mathsf{CLWX}$ 2-algebroid by $(E_{-1}, E_0, \partial, \rho, S, \diamond, \Omega)$, or simply by $\mathcal{E}$. Since the section space of a $\mathsf{CLWX}$ 2-algebroid is a Leibniz 2-algebra, the section space of a Courant algebroid is a Leibniz algebra and Leibniz 2-algebras are the categorification of Leibniz algebras, we can view $\mathsf{CLWX}$ 2-algebroids as the categorification of Courant algebroids.

As a higher analogue of Roytenberg's result about symplectic NQ manifolds of degree 2 and Courant algebroids (^[45]), we have

Theorem 5.2. (^[34]) Let $(T^*[3]A^*[2], \Theta)$ be a symplectic NQ manifold of degree $3$, where $A$ is an ordinary vector bundle and $\Theta$ is a degree $4$ function on $T^*[3]A^*[2]$ satisfying $\{\Theta, \Theta\} = 0$. Here $\{\cdot, \cdot\}$ is the canonical Poisson bracket on $T^*[3]A^*[2]$. Then $(A^*[1], A, \partial, \rho, S, \diamond, \Omega)$ is a $\mathsf{CLWX}$ $2$-algebroid, where the bilinear form $S$ is given by

and $\partial$, $\rho$, $\diamond$ and $\Omega$ are given by derived brackets. More precisely, we have

See ^[27,53] for more information of derived brackets. Note that various kinds of geometric structures were obtained in the study of QP manifolds of degree 3, e.g. Grutzmann's $H$-twisted Lie algebroids ^[21] and Ikeda-Uchino's Lie algebroids up to homotopy ^[23].

Definition 5.3. A metric graded double vector bundle is a graded double vector bundle $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$ equipped with a degree $1$ nondegenerate graded symmetric bilinear form $S$ on the graded bundle $D^B_{-1}\oplus D^B_0$ such that it induces an isomorphism between graded double vector bundles

where $\star B$ means dual over $B$.

Given a metric graded double vector bundle, we have

In the sequel, we will always identify $C_0$ with $A_{-1}^*$, $C_{-1}$ with $A_0^*$. Thus, a metric graded double vector bundle is of the following form:

Now we are ready to put a $\mathsf{CLWX}$ 2-algebroid structure on a graded double vector bundle.

Definition 5.4. A $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid is a metric graded double vector bundle

equipped with a $\mathsf{CLWX}$ $2$-algebroid structure $(D_{-1}^B, D_0^B, \partial, \rho, S, \diamond, \Omega)$ such that

${\rm(i)}$ $\partial$ is linear, i.e. there exists a unique bundle map $\overline{\partial}:A_{-1}\longrightarrow A_0$ such that $\partial: D_{-1}\longrightarrow D_0$ is a double vector bundle morphism over $\overline{\partial}:A_{-1}\longrightarrow A_0$ (see Diagram ${{\rm(iii)}}$);

${\rm(ii)}$ the anchor $\rho$ is a linear, i.e. there exists a unique bundle map $\overline{\rho}:A_{0}\longrightarrow TM$ such that $\rho:D_0\longrightarrow TB$ is a double vector bundle morphism over $\overline{\rho}:A_{0}\longrightarrow TM$ (see Diagram ${{\rm(iv)}}$);

${\rm(iii)}$ the operation $\diamond$ is linear;

${\rm(iv)}$ $\Omega$ is linear.

Since a $\mathsf{CLWX}$ $2$-algebroid can be viewed as the categorification of a Courant algebroid, we can view a $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid as the categorification of a $\mathsf{VB}$-Courant algebroid.

Example 1. Let $(A_{-1}, A_0, a, l_1, l_2, l_3)$ be a Lie 2-algebroid. Let $E_0 = A_0\oplus A^*_{-1}$, $E_{-1} = A_{-1}\oplus A^*_{0}$ and $\mathcal{E} = E_0\oplus E_{-1}$. Then $(E_{-1}, E_0, \partial, \rho, S, \diamond, \Omega)$ is a $\mathsf{CLWX}$ $2$-algebroid, where $\partial:E_{-1}\longrightarrow E_0$ is given by

$\rho:E_0\longrightarrow TM$ is given by

the symmetric bilinear form $S = (\cdot, \cdot)_+$ is given by

the operation $\diamond$ is given by

and the $E_{-1}$-valued $3$-form $\Omega$ is defined by

where $L^0, L^1, L^3$ are given by (3.1). It is straightforward to see that this $\mathsf{CLWX}$ 2-algebroid gives rise to a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid:

Example 2. For any manifold $M$, $(T^*[1]M, TM, \partial = 0, \rho = { \rm{id}}, S, \diamond, \Omega = 0)$ is a $\mathsf{CLWX}$ 2-algebroid, where $S$ is the natural symmetric pairing between $TM$ and $T^*M$, and $\diamond$ is the standard Dorfman bracket given by

See [34,Remark 3.4] for more details. In particular, for any vector bundle $E$, $(T^*E^*, TE^*, \partial = 0, \rho = { \rm{id}}, S, \diamond, \Omega = 0)$ is a $\mathsf{CLWX}$ 2-algebroid, which gives rise to a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid:

We have a higher analogue of Theorem 2.3:

Theorem 5.5. There is a one-to-one correspondence between split Lie $3$-algebroids and split $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroids.

Proof. Let $\mathcal{A} = (A_{-2}, A_{-1}, A_0, a, l_1, l_2, l_3, l_4)$ be a split Lie 3-algebroid. Then $T^*[3] \mathcal{A}[1]$ is a symplectic NQ manifold of degree 3. Note that

where $A_0\times_MA_{-1}^*\times_M A_{-2}^*$ is viewed as a vector bundle over the base $A_{-2}^*$ and $A_{-1}\times_M A_{0}^*\times_M A_{-2}^*$ is its dual bundle. Denote by $(x^i, \mu_j, \xi^k, \theta_l, p_i, \mu^j, \xi_k, \theta^l)$ a canonical (Darboux) coordinate on $T^*[3](A_0\times_MA_{-1}^*\times_M A_{-2}^*)[1]$, where $x^i$ is a smooth coordinate on $M$, $\mu_j\in\Gamma(A_{-2})$ is a fibre coordinate on $A_{-2}^*$, $\xi^k\in\Gamma(A_{0}^*)$ is a fibre coordinate on $A_{0}$, $\theta_l\in\Gamma(A_{-1})$ is a fibre coordinate on $A_{-1}^*$ and $(p_i, \mu^j, \xi_k, \theta^l)$ are the momentum coordinates for $(x^i, \mu_j, \xi^k, \theta_l)$. About their degrees, we have

The symplectic structure is given by

which is degree 3. The Lie 3-algebroid structure gives rise to a degree 4 function $\Theta$ satisfying $\{\Theta, \Theta\} = 0.$ By Theorem 5.2, we obtain a $\mathsf{CLWX}$ 2-algebroid $(D_{-1}, D_0, \partial, \rho, S, \diamond, \Omega)$, where $D_{-1} = A_{-1}\times_M A_0^* \times_M A_{-2}^*$ and $D_0 = A_0\times_MA_{-1}^*\times_M A_{-2}^*$ are vector bundles over $A_{-2}^*$. Obviously, they give the graded double vector bundle

The section space $\Gamma_{A_{-2}^*}(D_0)$ are generated by $\Gamma(A_{-1}^*)$ (the space of core sections) and $\Gamma(A_{-2}\otimes A_{-1}^*)\oplus \Gamma(A_0)$ (the space of linear sections) as $C^\infty(A_{-2}^*)$-module. Similarly, The section space $\Gamma_{A_{-2}^*}(D_{-1})$ are generated by $\Gamma(A_{0}^*)$ and $\Gamma(A_{-2}\otimes A_{0}^*)\oplus \Gamma(A_{-1})$ as $C^\infty(A_{-2}^*)$-module. Thus, in the sequel we only consider core sections and linear sections.

The graded symmetric bilinear form $S$ is given by

for all $e^0 = X^0+\psi^1+\alpha^1\in\Gamma_{A_{-2}^*}(D_0)$ and $e^1 = X^1+\psi^0+\alpha^0\in\Gamma_{A_{-2}^*}(D_{-1})$, where $X^i\in\Gamma(A_{-i})$, $\psi^i\in\Gamma(A_{-2}\otimes A_{-i}^*)$ and $\alpha^i\in\Gamma(A_{-i}^*)$. Then it is obvious that

is a metric graded double vector bundle.

The bundle map $\partial:D_{-1}\longrightarrow D_0$ is given by

Thus, $\partial:D_{-1}\longrightarrow D_0$ is a double vector bundle morphism over $l_1:A_{-1}\longrightarrow A_0$.

Note that functions on $A_{-2}^*$ are generated by fibrewise constant functions $C^{\infty}(M)$ and fibrewise linear functions $\Gamma(A_{-2})$. For all $f\in C^{\infty}(M)$ and $X^2\in\Gamma(A_{-2})$, the anchor $\rho:D_0\longrightarrow TA_{-2}^*$ is given by

Therefore, for a linear section $X^0+\psi^1\in\Gamma^l_{A_{-2}^*}(D_0)$, the image $\rho(X^0+\psi^1)$ is a linear vector field and for a core section $\alpha^1\in\Gamma(A_{-1}^*)$, the image $\rho(\alpha^1)$ is a constant vector field. Thus, $\rho$ is linear.

The bracket operation $\diamond$ is given by

Then it is straightforward to see that the operation $\diamond$ is linear.

Finally, $\Omega$ is given by

which implies that $\Omega$ is also linear.

Thus, a split Lie 3-algebroid gives rise to a split $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid:

Conversely, given a split $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid:

where $D_{-1} = A_{-1}\times_M A_0^* \times_M B$ and $D_0 = A_0\times_M A_{-1}^* \times_M B$, then we can deduce that the corresponding symplectic NQ-manifold of degree 3 is $T^*[3] \mathcal{A}[1]$, where $\mathcal{A} = A_0\oplus A_{-1}\oplus B$ is a graded vector bundle in which $B$ is of degree $-2$, and the $Q$-structure gives rise to a Lie $3$-algebroid structure on $\mathcal{A}$. We omit details.

Remark 3. Since every double vector bundle is splitable, every $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid is isomorphic to a split one. Meanwhile, by choosing a splitting, we obtain a split Lie $3$-algebroid from an NQ-manifold of degree $3$ (Lie $3$-algebroid). Thus, we can enhance the above result to be a one-to-one correspondence between Lie $3$-algebroids and $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroids. We omit such details.

Recall that the tangent prolongation of a Courant algebroid is a $\mathsf{VB}$-Courant algebroid ([32,Proposition 3.4.1]). Now we show that the tangent prolongation of a $\mathsf{CLWX}$ 2-algebroid is a $\mathsf{VB}$-$\mathsf{CLWX}$ 2-algebroid. The notations used below is the same as the ones used in Section 3.

Proposition 3. Let $(E_{-1}, E_0, \partial, \rho, S, \diamond, \Omega)$ be a $\mathsf{CLWX}$ $2$-algebroid. Then we obtain that $(TE_{-1}, TE_0, \widetilde{\partial}, \widetilde{\rho}, \widetilde{S}, \widetilde{\diamond}, \widetilde{\Omega})$ is a $\mathsf{CLWX}$ $2$-algebroid over $TM$, where the bundle map $\widetilde{\partial}:TE_{-1}\longrightarrow TE_0$ is given by

the bundle map $\widetilde{\rho}:TE_0\longrightarrow TTM$ is given by

the degree $1$ bilinear form $\widetilde{S}$ is given by

the bilinear operation $\widetilde{\diamond}$ is given by

and $\widetilde{\Omega}:\wedge^3TE_0\longrightarrow TE_{-1}$ is given by

for all $\sigma^0, \tau^0, \varsigma^0\in \Gamma(E_0)$ and $\sigma^1, \tau^1\in \Gamma(E_{-1})$.

Moreover, we have the following $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid:

Proof. Since $(E_{-1}, E_0, \partial, \rho, S, \diamond, \Omega)$ is a $\mathsf{CLWX}$ $2$-algebroid, it is straightforward to deduce that $(TE_{-1}, TE_0, \widetilde{\partial}, \widetilde{\rho}, \widetilde{S}, \widetilde{\diamond}, \widetilde{\Omega})$ is a $\mathsf{CLWX}$ $2$-algebroid over $TM$. Moveover, it is obvious that $\widetilde{\partial}, \widetilde{\rho}, \widetilde{S}, \widetilde{\diamond}, \widetilde{\Omega}$ are all linear, which implies that we have a $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid.

6.   $E$-$\mathsf{CLWX}$ 2-algebroid

In this section, we introduce the notion of an $E$-$\mathsf{CLWX}$ $2$-algebroid as the categorification of an $E$-Courant algebroid introduced in ^[11]. We show that associated to a $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid, there is an $E$-$\mathsf{CLWX}$ $2$-algebroid structure on the corresponding graded fat bundle.

There is an $E$-valued pairing $\langle {\cdot, \cdot}\rangle_E$ between the jet bundle $\mathfrak{J}{E}$ and the first order covariant differential operator bundle $\mathfrak{D}{E}$ defined by

Definition 6.1. Let $E$ be a vector bundle. An $E$-$\mathsf{CLWX}$ $2$-algebroid is a $6$-tuple $(\mathcal{K}, \partial, \rho, \mathcal{S}, \diamond, \Omega)$, where $\mathcal{K} = K_{-1}\oplus K_0$ is a graded vector bundle over $M$ and

$\bullet$ $\partial:K_{-1}\longrightarrow K_0$ is a bundle map;

$\bullet$ $\mathcal{S}: \mathcal{K}\otimes \mathcal{K}\longrightarrow E$ is a surjective graded symmetric nondegenerate $E$-valued pairing of degree $1$, which induces an embedding: $\mathcal{K} \hookrightarrow \mathrm{Hom}(\mathcal{K}, E)$;

$\bullet$ $\rho:K_0 \, \rightarrow \, \mathfrak{D} E$ is a bundle map, called the anchor, such that $\rho^{\star}(\mathfrak{J} E)\subset K_{-1}$, i.e.

$\bullet$ $\diamond:\Gamma(K_{-i})\times \Gamma(K_{-j})\longrightarrow \Gamma(K_{-(i+j)}), \; 0\leq i+j\leq 1$ is an $\mathbb R$-bilinear operation;

$\bullet$ $\Omega:\wedge^3 K_0\longrightarrow K_{-1}$ is a bundle map,

such that the following properties hold:

${\rm{(E1)}}$ $(\Gamma(\mathcal{K}), \partial, \diamond, \Omega)$ is a Leibniz $2$-algebra;

${\rm{(E2)}}$ for all $e\in\Gamma(\mathcal{K})$, $e\diamond e = \frac{1}{2} \mathcal{D} \mathcal{S}(e, e)$, where $\mathcal{D}:\Gamma(E)\longrightarrow \Gamma(K_{-1})$ is defined by

${\rm{ (E3)}}$ for all $e^1_1, e^1_2\in\Gamma(K_{-1})$, $\mathcal{S}(\partial(e^1_1), e^1_2) = \mathcal{S}(e^1_1, \partial(e^1_2))$;

${\rm{ (E4)}}$ for all $e_1, e_2, e_3\in\Gamma(\mathcal{K})$, $\rho(e_1) \mathcal{S}(e_2, e_3) = \mathcal{S}(e_1\diamond e_2, e_3)+ \mathcal{S}(e_2, e_1\diamond e_3)$;

${\rm{ (E5)}}$ for all $e^0_1, e^0_2, e^0_3, e^0_4\in\Gamma(K_0)$, $\mathcal{S}(\Omega(e^0_1, e^0_2, e^0_3), e^0_4) = - \mathcal{S}(e^0_3, \Omega(e^0_1, e^0_2, e^0_4))$;

${\rm{(E6) }}$ for all $e^0_1, e^0_2\in\Gamma(K_0)$, $\rho(e^0_1\diamond e^0_2) = [\rho(e^0_1), \rho(e^0_2)]_{ \mathfrak{D}}$, where $[\cdot, \cdot]_ \mathfrak{D}$ is the commutator bracket on $\Gamma(\mathfrak{D} E)$.

A $\mathsf{CLWX}$ $2$-algebroid can give rise to a Lie 3-algebra ([34,Theorem 3.11]). Similarly, an $E$-$\mathsf{CLWX}$ $2$-algebroid can also give rise to a Lie 3-algebra. Consider the graded vector space $\mathfrak e = \mathfrak e_{-2}\oplus \mathfrak e_{-1}\oplus \mathfrak e_0$, where $\mathfrak e_{-2} = \Gamma(E)$, $\mathfrak e_{-1} = \Gamma(K_{-1})$ and $\mathfrak e_0 = \Gamma(K_0)$. We introduce a skew-symmetric bracket on $\Gamma(\mathcal{K})$,

which is the skew-symmetrization of $\diamond$.

Theorem 6.2. An $E$-$\mathsf{CLWX}$ $2$-algebroid $(\mathcal{K}, \partial, \rho, \mathcal{S}, \diamond, \Omega)$ gives rise to a Lie $3$-algebra $(\mathfrak e, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3, \mathfrak l_4)$, where $\mathfrak l_i$ are given by

where the totally skew-symmetric $T:\Gamma(K_0)\times \Gamma(K_0)\times \Gamma(K_{-1})\longrightarrow \Gamma(E)$ is given by

and $\overline{\Omega}:\wedge^4\Gamma(K_0) \longrightarrow \Gamma(E)$ is given by

Proof. The proof is totally parallel to the proof of [34,Theorem 3.11], we omit the details.

Let $(D_{-1}^B, D_0^B, \partial, \rho, S, \diamond, \Omega)$ be a $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid on the graded double vector bundle $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right).$ Then we have the associated graded fat bundles $\hat{A}_{-1}\oplus \hat{A}_0$, which fit the exact sequences:

Since the bundle map $\partial$ is linear, it induces a bundle map $\hat{\partial}:\hat{A}_{-1}\longrightarrow \hat{A}_{0}$. Since the anchor $\rho$ is linear, it induces a bundle map $\hat{\rho}:\hat{A}_{0}\longrightarrow \mathfrak{D} B^*$, where sections of $\mathfrak{D} B^*$ are viewed as linear vector fields on $B$. Furthermore, the restriction of $S$ on linear sections will give rise to linear functions on $B$. Thus, we obtain a $B^*$-valued degree 1 graded symmetric bilinear form $\hat{S}$ on the graded fat bundle $\hat{A}_{-1}\oplus \hat{A}_0$. Since the operation $\diamond$ is linear, it induces an operation $\hat{\diamond}:\hat{A}_{-i}\times \hat{A}_{-j}\longrightarrow \hat{A}_{-(i+j)}$, $0\leq i+j\leq 1$. Finally, since $\Omega$ is linear, it induces an $\hat{\Omega}:\Gamma(\wedge^3\hat{A_0})\longrightarrow \hat{A}_{-1}$. Then we obtain:

Theorem 6.3. A $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid gives rise to a $B^*$-$\mathsf{CLWX}$ $2$-algebroid structure on the corresponding graded fat bundle. More precisely, let $(D_{-1}^B, D_0^B, \partial, \rho, S, \diamond, \Omega)$ be a $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid on the graded double vector bundle $\left(\begin{array}{ccc} D_{-1};&A_{-1}, B; &M\\ D_{0};&A_{0}, B; &M \end{array}\right)$ with the associated graded fat bundle $\hat{A}_{-1}\oplus \hat{A}_0$. Then $(\hat{A}_{-1}, \hat{A}_0, \hat{\partial}, \hat{\rho}, \hat{S}, \hat{\diamond}, \hat{\Omega})$ is a $B^*$-$\mathsf{CLWX}$ $2$-algebroid.

Proof. Since all the structures defined on the graded fat bundle $\hat{A}_{-1}\oplus \hat{A}_0$ are the restriction of the structures in the $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid, it is straightforward to see that all the axioms in Definition 6.1 hold.

Example 3. Consider the $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid given in Example 2, the corresponding $E$-$\mathsf{CLWX}$ $2$-algebroid is $((\mathfrak{J} E)[1], \mathfrak{D} E, \partial = 0, \rho = { \rm{id}}, \mathcal{S} = \left ({\cdot, \cdot}\right)_E, \diamond, \Omega = 0)$, where the graded symmetric nondegenerate $E$-valued pairing $\left ({\cdot, \cdot}\right)_E$ is given by

and $\diamond$ is given by

See ^[10] for more details.

Example 4. Consider the $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid given in Proposition 3. The graded fat bundle is $\mathfrak{J} E_{-1}\oplus \mathfrak{J} E_0$. It follows that the graded jet bundle associated to a $\mathsf{CLWX}$ $2$-algebroid is a $T^*M$-$\mathsf{CLWX}$ $2$-algebroid. This is the higher analogue of the result that the jet bundle of a Courant algebroid is $T^*M$-Courant algebroid given in ^[11]. See also ^[24] for more details. $

7.   Constructions of Lie 3-algebras

As applications of $E$-$\mathsf{CLWX}$ 2-algebroids introduced in the last section, we construct Lie 3-algebras from Lie 3-algebras in this section. Let $(\mathfrak g_{-2}, \mathfrak g_{-1}, \mathfrak g_0, l_1, l_2, l_3, l_4)$ be a Lie $3$-algebra. By Theorem 5.5, the corresponding $\mathsf{VB}$-$\mathsf{CLWX}$ $2$-algebroid is given by

where $D_{-1} = \mathfrak g_{-1} \oplus \mathfrak g_0^* \oplus \mathfrak g_{-2}^*$ and $D_0 = \mathfrak g_0 \oplus \mathfrak g_{-1}^*\oplus \mathfrak g_{-2}^*$.

By Theorem 6.3, we obtain:

Proposition 4. Let $(\mathfrak g_{-2}, \mathfrak g_{-1}, \mathfrak g_0, l_1, l_2, l_3, l_4)$ be a Lie $3$-algebra. Then there is an $E$-$\mathsf{CLWX}$ $2$-algebroid $(\mathrm{Hom}(\mathfrak g_0, \mathfrak g_{-2})\oplus \mathfrak g_{-1}, \mathrm{Hom}(\mathfrak g_{-1}, \mathfrak g_{-2})\oplus \mathfrak g_{0}, \partial, \rho, \mathcal{S}, \diamond, \Omega)$, where for all $x^i, y^i, z^i\in \mathfrak g_{-i}$, $\phi^i, \psi^i, \varphi^i\in \mathrm{Hom}(\mathfrak g_{-i}, \mathfrak g_{-2})$, $\partial: \mathrm{Hom}(\mathfrak g_0, \mathfrak g_{-2})\oplus \mathfrak g_{-1}\longrightarrow \mathrm{Hom}(\mathfrak g_{-1}, \mathfrak g_{-2})\oplus \mathfrak g_{0}$ is given by

$\rho: \mathrm{Hom}(\mathfrak g_{-1}, \mathfrak g_{-2})\oplus \mathfrak g_{0}\longrightarrow \mathfrak {gl}(\mathfrak g_{-2})$ is given by

the $\mathfrak g_{-2}$-valued pairing $\mathcal{S}$ is given by

the operation $\diamond$ is given by

and $\Omega$ is given by

By (7.2), it is straightforward to deduce that the corresponding $\mathcal{D}: \mathfrak g_{-2}\longrightarrow \mathrm{Hom}(\mathfrak g_0, \mathfrak g_{-2})\oplus \mathfrak g_{-1}$ is given by

Then by Theorem 6.2, we obtain:

Proposition 5. Let $(\mathfrak g_{-2}, \mathfrak g_{-1}, \mathfrak g_0, l_1, l_2, l_3, l_4)$ be a Lie $3$-algebra. Then there is a Lie $3$-algebra $(\overline{ \mathfrak g}_{-2}, \overline{ \mathfrak g}_{-1}, \overline{ \mathfrak g}_0, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3, \mathfrak l_4)$, where $\overline{ \mathfrak g}_{-2} = \mathfrak g_{-2}$, $\overline{ \mathfrak g}_{-1} = \mathrm{Hom}(\mathfrak g_0, \mathfrak g_{-2})\oplus \mathfrak g_{-1}$, $\overline{ \mathfrak g}_0 = \mathrm{Hom}(\mathfrak g_{-1}, \mathfrak g_{-2})\oplus \mathfrak g_{0}$, and $\mathfrak l_i$ are given by

where the operation $\mathcal{D}$, $\diamond$, $\Omega$ are given by (7.6), (7.4), (7.5) respectively, $T:\overline{ \mathfrak g}_0\times\overline{ \mathfrak g}_0\times\overline{ \mathfrak g}_{-1}\longrightarrow \mathfrak g_{-2}$ is given by

and $\overline{\Omega}:\wedge^4\overline{ \mathfrak g}_{0} \longrightarrow \mathfrak g_{-2}$ is given by

By Proposition 5, we can give interesting examples of Lie 3-algebras.

Example 5. We view a 3-term complex of vector spaces $V_{-2}\stackrel{l_1}{\longrightarrow}V_{-1}\stackrel{l_1}{\longrightarrow}V_0$ as an abelian Lie 3-algebra. By Proposition 5, we obtain the Lie 3-algebra

where $\mathfrak l_i, i = 1, 2, 3$ are given by

for all $x^2\in V_{-2}, \; \psi^0+x^1, \phi^0+y^1, \varphi^0+z^1\in \mathrm{Hom}(V_0, V_{-2})\oplus V_{-1}, \; \psi^1+x^0, \phi^1+y^0, \varphi^1+z^0\in \mathrm{Hom}(V_{-1}, V_{-2})\oplus V_0.$

Example 6. (Higher analogue of the Lie 2-algebra of string type)

A Lie $2$-algebra $(\mathfrak g_{-1}, \mathfrak g_0, \widetilde{l_1}, \widetilde{l_2}, \widetilde{l_3})$ gives rise to a Lie $3$-algebra $(\mathbb R, \mathfrak g_{-1}, \mathfrak g_0, {l_1}, {l_2}, {l_3}, {l_4} = 0)$ naturally, where ${l_i}$, $i = 1, 2, 3$ is given by

for all $x^0, y^0, z^0\in \mathfrak g_0$, $x^1, y^1, z^1\in \mathfrak g_{-1}$, and $r, s\in\mathbb R$. By Proposition 5, we obtain the Lie 3-algebra $(\mathbb R, \mathfrak g_{-1}\oplus \mathfrak g_0^*, \mathfrak g_0\oplus \mathfrak g_{-1}^*, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3, \mathfrak l_4)$, where $\mathfrak l_i$, $i = 1, 2, 3, 4$ are given by

for all $x^0, y^0, z^0, u^0\in \mathfrak g_{0},$ $x^1, y^1, z^1\in \mathfrak g_{-1},$ $\alpha^1, \beta^1, \zeta^1, \gamma^1\in \mathfrak g^*_{-1},$ $\alpha^0, \beta^0\in \mathfrak g^*_{0}$, where ${ \mathrm{ad}^0}^*_{x^0}: \mathfrak g^*_{-i}\longrightarrow \mathfrak g^*_{-i}$, ${ \mathrm{ad}^1}^*_{x^1}: \mathfrak g^*_{-1}\longrightarrow \mathfrak g^*_{0}$ and ${ \mathrm{ad}^3}^*_{x^0, y^0}: \mathfrak g^*_{-1}\longrightarrow \mathfrak g^*_{0}$ are defined respectively by

Remark 4. For any Lie algebra $(\mathfrak h, [\cdot, \cdot]_ \mathfrak h)$, we have the semidirect product Lie algebra $(\mathfrak h\ltimes_{ \mathrm{ad}^*} \mathfrak h^*, [\cdot, \cdot]_{ \mathrm{ad}^*})$, which is a quadratic Lie algebra naturally. Consequently, one can construct the corresponding Lie $2$-algebra $(\mathbb R, \mathfrak h\ltimes_{ \mathrm{ad}^*} \mathfrak h^*, l_1 = 0, l_2 = [\cdot, \cdot]_{ \mathrm{ad}^*}, l_3)$, where $l_3$ is given by

This Lie $2$-algebra is called the Lie $2$-algebra of string type in ^[51]. On the other hand, associated to a Lie $2$-algebra $(\mathfrak g_{-1}, \mathfrak g_0, \widetilde{l_1}, \widetilde{l_2}, \widetilde{l_3})$, there is a naturally a quadratic Lie $2$-algebra structure on $\big(\mathfrak g_{-1}\oplus \mathfrak g_0^*\big)\oplus\big(\mathfrak g_0\oplus \mathfrak g_{-1}^*\big)$ ([34,Example 4.8]). Thus, the Lie $3$-algebra given in the above example can be viewed as the higher analogue of the Lie $2$-algebra of string type.

Motivated by the above example, we show that one can obtain a Lie 3-algebra associated to a quadratic Lie 2-algebra in the sequel. This result is the higher analogue of the fact that there is a Lie 2-algebra, called the string Lie 2-algebra, associated to a quadratic Lie algebra.

A quadratic Lie 2-algebra is a Lie 2-algebra $(\mathfrak g_{-1}, \mathfrak g_0, l_1, l_2, l_3)$ equipped with a degree $1$ graded symmetric nondegenerate bilinear form $S$ which induces an isomorphism between $\mathfrak g_{-1}$ and $\mathfrak g_0^*$, such that the following invariant conditions hold:

for all $x^0, y^0, z^0, u^0\in \mathfrak g_0,$ $x^1, y^1\in \mathfrak g_{-1}$.

Let $(\mathfrak g_{-1}, \mathfrak g_0, l_1, l_2, l_3, S)$ be a quadratic Lie 2-algebra. On the 3-term complex of vector spaces $\mathbb R\oplus \mathfrak g_{-1}\oplus \mathfrak g_0$, where $\mathbb R$ is of degree $-2$, we define $\mathfrak l_i$, $i = 1, 2, 3, 4$, by

for all $x^0, y^0, z^0, u^0\in \mathfrak g_0,$ $x^1, y^1, z^1\in \mathfrak g_{-1}$ and $r\in\mathbb R$.

Theorem 7.1. With above notations, $(\mathbb R, \mathfrak g_{-1}, \mathfrak g_0, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3, \mathfrak l_4)$ is a Lie $3$-algebra, called the higher analogue of the string Lie $2$-algebra.

Proof. It follows from direct verification of the coherence conditions for $\mathfrak l_3$ and $\mathfrak l_4$ using the invariant conditions (7.7)-(7.9). We omit details.