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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.
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
An NQ-manifold of degree $n$ corresponds to a Lie $n$-algebroid. |
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.
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].
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.
See [40,Definition 9.1.1] for the precise definition of a double vector bundle. We denote a double vector bundle
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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:
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A morphism of double vector bundles
$ (\varphi;f_A, f_B;f_M):(D;A, B;M) \rightarrow (D';A', B';M') $ |
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
$ c^\dagger(b_m) = \tilde{0}_{b_m}+_A \overline{c(m)}, \quad \forall\; m\in M, \; b_m\in B_m, $ |
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
$ \widetilde{\psi}(b_m) = \tilde{0}_{b_m}+_A \overline{\psi(b_m)}. $ |
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 $
$ 0→B∗⊗C⟶ˆApr⟶A→0. $
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(2.1) |
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:
$ [\Gamma_B^l(D), \Gamma_B^l(D)]_D\subset \Gamma_B^l(D), \; [\Gamma_B^l(D), \Gamma_B^c(D)]_D\subset \Gamma_B^c(D), \; [ \Gamma_B^c(D), \Gamma_B^c(D)]_D = 0. $ |
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
$ \left[\!\left[ {\Gamma_B^l(D), \Gamma_B^l(D)} \right]\!\right]\subseteq \Gamma_B^l(D), \ \ \ \left[\!\left[ {\Gamma_B^l(D), \Gamma_B^c(D)} \right]\!\right]\subseteq \Gamma_B^c(D), \quad \left[\!\left[ {\Gamma_B^c(D), \Gamma_B^c(D)} \right]\!\right] = 0. $ |
Theorem 2.3. ([32,Proposition 3.2.1]) There is a one-to-one correspondence between Lie $ 2 $-algebroids and $ \mathsf{VB} $-Courant 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:
$ ∑i+j=k+1(−1)i(j−1)∑σ∈Sh−1i,k−isgn(σ)Ksgn(σ)lj(li(Xσ(1),⋯,Xσ(i)),Xσ(i+1),⋯,Xσ(k))=0, $
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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.
$ X_1\wedge \cdots\wedge X_k = \mathrm{Ksgn}(\sigma)X_{\sigma(1)}\wedge \cdots\wedge X_{\sigma(k)}. $ |
$ 2. $ $ l_2 $ satisfies the Leibniz rule with respect to the anchor $ a $:
$ l_2(X^0, fX) = fl_2(X^0, X)+a(X^0)(f)X, \quad\forall\; X^0\in\Gamma(A_0), \; f\in C^\infty(M), \; X\in\Gamma( \mathcal{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
$ a′∘F0=a,l′1∘F1=F0∘l1,F0l2(X0,Y0)−l′2(F0(X0),F0(Y0))=l′1F20(X0,Y0),F1l2(X0,Y1)−l′2(F0(X0),F1(Y1))=F20(X0,l1(Y1))−l′1F21(X0,Y1),l′2(F1(X1),F1(Y1))=F21(l1(X1),Y1)−F21(X1,l1(Y1)),l′2(F0(X0),F2(Y0,Z0))−F20(l2(X0,Y0),Z0)+c.p.=F1(l3(X0,Y0,Z0))+l′1F3(X0,Y0,Z0),l′2(F0(X0),F21(Y0,Z1))+l′2(F0(Y0),F21(Z1,X0))+l′2(F1(Z1),F20(X0,Y0))=F21(l2(X0,Y0),Z1)+c.p.+F3(X0,Y0,l1(Z1)), $
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and
$ 4∑i=1(−1)i+1(F21(X0i,l3(X01,⋯,^X0i,⋯X04))+l′2(F0(X0i),F3(X01,⋯,^X0i,⋯X04)))+∑i<j(−1)i+j(F3(l2(X0i,X0j),X0k,X0l)+c.p.−12l′2(F20(X0i,X0j),F20(X0k,X0l)))=0, $
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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
$ {⟨L0X0α0,Y0⟩=ρ(X0)⟨Y0,α0⟩−⟨α0,l2(X0,Y0)⟩,⟨L0X0α1,Y1⟩=ρ(X0)⟨Y1,α1⟩−⟨α1,l2(X0,Y1)⟩,⟨L1X1α1,Y0⟩=−⟨α1,l2(X1,Y0)⟩,⟨L3X0,Y0α1,Z0⟩=−⟨α1,l3(X0,Y0,Z0)⟩, $
|
(3.1) |
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(D−1;A−1,B−1;M−1D0;A0,B0;M0
Definition 3.3. A $ \mathsf{VB} $-Lie $ 2 $-algebroid is a graded double vector bundle
$ \left(D−1;A−1,B;MD0;A0,B;M \right) $
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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.
$ l2(ΓlB(D0),ΓlB(D0))⊂ΓlB(D0),l2(ΓlB(D0),ΓcB(D0))⊂ΓcB(D0),l2(ΓlB(D0),ΓlB(D−1))⊂ΓlB(D−1),l2(ΓlB(D0),ΓcB(D−1))⊂ΓcB(D−1),l2(ΓcB(D0),ΓlB(D−1))⊂ΓcB(D−1),l2(ΓcB(D0),ΓcB(D−1))=0;l2(ΓcB(D0),ΓcB(D0))=0. $
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$ {\rm(iv)} $ $ l_3 $ is linear, i.e.
$ l3(ΓlB(D0),ΓlB(D0),ΓlB(D0))⊂ΓlB(D−1),l3(ΓlB(D0),ΓlB(D0),ΓcB(D0))⊂ΓcB(D−1),l3(ΓcB(D0),ΓcB(D0),⋅)=0. $
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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(D−1;A−1,B;MD0;A0,B;M
$ (A_{-1}, A_0, \mathfrak a, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3) $ |
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
$ \pi^{A_{-(i+j)}}(l_2(\xi^i, \eta^j)) = 0, \quad\forall\; \eta^j\in\Gamma^l_B(D_{-j}). $ |
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
$ l1(l2(X0,X1))=l2(X0,l1(X1)),∀X0∈Γ(A0),X1∈Γ(A−1). $
|
(3.2) |
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
$ \pi^{A_{0}}l_1(l_2(\xi^0, \xi^1)) = \pi^{A_{0}}l_2(\xi^0, l_1(\xi^1)). $ |
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
$ \pi^{A_{0}}l_1(l_2(\xi^0, \xi^1)) = \mathfrak l_1\pi^{A_{-1}}l_2(\xi^0, \xi^1) = \mathfrak l_1 \mathfrak l_2(X^0, X^1), $ |
and the right hand side is equal to
$ \pi^{A_{0}}l_2(\xi^0, l_1(\xi^1)) = \mathfrak l_2(\pi^{A_{0}}(\xi^0), \pi^{A_{0}}(l_1(\xi^1))) = \mathfrak l_2(X_0, \mathfrak l_1(X^1)). $ |
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
$ l2(X0,fYi)=πA−il2(ξ0,q∗B(f)ηi)=πA−i(q∗B(f)l2(ξ0,ηi)+a(ξ0)(q∗B(f))ηi)=fl2(X0,Yi)+a(X0)(f)Yi. $
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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(D−1;A−1,B;MD0;A0,B;M
Consequently, we have the following exact sequences of split Lie $ 2 $-algebroids:
![]() |
(3.3) |
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
$ ˆl1(ϕ1)=lC1∘ϕ1,∀ϕ1∈Γ(B∗⊗C−1)=Γ(Hom(B,C−1)). $
|
(3.4) |
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
$ ⟨ϱ(c0),ξ⟩=−a(c0)(ξ),∀c0∈Γ(C0),ξ∈Γ(B∗). $
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(3.5) |
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
$ ˆl2(ϕ0,ψ0)=ϕ0∘ϱ∘ψ0−ψ0∘ϱ∘ϕ0, $
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(3.6) |
$ ˆl2(ϕ0,ψ1)=−ˆl2(ψ1,ϕ0)=−ψ1∘ϱ∘ϕ0, $
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(3.7) |
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
$ f_C = q^*f, \quad f_T = df, $ |
where $ q:TM\longrightarrow M $ is the projection. We have the following relations about the module structure:
$ (fσ)C=fCσC,(fσ)T=fTσC+fCσT. $
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(3.8) |
In particular, for $ A = TM $, we have
$ XT(fT)=X(f)T,XT(fC)=X(f)C,XC(fT)=X(f)C,XC(fC)=0, $
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(3.9) |
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
$ (TA_{-1}, TA_0, a, l_1, l_2, l_3) $ |
is a split Lie $ 2 $-algebroid over $ TM $, where $ a:TA_0\longrightarrow TTM $ is given by
$ a(σ0T)=a(σ0)T,a(σ0C)=a(σ0)C, $
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(3.10) |
$ l_1:\Gamma_{TM}(TA_{-1})\longrightarrow \Gamma_{TM}(TA_0) $ is given by
$ l1(σ1T)=l1(σ1)T,l1(σ1C)=l1(σ1)C, $
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(3.11) |
$ l_2:\Gamma_{TM}(TA_{-i})\times \Gamma_{TM}(TA_{-j})\longrightarrow \Gamma_{TM}(TA_{-(i+j)}) $ is given by
$ l2(σ0T,τ0T)=l2(σ0,τ0)T,l2(σ0T,τ0C)=l2(σ0,τ0)C,l2(σ0C,τ0C)=0,l2(σ0T,τ1T)=l2(σ0,τ1)T,l2(σ0T,τ1C)=l2(σ0,τ1)C,l2(σ0C,τ1T)=l2(σ0,τ1)C,l2(σ0C,τ1C)=0, $
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and $ l_3:\wedge^3\Gamma_{TM}(TA_{0})\longrightarrow \Gamma_{TM}(TA_{-1}) $ is given by
$ l3(σ0T,τ0T,ς0T)=l3(σ0,τ0,ς0)T,l3(σ0T,τ0T,ς0C)=l3(σ0,τ0,ς0)C, $
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(3.12) |
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
$ ˆa(σ0T)=a(σ0),ˆl2(σ0T,τ0T)=l2(σ0,τ0)T,ˆl2(σ0T,τ1T)=l2(σ0,τ1)T,ˆl3(σ0T,τ0T,ζ0T)=l2(σ0,τ0,ζ0)T, $
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for all $ \sigma^0, \; \tau^0, \; \zeta^0\in\Gamma(A_0) $ and $ \tau^1\in\Gamma(A_{-1}) $.
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
$ \mathfrak d_i(fe^i) = f \mathfrak d_i(e^i)+X(f)e^i, \quad \forall f\in C^{\infty}(M), \; e^i\in\Gamma(E_{-i}). $ |
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
$ p(d0,d1,d2)=X. $
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(4.1) |
Then the covariant differential operator bundle $ \mathfrak{D}(\mathcal{E}) $ fits the following exact sequence:
$ 0⟶End(E0)⊕End(E−1)⊕End(E−2)⟶D(E)⟶TM⟶0. $
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(4.2) |
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
$ d(θ2)=π∘θ2−θ2∘π,∀θ2∈Γ(Hom(E0,E−2)), $
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(4.3) |
and define $ \mathrm{d}: \mathrm{End}^{-1}(\mathcal{E})\longrightarrow \mathfrak{D}(\mathcal{E}) $ by
$ d(θ1)=π∘θ1+θ1∘π,∀θ1∈Γ(Hom(E0,E−1)⊕Hom(E−1,E−2)). $
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(4.4) |
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
$ [d,t]C=d∘t−t∘d,∀d,t∈Γ(D(E)), $
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(4.5) |
$ [d,θi]C=d∘θi−θi∘d,∀d∈Γ(D(E)),θi∈Γ(End−i(E)), $
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(4.6) |
$ [θ1,ϑ1]C=θ1∘ϑ1+ϑ1∘θ1,∀θ1,ϑ1∈Γ(End−1(E)). $
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(4.7) |
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,
$ \mathfrak d_0\circ\pi = \pi\circ \mathfrak d_1, \quad \mathfrak d_1\circ\pi = \pi\circ \mathfrak d_2. $ |
It is obvious that $ \Gamma(\mathfrak{D}_\pi(\mathcal{E})) $ is closed under the bracket operation $ [\cdot, \cdot]_C $ and
$ \mathrm{d}( \mathrm{End}^{-1}( \mathcal{E}))\subset \mathfrak{D}_\pi( \mathcal{E}). $ |
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(D−1;A−1,B;MD0;A0,B;M
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
$ \langle \nabla^B_{X^0}b, \xi\rangle = \mathfrak a(X^0)\langle\xi, b\rangle-\langle b, \hat{a}(s_0(X^0))(\xi)\rangle, \quad\forall X^0\in\Gamma(A_0), \; b\in\Gamma(B), \; \xi\in\Gamma(B^*). $ |
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
$ ∇0X0c0=l2(s0(X0),c0),∇1X0c1=l2(s0(X0),c1), $
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(4.8) |
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
$ Υ1X1=s0(l1(X1))−ˆl1(s1(X1)),Υ2X1c0=l2(s1(X1),c0), $
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(4.9) |
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
$ R0(X0,Y0)=s0l2(X0,Y0)−ˆl2(s0(X0),s0(Y0)), $
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(4.10) |
$ Λ(X0,Y0)(c0)=−l3(s0(X0),s0(Y0),c0), $
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(4.11) |
$ R1(X0,Y1)=s1l2(X0,Y1)−ˆl2(s0(X0),s1(Y1)), $
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(4.12) |
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
$ Ξ(X0,Y0,Z0))=s1l3(X0,Y0,Z0)−ˆl3(s0(X0),s0(Y0),s0(Z0)). $
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(4.13) |
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
$ lC1∘∇1X0=∇0X0∘lC1. $
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(4.14) |
By the fact that $ a:D_0\longrightarrow TB $ preserves the bracket operation, we obtain
$ ⟨∇BX0ϱ(c0),ξ⟩=a(X0)⟨ϱ(c0),ξ⟩−⟨ϱ(c0),a(s0(X0))(ξ)⟩=−[a(s0(X0)),a(c0)]TB(ξ)=−a(l2(s0(X0),c0))(ξ)=⟨ϱ∇0X0c0,ξ⟩, $
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which implies that
$ ∇BX0∘ϱ=ϱ∘∇0X0. $
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(4.15) |
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
$ F^0 = (\nabla^B, \nabla^0, \nabla^1), \quad F^1 = (\Upsilon^1, \Upsilon^2), \quad F^2_0 = (R^0, \Lambda), \quad F^2_1 = R^1, \quad F^3 = \Xi. $ |
Theorem 4.3. There is a one-to-one correspondence between $ \mathsf{VB} $-Lie $ 2 $-algebroids $ \left(D−1;A−1,B;MD0;A0,B;M
Proof. First it is obvious that
$ p∘F0=a. $
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(4.16) |
Using equalities $ \mathfrak a\circ \mathfrak l_1 = 0 $ and $ a\circ l_1 = 0 $, we have
$ ⟨∇Bl1X1b,ξ⟩=a(l1(X1))⟨b,ξ⟩−⟨b,a(s0(l1(X1)))(ξ)⟩=−⟨b,a(Υ1X1)(ξ)⟩, $
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which implies that
$ ∇Bl1X1=ϱ∘Υ1X1. $
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(4.17) |
For $ \nabla^0 $, we can obtain
$ ∇0l1(X1)=l2(s0l1(X1),⋅)|C0=l2(l1(s1(X1))+Υ1X1,⋅)|C0=lC1∘Υ2X1+Υ1X1∘ϱ. $
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(4.18) |
For $ \nabla^1 $, we have
$ ∇1l1(X1)=l2(s0l1(X1),⋅)|C1=l2(l1(s1(X1))+Υ1X1,⋅)|C1=Υ2X1∘lC1. $
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(4.19) |
By (4.17), (4.18) and (4.19), we deduce that
$ F0∘l1=d∘F1. $
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(4.20) |
By straightforward computation, we have
$ ⟨∇Bl2(X0,Y0)b−∇BX0∇BY0b+∇BY0∇BX0b,ξ⟩=⟨b,a(ˆl2(s0(X0),s0(Y0))−s0l2(X0,Y0))(ξ)⟩=⟨b,−a(R0(X0,Y0))(ξ)⟩, $
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which implies that
$ ∇Bl2(X0,Y0)−∇BX0∇BY0+∇BY0∇BX0=ϱ∘R0(X0,Y0). $
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(4.21) |
Similarly, we have
$ ∇0l2(X0,Y0)c0−∇0X0∇0Y0c0+∇0Y0∇0X0c0=l2(s0l2(X0,Y0),c0)−l2(s0(X0),l2(s0(Y0),c0))+l2(s0(Y0),l2(s0(X0),c0))=−l1l3(s0(X0),s0(Y0),c0)+l2(R0(X0,Y0),c0), $
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which implies that
$ ∇0l2(X0,Y0)−∇0X0∇0Y0+∇0Y0∇0X0=lC1∘Λ(X0,Y0)+R0(X0,Y0)∘ϱ, $
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(4.22) |
and
$ ∇1l2(X0,Y0)c1−∇1X0∇1Y0c1+∇1Y0∇1X0c1=l2(s0l2(X0,Y0),c1)−l2(s0(X0),l2(s0(Y0),c1))+l2(s0(Y0),l2(s0(X0),c1))=−l3(s0(X0),s0(Y0),l1(c1))+l2(R0(X0,Y0),c1), $
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which implies that
$ ∇1l2(X0,Y0)−∇1X0∇1Y0+∇1Y0∇1X0=Λ(X0,Y0)∘lC1. $
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(4.23) |
By (4.21), (4.22) and (4.23), we obtain
$ F0(l2(X0,Y0))−[F0(X0),F0(Y0)]C=dF20(X0,Y0). $
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(4.24) |
By the equality
$ {l}_2(s_0(X^0), {l}_2(s_1(Y^1), c^0))+c.p. = \hat{l}_3(s_0(X^0), l_1(s_1(Y^1)), c^0), $ |
we obtain
$ [F0(X0),Υ2Y1]C−Υ2l2(X0,Y1)=−Λ(X0,l1(Y1))−R1(X0,Y1)∘ϱ. $
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(4.25) |
Furthermore, we have
$ Υ1l2(X0,Y1)=s0l1(l2(X0,Y1))−ˆl1s1(l2(X0,Y1))=s0l2(X0,l1(Y1))−ˆl1ˆl2(s0(X0),s1(Y1))−ˆl1R1(X0,Y1)=s0l2(X0,l1(Y1))−ˆl2(s0(X0),ˆl1s1(Y1))−lC1∘R1(X0,Y1)=s0l2(X0,l1(Y1))−ˆl2(s0(X0),s0l1(Y1)−Υ1Y1)−lC1∘R1(X0,Y1)=[F0(X0),Υ1Y1]C+R0(X0,l1(Y1))−lC1∘R1(X0,Y1). $
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(4.26) |
By (4.25) and (4.26), we deduce that
$ F1(l2(X0,Y1))−[F0(X0),F1(Y1)]C=F20(X0,l1(Y1))−dF21(X0,Y1). $
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(4.27) |
By straightforward computation, we have
$ R1(l1(X1),Y1)−R1(X1,l1(Y1))=s1l2(l1(X1),Y1)−ˆl2(s0l1(X1),s1(Y1))−s1l2(X1,l1(Y1))+ˆl2(s1(X1),s0l1(Y1))=ˆl2(s1(X1),ˆl1s1(Y1))+ˆl2(s1(X1),Υ1Y1)−ˆl2(s0l1(X1),s1(Y1))=−ˆl2(Υ1X1,s1(Y1))+ˆl2(s1(X1),Υ1Y1)=[Υ1X1+Υ2X1,Υ1Y1+Υ2Y1]C. $
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(4.28) |
By the equality
$ \hat{l}_2(s_0(X^0), \hat{l}_2(s_0(Y^0), s_0(Z^0)))+c.p. = \hat{l}_1\hat{l}_3(s_0(X^0), s_0(Y^0), s_0(Z^0)), $ |
we deduce that
$ [F0(X0),R0(Y0,Z0)]C+R0(X0,l2(Y0,Z0))+c.p.=Υ1l3(X0,Y0,Z0)+lC1∘Ξ(X0,Y0,Z0). $
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(4.29) |
By the equality
$ l_2(s_0(X^0), l_3(s_0(Y^0), s_0(Z^0), c^0))-l_3(l_2(s_0(X^0), s_0(Y^0)), s_0(Z^0), c^0)+c.p. = 0, $ |
we deduce that
$ −[F0(X0),Λ(Y0,Z0)]C+Λ(l2(X0,Y0),Z0)+c.p.+Υ2l3(X0,Y0,Z0)−Ξ(X0,Y0,Z0)∘ϱ=0. $
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(4.30) |
By (4.29) and (4.30), we obtain
$ [F0(X0),F20(Y0,Z0)]C+F20(X0,l2(Y0,Z0))+c.p.=F1(l3(X0,Y0,Z0))+dF3(X0,Y0,Z0). $
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(4.31) |
Then by the equality
$ \hat{l}_2(s_0(X^0), \hat{l}_2(s_0(Y^0), s_1(Z^1)))+c.p. = \hat{l}_3(s_0(X^0), s_0(Y^0), \hat{l}_1( s_1(Z^1))), $ |
we deduce that
$ [F0(X0),R1(Y0,Z1)]C+[F0(Y0),R1(Z1,X0)]C+[Υ2Z1,R0(X0,Y0)]C+R1(X0,l2(Y0,Z1))+R1(Y0,l2(Z1,X0))+R1(Z1,l2(X0,Y0))=Ξ(X0,Y0,l1(Z1))−[Λ(X0,Y0),Υ1Z1]C. $
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(4.32) |
Finally, by the equality
$ 4∑i=1(−1)i+1ˆl2(s0(X0i),ˆl3(s0(X01),⋯,^s0(X0i),⋯,s0(X04)))+∑i<j,k<l(−1)i+jˆl3(ˆl2(s0(X0i),s0(X0j)),s0(X0k),s0(X0l))=0, $
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we deduce that
$ 4∑i=1(−1)i+1([F0(X0i),Ξ(X01,⋯,^X0i,⋯,X04)]C+R1(X0i,l3(X01,⋯,^X0i,⋯,X04)))+∑i<j(−1)i+j(Ξ(l2(X0i,X0j),X01,⋯,^X0i,⋯,^X0j,⋯,X04)−[R0(X0i,X0j),Λ(X01,⋯,^X0i,⋯,^X0j,⋯,X04)]C)=0. $
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(4.33) |
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
$ ( \mathrm{End}^{-2}( \mathcal{E}), \mathrm{End}^{-1}( \mathcal{E}), \mathfrak{D}_\pi( \mathcal{E}), \mathfrak p, \mathrm{d}, [\cdot, \cdot]_C). $ |
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(A−1⊕B⊕C−1;A−1,B;MA0⊕B⊕C0;A0,B;M
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
$ S(Df,e0)=ρ(e0)(f),∀f∈C∞(M),e0∈Γ(E0); $
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(5.1) |
$ {\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
$ S(X+\alpha, Y+\beta) = \langle X, \beta\rangle+\langle Y, \alpha \rangle, \quad \forall\; X, Y\in\Gamma(A), \alpha, \beta\in\Gamma(A^*), $ |
and $ \partial $, $ \rho $, $ \diamond $ and $ \Omega $ are given by derived brackets. More precisely, we have
$ ∂α={α,Θ},∀α∈Γ(A∗),ρ(X)(f)={f,{X,Θ}},∀X∈Γ(A),f∈C∞(M),X⋄Y={Y,{X,Θ}},∀X,Y∈Γ(A),X⋄α={α,{X,Θ}},∀X∈Γ(A),α∈Γ(A∗),α⋄X=−{X,{α,Θ}},∀X∈Γ(A),α∈Γ(A∗),Ω(X,Y,Z)={Z,{Y,{X,Θ}}},∀X,Y,Z∈Γ(A). $
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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(D−1;A−1,B;MD0;A0,B;M
![]() |
where $ \star B $ means dual over $ B $.
Given a metric graded double vector bundle, we have
$ C_0\cong A_{-1}^*, \quad C_{-1}\cong A_0^*. $ |
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
$ \left(\left(D−1;A−1,B;MD0;A0,B;M \right), S\right), $
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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
$ \partial(X^1+\alpha^0) = l_1(X^1)+l^*_1(\alpha^0), \quad \forall X^1\in\Gamma(A_{-1}), \; \alpha^0\in\Gamma(A^*_0), $ |
$ \rho:E_0\longrightarrow TM $ is given by
$ \rho(X^0+\alpha^1) = a(X^0), \quad \forall X^0\in\Gamma(A_0), \; \alpha^1\in\Gamma(A_{-1}^*), $ |
the symmetric bilinear form $ S = (\cdot, \cdot)_+ $ is given by
$ (X^0+\alpha^1+X^1+\alpha^0, Y^0+\beta^1+Y^1+\beta^0)_+ = \langle X^0, \beta^0 \rangle+\langle Y^0, \alpha^0 \rangle+\langle X^1, \beta^1 \rangle+\langle Y^1, \alpha^1 \rangle, $ |
the operation $ \diamond $ is given by
$ {(X0+α1)⋄(Y0+β1)=l2(X0,Y0)+L0X0β1−L0Y0α1,(X0+α1)⋄(X1+α0)=l2(X0,X1)+L0X0α0+ιX1δ(α1),(X1+α0)⋄(X0+α1)=l2(X1,X0)+L1X1α1−ιX0δ(α0), $
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(5.2) |
and the $ E_{-1} $-valued $ 3 $-form $ \Omega $ is defined by
$ \Omega(X^0+\alpha^1, Y^0+\beta^1, Z^0+\zeta^1) = l_3(X^0, Y^0, Z^0)+L^3_{X^0, Y^0}\zeta^1+L^3_{Z^0, X^0}\beta^1+L^3_{Y^0, Z^0}\alpha^1, $ |
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
$ (X+α)⋄(Y+β)=[X,Y]+LXβ−ιYdα,∀X,Y∈X(M),α,β∈Ω1(M). $
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(5.3) |
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
$ T^*[3] \mathcal{A}[1] = T^*[3]( A_0\times_MA_{-1}^*\times_M A_{-2}^*)[1], $ |
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
$ \left(xiμjξkθlpiμjξkθl00113322 \right) $
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The symplectic structure is given by
$ \omega = dx^idp_i+d\mu_jd\mu^j+d\xi^kd\xi_k+d\theta_ld\theta^l, $ |
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
$ \left(A−1×MA∗0×MA∗−2;A−1,A∗−2;MA0×MA∗−1×MA∗−2;A0,A∗−2;M \right). $
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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
$ S(e0,e1)=S(X0+ψ1+α1,X1+ψ0+α0)=⟨α1,X1⟩+⟨α0,X0⟩+ψ1(X1)+ψ0(X0), $
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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
$ \left(\left(A−1×MA∗0×MA∗−2;A−1,A∗−2;MA0×MA∗−1×MA∗−2;A0,A∗−2;M \right), S\right) $
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is a metric graded double vector bundle.
The bundle map $ \partial:D_{-1}\longrightarrow D_0 $ is given by
$ \partial(X^1+\psi^0+\alpha^0) = l_1(X^1)+l_2(X^1, \cdot)|_{A_{-1}}+\psi^0\circ l_1+l_1^*(\alpha^0). $ |
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
$ \rho(X^0+\psi^1+\alpha^1)(f+X^2) = a(X^0)(f)+\langle \alpha^1, l_1(X^2)\rangle+l_2(X^0, X^2)+\psi^1 (l_1(X^2)). $ |
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
$ (X0+ψ1+α1)⋄(Y0+ϕ1+β1)=l2(X0,Y0)+l3(X0,Y0,⋅)|A−1+l2(X0,ϕ1(⋅))−ϕ1∘l2(X0,⋅)|A−1+L0X0β1+ψ1∘l2(Y0,⋅)|A−1−l2(Y0,ψ1(⋅))+ψ1∘l1∘ϕ1−ϕ1∘l1∘ψ1−β1∘l1∘ψ1−L0Y0α1+α1∘l1∘ϕ1,(X0+ψ1+α1)⋄(Y1+ϕ0+β0)=l2(X0,Y1)+l3(X0,⋅,Y1)|A0+l2(X0,ϕ0(⋅))−ϕ0∘l2(X0,⋅)|A0+L0X0β0−ψ1l2(⋅,Y1)|A0+δ(ψ1(Y1))+ψ1∘l1∘ϕ0+ιY1δα1+α1∘l1∘ϕ0,(Y1+ϕ0+β0)⋄(X0+ψ1+α1)=l2(Y1,X0)−l3(X0,⋅,Y1)|A0−l2(X0,ϕ0(⋅))+ϕ0∘l2(X0,⋅)|A0+δ(ϕ0(X0))−ιX0δβ0+ψ1l2(⋅,Y1)|A0−ψ1∘l1∘ϕ0+L1Y1α1−α1∘l1∘ϕ0. $
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Then it is straightforward to see that the operation $ \diamond $ is linear.
Finally, $ \Omega $ is given by
$ Ω(X0+ψ1+α1,Y0+ϕ1+β1,Z0+φ1+γ1)=l3(X0,Y0,Z0)+l4(X0,Y0,Z0,⋅)−φ1∘l3(X0,Y0,⋅)|A0−ϕ1∘l3(Z0,X0,⋅)|A0−ψ1∘l3(Y0,Z0,⋅)|A0+L3X0,Y0γ1+L3Y0,Z0α1+L3Z0,X0β1, $
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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:
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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
$ \widetilde{\partial}(\sigma^1_T) = \partial(\sigma^1)_T, \quad \widetilde{\partial}(\sigma^1_C) = \partial(\sigma^1)_C, $ |
the bundle map $ \widetilde{\rho}:TE_0\longrightarrow TTM $ is given by
$ \widetilde{\rho}(\sigma^0_T) = \rho(\sigma^0)_T, \quad \widetilde{\rho}(\sigma^0_C) = \rho(\sigma^0)_C, $ |
the degree $ 1 $ bilinear form $ \widetilde{S} $ is given by
$ ˜S(σ0T,τ1T)=S(σ0,τ1)T,˜S(σ0T,τ1C)=S(σ0,τ1)C,˜S(σ0C,τ1T)=S(σ0,τ1)C,˜S(σ0C,τ1C)=0, $
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the bilinear operation $ \widetilde{\diamond} $ is given by
$ σ0T˜⋄τ0T=(σ0⋄τ0)T,σ0T˜⋄τ0C=−τ0C˜⋄σ0T=(σ0⋄τ0)C,σ0C˜⋄τ0C=0,σ0T˜⋄τ1T=(σ0⋄τ1)T,σ0T˜⋄τ1C=σ0C˜⋄τ1T=(σ0⋄τ1)C,σ0C˜⋄τ1C=0,τ1T˜⋄σ0T=(τ1⋄σ0)T,τ1C˜⋄σ0T=τ1T˜⋄σ0C=(τ1⋄σ0)C,τ1C˜⋄σ0C=0, $
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and $ \widetilde{\Omega}:\wedge^3TE_0\longrightarrow TE_{-1} $ is given by
$ ˜Ω(σ0T,τ0T,ς0T)=Ω(σ0,τ0,ς0)T,˜Ω(σ0T,τ0T,ς0C)=Ω(σ0,τ0,ς0)C,˜Ω(σ0T,τ0C,ς0C)=0, $
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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:
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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.
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
$ \langle {\mu, \mathfrak d}\rangle_E\; \triangleq \mathfrak d(u), \quad\forall\; \; \mathfrak d\in( \mathfrak{D}{E})_m, \; \mu\in ( \mathfrak{J}{E})_m, \; u\in \Gamma(E)\; \; \mbox{statisfying}\; \; \mu = [u]_m. $ |
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.
$ \mathcal{S}( \rho^{\star}(\mu), \; e^0) = \langle {\mu, \rho(e^0)}\rangle_E, \; \; \; \forall\; \mu\in\Gamma( \mathfrak{J} E), \; e^0\in\Gamma(K_0); $ |
$\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
$ S(Du,e0)=ρ(e0)(u),∀u∈Γ(E),e0∈Γ(K0); $
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(6.1) |
$ {\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}) $,
$ [[e1,e2]]=12(e1⋄e2−e2⋄e1),∀e1,e2∈Γ(K), $
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(6.2) |
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
$ l1(u)=D(u),∀u∈Γ(E),l1(e1)=∂(e1),∀e1∈Γ(K−1),l2(e01,e02)=[[e01,e02]],∀e01,e02∈Γ(K0),l2(e0,e1)=[[e0,e1]],∀e0∈Γ(K0),e1∈Γ(K−1),l2(e0,f)=12S(e0,Df),∀e0∈Γ(K0),f∈Γ(E),l2(e11,e12)=0,∀e11,e12∈Γ(K−1),l3(e01,e02,e03)=Ω(e01,e02,e03),∀e01,e02,e03∈Γ(K0),l3(e01,e02,e1)=−T(e01,e02,e1),∀e01,e02∈Γ(K0),e1∈Γ(K−1),l4(e01,e02,e03,e04)=¯Ω(e01,e02,e03,e04),∀e01,e02,e03,e04∈Γ(K0), $
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where the totally skew-symmetric $ T:\Gamma(K_0)\times \Gamma(K_0)\times \Gamma(K_{-1})\longrightarrow \Gamma(E) $ is given by
$ T(e01,e02,e1)=16(S(e01,[[e02,e1]])+S(e1,[[e01,e02]])+S(e02,[[e1,e01]])), $
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(6.3) |
and $ \overline{\Omega}:\wedge^4\Gamma(K_0) \longrightarrow \Gamma(E) $ is given by
$ \overline{\Omega}(e^0_1, e^0_2, e^0_3, e^0_4) = \mathcal{S}(\Omega(e^0_1, e^0_2, e^0_3), e^0_4). $ |
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(D−1;A−1,B;MD0;A0,B;M
$ 0→B∗⊗A∗0⟶ˆA−1⟶A−1→0,0→B∗⊗A∗−1⟶ˆA0⟶A0→0. $
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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(D−1;A−1,B;MD0;A0,B;M
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
$ \left ( { \mathfrak d+\mu, \mathfrak t+\nu}\right )_E = \langle {\mu, \mathfrak t}\rangle_E+\langle {\nu, \mathfrak d}\rangle_E, \quad\forall\; \mathfrak d+\mu, \; \mathfrak t+\nu\in \mathfrak{D} E\oplus \mathfrak{J} E, $ |
and $ \diamond $ is given by
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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. $
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
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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
$ ∂(ϕ0+x1)=ϕ0∘l1+l2(x1,⋅)|g−1+l1(x1), $
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(7.1) |
$ \rho: \mathrm{Hom}(\mathfrak g_{-1}, \mathfrak g_{-2})\oplus \mathfrak g_{0}\longrightarrow \mathfrak {gl}(\mathfrak g_{-2}) $ is given by
$ ρ(ϕ1+x0)=ϕ1∘l1+l2(x0,⋅)|g−2, $
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(7.2) |
the $ \mathfrak g_{-2} $-valued pairing $ \mathcal{S} $ is given by
$ S(ϕ1+x0,ψ0+y1)=ϕ1(y1)+ψ0(x0), $
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(7.3) |
the operation $ \diamond $ is given by
$ {(x0+ψ1)⋄(y0+ϕ1)=l2(x0,y0)+l3(x0,y0,⋅)|g−1+l2(x0,ϕ1(⋅))−ϕ1∘l1∘ψ1−ϕ1∘l2(x0,⋅)|g−1+ψ1∘l2(y0,⋅)|g−1−l2(y0,ψ1(⋅))+ψ1∘l1∘ϕ1,(x0+ψ1)⋄(y1+ϕ0)=l2(x0,y1)+l3(x0,⋅,y1)|g0+l2(x0,ϕ0(⋅))−ϕ0∘l2(x0,⋅)|g0−ψ1l2(⋅,y1)|g0+δ(ψ1(y1))+ψ1∘l1∘ϕ0,(y1+ϕ0)⋄(x0+ψ1)=l2(y1,x0)−l3(x0,⋅,y1)|g0−l2(x0,ϕ0(⋅))+ϕ0∘l2(x0,⋅)|g0+δ(ϕ0(x0))+ψ1l2(⋅,y1)|g0−ψ1∘l1∘ϕ0, $
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(7.4) |
and $ \Omega $ is given by
$ Ω(ϕ1+x0,ψ1+y0+φ1+z0)=l3(x0,y0,z0)+l4(x0,y0,z0,⋅)−φ1∘l3(x0,y0,⋅)|g0−ϕ1∘l3(z0,x0,⋅)|g0−ψ1∘l3(y0,z0,⋅)|g0. $
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(7.5) |
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
$ D(x2)=l2(⋅,x2)+l1(x2) $
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(7.6) |
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
$ l1(x2)=D(x2),∀x2∈g−2,l1(ϕ0+x1)=ϕ0∘l1+l2(x1,⋅)|g−1+l1(x1),∀ϕ0+x1∈¯g−1,l2(e01,e02)=e01⋄e02,∀e01,e02∈¯g0,l2(e0,e1)=12(e0⋄e1−e1⋄e0),∀e0∈¯g0,e1∈¯g−1,l2(e0,x2)=12S(e0,Dx2),∀e0∈¯g0,x2∈g−2,l2(e11,e12)=0,∀e11,e12∈¯g−1,l3(e01,e02,e03)=Ω(e01,e02,e03),∀e01,e02,e03∈¯g0,l3(e01,e02,e1)=−T(e01,e02,e1),∀e01,e02∈¯g0,e1∈¯g−1,l4(e01,e02,e03,e04)=¯Ω(e01,e02,e03,e04),∀e01,e02,e03,e04∈¯g0, $
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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
$ T(e^0_1, e^0_2, e^1) = \frac{1}{6}\big( \mathcal{S}(e^0_1, \mathfrak l_2(e^0_2, e^1))+ \mathcal{S}(e^1, \mathfrak l_2(e^0_1, e^0_2))+ \mathcal{S}(e^0_2, \mathfrak l_2(e^1, e^0_1))\big), $ |
and $ \overline{\Omega}:\wedge^4\overline{ \mathfrak g}_{0} \longrightarrow \mathfrak g_{-2} $ is given by
$ \overline{\Omega}(e^0_1, e^0_2, e^0_3, e^0_4) = \mathcal{S}(\Omega(e^0_1, e^0_2, e^0_3), e^0_4). $ |
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
$ (V_{-2}, \mathrm{Hom}(V_0, V_{-2})\oplus V_{-1}, \mathrm{Hom}(V_{-1}, V_{-2})\oplus V_0, \mathfrak l_1, \mathfrak l_2, \mathfrak l_3, \mathfrak l_4 = 0), $ |
where $ \mathfrak l_i, i = 1, 2, 3 $ are given by
$ l1(x2)=l1(x2),l1(ϕ0+y1)=ϕ0∘l1+l1(y1),l2(ψ1+x0,ϕ1+y0)=ψ1∘l1∘ϕ1−ϕ1∘l1∘ψ1,l2(ψ1+x0,ϕ0+y1)=12l1(ψ1(y1)−ϕ0(x0))+ψ1∘l1∘ϕ0,l2(ψ1+x0,x2)=12ψ1(l1(x2)),l2(ψ0+x1,ϕ0+y1)=0,l3(ψ1+x0,ϕ1+y0,φ1+z0)=0,l3(ψ1+x0,ϕ1+y0,φ0+z1)=−14(ψ1∘l1∘ϕ1(z1)−ϕ1∘l1∘ψ1(z1)−ψ1∘l1∘φ0(y0)+ϕ1∘l1∘φ0(x0)), $
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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
$ l1(r)=0,l1(x1)=~l1(x1),l2(x0,y0)=~l2(x0,y0),l2(x0,y1)=~l2(x0,y1),l2(x0,r)=0,l2(x1,y1)=0,l3(x0,y0,z0)=~l3(x0,y0,z0),l3(x0,y0,z1)=0, $
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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
$ l1(r)=0,l1(x1+α0)=l1(x1)+l∗1(α0),l2(x0+α1,y0+β1)=l2(x0,y0)+ad0∗x0β1−ad0∗y0α1,l2(x0+α1,y1+β0)=l2(x0,y1)+ad0∗x0β0−ad1∗y1α1,l2(x1+α0,y1+β0)=0,l2(x0+α1,r)=0,l3(x0+α1,y0+β1,z0+ζ1)=l3(x0,y0,z0)+ad3∗x0,y0ζ1+ad3∗y0,z0α1+ad3∗z0,x0β1,l3(x0+α1,y0+β1,z1+ζ0)=12(⟨α1,l2(y0,z1)⟩+⟨β1,l2(z1,x0)⟩+⟨ζ0,l2(x0,y0)⟩),l4(x0+α1,y0+β1,z0+ζ1,u0+γ1)=⟨γ1,l3(x0,y0,z0)⟩−⟨ζ1,l3(x0,y0,u0)⟩−⟨α1,l3(y0,z0,u0)⟩−⟨β1,l3(z0,x0,u0)⟩ $
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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
$ ⟨ad0∗x0α1,x1⟩=−⟨α1,l2(x0,x1)⟩,⟨ad0∗x0α0,y0⟩=−⟨α0,l2(x0,y0)⟩,⟨ad1∗x1α1,y0⟩=−⟨α1,l2(x1,y0)⟩,⟨ad3∗x0,y0α1,z0⟩=−⟨α1,l3(x0,y0,z0)⟩. $
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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
$ l_3(x+\alpha, y+\beta, z+\gamma) = \langle\gamma, [x, y]_ \mathfrak h\rangle+\langle\beta, [z, x]_ \mathfrak h\rangle+\langle\alpha, [y, z]_ \mathfrak h\rangle, \quad \forall x, y, z\in \mathfrak h, \alpha, \beta, \gamma\in \mathfrak h^*. $ |
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:
$ S(l1(x1),y1)=S(l1(y1),x1), $
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(7.7) |
$ S(l2(x0,y0),z1)=−S(l2(x0,z1),y0), $
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(7.8) |
$ S(l3(x0,y0,z0),u0)=−S(l3(x0,y0,u0),z0), $
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(7.9) |
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
$ {l1(r)=0,l1(x1)=l1(x1),l2(x0,y0)=l2(x0,y0),l2(x0,y1)=l2(x0,y1),l2(x0,r)=0,l2(x1,y1)=0,l3(x0,y0,z0)=l3(x0,y0,z0),l3(x0,y0,z1)=12S(z1,l2(x0,y0)),l4(x0,y0,z0,u0)=S(l3(x0,y0,z0),u0), $
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(7.10) |
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.
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