---
title: 'Lie 2-Algebroid: Structures & Applications'
url: https://www.emergentmind.com/topics/lie-2-algebroid
type: topic
---

# Lie 2-Algebroid: Structures & Applications

A Lie 2-algebroid is a 2-term \(L_\infty\)-algebroid, or equivalently a categorified Lie algebroid, whose basic presentations in the cited literature are a graded vector bundle \(A_{-1}\oplus A_0\to M\) with structure maps \(l_1,l_2,l_3\), and the split degree-2 NQ-manifold \(Q[-1]\oplus B^*[-2]\) encoded by \((Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)\). In both descriptions, the ordinary Lie algebroid axioms are replaced by higher Jacobi identities: the binary bracket is controlled by an anchor and a Leibniz rule, while Jacobi failure is measured by a ternary bracket or, in split form, by a \(B^*\)-valued \(3\)-form \(\omega\) [1807.11789] [1712.07035].

## 1. Foundational definitions and split presentations

In the split \(L_\infty\)-algebroid language, the underlying graded vector bundle is
\[
A=A_{-1}\oplus A_0 \longrightarrow M,
\]
equipped with an anchor
\[
\rho:A_0\to TM
\]
and structure maps
\[
l_1:A_{-1}\to A_0,\qquad l_2:\wedge^2 A\to A,\qquad l_3:\wedge^3 A_0\to A_{-1},
\]
of degrees \(+1,0,-1\), respectively. The bundle of sections \(T(A)\) carries a 2-term \(L_\infty\)-algebra structure \((l_1,l_2,l_3)\); \(l_2\) is a derivation with respect to functions,
\[
l_2(X_0,fY)=f\,l_2(X_0,Y)+\rho(X_0)(f)\,Y,
\]
and \(l_3\) is \(C^\infty(M)\)-linear. The anchor compatibilities include
\[
\rho\circ l_1=0,\qquad \rho(l_2(X_0,Y_0))=[\rho(X_0),\rho(Y_0)].
\]
This is the basic form in which Lie 2-algebroids appear in the study of transitive and quadratic cases [1807.11789].

A second, equivalent split presentation uses the graded manifold
\[
\mathcal M=Q[-1]\oplus B^*[-2].
\]
A split Lie 2-algebroid in this form consists of an anchored vector bundle \((Q,\rho_Q)\), another vector bundle \(B\to M\), a bundle map
\[
\ell:B^*\to Q,
\]
a skew-symmetric dull bracket
\[
[\cdot,\cdot]:\Gamma(Q)\times\Gamma(Q)\to\Gamma(Q),
\]
a \(Q\)-connection
\[
\nabla:\Gamma(Q)\times\Gamma(B)\to\Gamma(B),
\]
and a \(B^*\)-valued \(3\)-form
\[
\omega\in \Omega^3(Q,B^*),
\]
subject to five conditions: the dual connection condition on \(\ell\), compatibility of \(\ell\) with the dull bracket, Jacobi identity up to \(\ell\omega\),
\[
\mathrm{Jac}_{[\cdot,\cdot]}(q_1,q_2,q_3)=\ell(\omega(q_1,q_2,q_3)),
\]
a curvature identity for \(\nabla\) controlled by \(\omega\), and the Bianchi identity
\[
d_{\nabla^*}\omega=0.
\]
The associated homological vector field \(\mathcal Q\) on \(\mathcal M\) satisfies \(\mathcal Q^2=0\), and the cited work proves that these split data are exactly equivalent to that homological condition [1712.07035].

These two descriptions are not competing definitions. They are equivalent languages for the same object: one emphasizes \(L_\infty\)-algebroid operations \(l_1,l_2,l_3\), the other exposes ordinary bundle-theoretic data \((\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)\).

## 2. Equivalent geometric models through VB-Courant geometry

A central structural result is that Lie 2-algebroids admit a purely classical reformulation in terms of VB-Courant algebroids. A VB-Courant algebroid is a metric double vector bundle
\[
\begin{matrix}
E &\to& B\\
\downarrow && \downarrow\\
Q &\to& M
\end{matrix}
\]
with core \(Q^*\), together with a Courant algebroid structure on the top bundle \(E\to B\), whose anchor is linear and whose bracket preserves linear and core sections in the specified sense. The cited paper proves two equivalences: decomposed VB-Courant algebroids are equivalent to split Lie 2-algebroids, and more generally the category of all Lie 2-algebroids is equivalent to the category of all VB-Courant algebroids [1712.07035].

Given a VB-Courant algebroid and a chosen Lagrangian splitting
\[
\Sigma:Q\times_M B\to E,
\]
one recovers the split Lie 2-algebroid data as follows. The anchor of a lifted linear section \(\sigma_Q(q)\) defines the \(Q\)-connection \(\nabla_q\) on \(B\). The core anchor
\[
\rho_B:Q^*\to B
\]
has dual
\[
\ell=\rho_B^*:B^*\to Q.
\]
The bracket of two lifted linear sections determines the dull bracket on \(Q\) and a curvature term \(R_\omega\), hence a form \(\omega\in \Omega^3(Q,B^*)\). Conversely, from split data \((Q,B^*,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)\), one reconstructs a decomposed metric double vector bundle
\[
E=Q\oplus \operatorname{Hom}(B,Q^*)\to B
\]
with explicit anchor, derivation map, and bracket, and this yields a VB-Courant algebroid [1712.07035].

This equivalence clarifies a common point of confusion. The phrases “degree-2 NQ-manifold,” “split Lie 2-algebroid,” and “decomposed VB-Courant algebroid” refer to different presentations, not different kinds of objects. Their distinction is one of formalism: supergeometric, \(L_\infty\)-algebroid, or classical double vector bundle language.

## 3. Transitive Lie 2-algebroids and derivation-valued description

A Lie 2-algebroid
\[
(A_{-1}\oplus A_0;\rho,l_1,l_2,l_3)
\]
is transitive when the anchor \(\rho:A_0\to TM\) is surjective. Writing
\[
\mathcal G:=\ker(\rho)\subset A_0,
\]
the restrictions of \(l_2\) and \(l_3\) imply that
\[
(A_{-1}\oplus \mathcal G;l_1,l_2,l_3)
\]
is a graded bundle of Lie 2-algebras over \(M\). In this sense, a transitive Lie 2-algebroid is a nonabelian extension of the tangent Lie algebroid \(TM\) by the isotropy Lie 2-algebra bundle \(A_{-1}\oplus \mathcal G\) [1807.11789].

Choosing a splitting of
\[
0\to \mathcal G \xrightarrow{i} A_0 \xrightarrow{\rho} TM\to 0
\]
amounts to a section \(\sigma:TM\to A_0\) with \(\rho\circ \sigma=\mathrm{id}_{TM}\), together with a bundle map
\[
\gamma:\wedge^2 TM\to A_{-1}.
\]
This identifies \(A_0\cong TM\oplus \mathcal G\) and produces the operators
\[
V_0:TM\to \mathcal D(\mathcal G),\qquad V_1:TM\to \mathcal D(A_{-1}),
\]
the curvature \(2\)-form
\[
R:\wedge^2 TM\to \mathcal G,
\]
the higher curvature \(3\)-form
\[
I:\wedge^3 TM\to A_{-1},
\]
and mixed tensors
\[
J:\wedge^2 TM\otimes \mathcal G\to A_{-1},\qquad K:TM\otimes \wedge^2\mathcal G\to A_{-1}.
\]
In terms of these data, the structure maps of the transitive Lie 2-algebroid are written explicitly, with the degree-\(0\) bracket on \(TM\oplus \mathcal G\), the action on \(A_{-1}\), and the ternary bracket expressed through \(R_\gamma\), \(I_\gamma\), \(J\), and \(K\) [1807.11789].

The higher Jacobi identities translate into a list of compatibility equations among \(V_0,V_1,R,I,J,K\), including
\[
V_0(X)l_1=l_1V_1(X),\qquad d^{V_0}R_\gamma=I_\gamma,
\]
together with higher Bianchi and coherence identities. Rather than treating these as isolated relations, the cited paper organizes them through derivations of the isotropy Lie 2-algebra bundle. It defines degree \(0\) and degree \(-1\) derivations, forms a strict Lie 2-algebroid \(\mathrm{Der}(\mathcal G)\), and then a strict Lie 3-algebroid \(\mathrm{DER}(\mathcal G)\). The main structural theorem states that the data of a transitive Lie 2-algebroid are equivalent to a morphism of Lie algebroids
\[
F:TM\longrightarrow \mathrm{DER}(\mathcal G),
\]
with components
\[
F_1(X)=(V_0(X),V_1(X),K(X,\cdot,\cdot)),\quad
F_2(X,Y)=J(X,Y,\cdot)+R_\gamma(X,Y),\quad
F_3(X,Y,Z)=I(X,Y,Z).
\]
Conversely, such a morphism reproduces the structure of the transitive Lie 2-algebroid [1807.11789].

This derivation-valued description isolates the extension-theoretic content of transitivity. The tangent bundle contributes the base Lie algebroid, while the isotropy bundle contributes the fiberwise Lie 2-algebra, and the entire coupling is encoded by a morphism into a strict Lie 3-algebroid of derivations.

## 4. Quadratic structures and the first Pontryagin class

A quadratic Lie 2-algebroid is a transitive Lie 2-algebroid equipped with a degree \(1\) graded symmetric nondegenerate bilinear form
\[
\mathcal S:(A_{-1}\oplus \mathcal G)\otimes (A_{-1}\oplus \mathcal G)\to \mathbb R,
\]
satisfying invariance with respect to \(l_1\), the \(l_2\)-action of \(A_0\), and the \(3\)-bracket \(l_3\). Nondegeneracy identifies \(A_{-1}\) with \(\mathcal G^*\), so one may write \(A_{-1}=\mathcal G^*\) and use the standard pairing
\[
\mathcal S(u+m,v+p)=\langle u,p\rangle+\langle v,m\rangle.
\]
The resulting fiberwise invariance relations are expressed by
\[
\langle l_1(m),n\rangle=\langle m,l_1(n)\rangle,
\]
\[
\langle l_2(u,v),m\rangle=-\langle v,l_2(u,m)\rangle,
\]
\[
\langle l_3(u,v,w),x\rangle=-\langle w,l_3(u,v,x)\rangle,
\]
and, after choosing a splitting, by additional compatibility identities involving \(V_0,V_1,J,K\) [1807.11789].

From a chosen splitting \((\sigma,\gamma)\), one obtains the curvature \(2\)-form
\[
R_\gamma:\wedge^2 TM\to \mathcal G
\]
and the \(3\)-curvature
\[
I_\gamma:\wedge^3 TM\to \mathcal G^*.
\]
Pairing them by \(\mathcal S\) gives a \(5\)-form
\[
\mathcal S(R_\gamma,I_\gamma)\in \Omega^5(M).
\]
The cited work proves that this form is closed and that its cohomology class is independent of the chosen splitting. This defines the first Pontryagin class
\[
p_1(A):=[\mathcal S(R_\gamma,I_\gamma)]\in H^5(M).
\]
The class is a higher characteristic class: it lives in degree \(5\), not degree \(4\), and pairs a \(2\)-curvature with a compatible \(3\)-curvature [1807.11789].

The same paper relates this invariant to CLWX 2-algebroids, described there as 2-categorified analogs of Courant algebroids. An exact CLWX 2-algebroid has an associated ample Lie 2-algebroid, and that ample object is a quadratic transitive Lie 2-algebroid. Conversely, for a quadratic Lie 2-algebroid with \(l_1\) injective, the first Pontryagin class is exactly the obstruction to the existence of a CLWX extension: the extension exists if and only if
\[
p_1(A)=0.
\]
In this sense, \(p_1(A)\) plays the role of the obstruction class for lifting a higher infinitesimal structure to a higher Courant-type structure [1807.11789].

The paper also constructs quadratic Lie 2-algebroids from trivial principal \(2\)-bundles with \(T\)-connection \((A,B)\), where \(T\) is a strict Lie \(2\)-group. In the quadratic strict case, the resulting first Pontryagin class vanishes; the corresponding \(5\)-cocycle
\[
-\mathcal S(\operatorname{fcurv}(A,B),\operatorname{curv}(A,B))
\]
is exact [1807.11789].

## 5. Matched pairs, 2-representations, and double Lie algebroids

A major source of split Lie 2-algebroids comes from double Lie algebroids and matched pairs of \(2\)-representations. A double Lie algebroid is a double vector bundle
\[
\begin{array}{ccc}
D & \to & A\\
\downarrow && \downarrow\\
B & \to & M
\end{array}
\]
with Lie algebroid structures on all four sides, subject to VB-algebroid conditions and a Lie bialgebroid condition on appropriate duals. Once a linear splitting is chosen, each VB-algebroid side is encoded by a \(2\)-term representation up to homotopy, and the cited work proves that a double Lie algebroid with chosen linear splitting is equivalent to a pair of \(2\)-term representations up to homotopy satisfying compatibility conditions that extend the notion of matched pair of Lie algebroids [1409.1502].

Concretely, if \(C\) is the core, a splitting yields a \(2\)-representation of \(A\) on the complex
\[
C\xrightarrow{\partial_B} B
\]
with data \((\partial_B,\nabla^{AB},\nabla^{AC},R_A)\), and a \(2\)-representation of \(B\) on
\[
C\xrightarrow{\partial_A} A
\]
with data \((\partial_A,\nabla^{BA},\nabla^{BC},R_B)\). The main theorem identifies the double Lie algebroid condition with explicit matched-pair identities, labeled \((\mathrm M1)-(\mathrm M9)\), including a common anchor condition, induced Lie algebroid structure on the core, compatibility of the side brackets with \(\partial_A,\partial_B\), anchor compatibility, mixed curvature identities, and a coherence equation
\[
d_{\nabla^A}R_B=d_{\nabla^B}R_A.
\]
The tangent double of a Lie algebroid is the canonical example: its two induced \(2\)-representations form a matched pair [1409.1502].

The geometric payoff appears in the bicrossproduct construction. For a matched pair of \(2\)-representations of Lie algebroids \(A\) and \(B\) on a core \(C\), the cited work constructs a split Lie 2-algebroid on the graded bundle \(C[-2]\oplus (A\oplus B)[-1]\). The degree-\(0\) bundle is \(A\oplus B\), its anchor is
\[
\rho_{A\bowtie B}(a,b)=\rho_A(a)+\rho_B(b),
\]
the lower differential is
\[
\ell:C^*\to A\oplus B,\qquad \ell=(-\partial_A^*,\partial_B^*),
\]
the dull bracket is
\[
[(a,b),(a',b')]
=
\big([a,a']_A+\nabla^B_b a'-\nabla^B_{b'}a,\,
[b,b']_B+\nabla^A_a b'-\nabla^A_{a'}b\big),
\]
the connection on \(C\) is the sum of the two core actions, and the \(3\)-form \(\omega\) is built from \(R_A\) and \(R_B\) by the cyclic formula given in the source [1712.07035].

This bicrossproduct theorem has a converse: under the conditions that \(A\) and \(B\) are Lie subalgebroids and that \(\omega\) vanishes on triples from \(A\) and on triples from \(B\), the split Lie 2-algebroid arises from a matched pair of \(2\)-representations. The result explains how the “double” of a matched pair in double Lie algebroid geometry and the “bicrossproduct” in Lie 2-algebroid language are related [1712.07035].

## 6. Algebraic limits, stacky counterparts, and surrounding higher structures

Over a point, a Lie 2-algebroid reduces to a Lie 2-algebra, or equivalently a \(2\)-term \(L_\infty\)-algebra. The big-bracket formalism for weak Lie 2-bialgebras makes this explicit: a weak Lie \(2\)-algebra, its dual weak Lie \(2\)-coalgebra, and their compatibility are encoded by a master element \(\Theta\) of degree \(-4\) satisfying
\[
\{\Theta,\Theta\}=0.
\]
In the strict case, Lie \(2\)-bialgebras are in bijection with crossed modules of Lie bialgebras. This is the linear prototype for the corresponding bundle-valued notions of Lie 2-bialgebroid and Poisson \(2\)-groupoid [1109.2290].

A complementary limiting picture comes from vector fields on stacks. For a Lie groupoid \(G\rightrightarrows M\), the category \(X(G)\) of multiplicative vector fields and natural transformations is a strict Lie \(2\)-algebra. Its degree-\(0\) part is the Lie algebra of multiplicative vector fields, its degree-\(1\) part is canonically \(\Gamma(A(G))\), and the crossed-module structure is defined using the usual bracket of multiplicative vector fields, the Lie algebroid bracket on sections, and the Mackenzie–Xu action. The construction is Morita invariant and descends to a canonical Lie \(2\)-algebra \(X(\mathcal X)\) of vector fields on any geometric stack \(\mathcal X\) [1609.03944]. The paper is explicit that it remains at the Lie \(2\)-algebra level, but it also states that this is exactly the kind of \(2\)-term infinitesimal structure one expects in a Lie 2-algebroid theory on stacks.

Another algebraic source of \(2\)-term structures is the theory of enhanced Leibniz algebras. An enhanced Leibniz algebra \((t:W\to V,[\cdot,\cdot],\circ)\) induces a semi-strict Lie \(2\)-algebra with
\[
\ell_1=t,\qquad \ell_2(v_1,v_2)=[v_1,v_2]-t(v_1\circ v_2),\qquad \ell_2(v,w)=t(w)\circ v,
\]
and a ternary bracket \(y\) measuring Jacobi failure. The cited paper constructs a functor from enhanced Leibniz algebras to Lie \(2\)-algebras and emphasizes their role in higher gauge theories with \(1\)-form and \(2\)-form gauge fields [1901.01014]. Since Lie \(2\)-algebras are Lie 2-algebroids over a point, these constructions supply algebraic models for the local \(2\)-term \(L_\infty\) structures that Lie 2-algebroids globalize.

Taken together, these results fix the present mathematical profile of the subject. A Lie 2-algebroid is simultaneously a degree-\(2\) NQ-manifold, a split \(2\)-term \(L_\infty\)-algebroid, and—after choosing decompositions—a VB-Courant-type classical object. Its transitive case is controlled by derivation-valued curvature data, its quadratic case carries a degree-\(5\) Pontryagin class, and its relation to matched pairs, doubles, bialgebraic big brackets, and stacky vector fields situates it as a central object in higher Lie theory [1712.07035] [1807.11789].

Source: https://www.emergentmind.com/topics/lie-2-algebroid