Lie 2-Algebroid: Structures & Applications

• Lie 2-algebroid is defined as a 2-term L∞-algebroid with structure maps (l1, l2, l3) that replace standard Lie algebroid axioms with higher Jacobi identities.
• They are equivalently described via split NQ-manifolds and classical VB-Courant algebroids, bridging supergeometric formulations with classical differential geometry.
• Lie 2-algebroids encompass transitive, quadratic, and matched pair structures, leading to higher curvature invariants such as a degree-5 Pontryagin class.

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 $\omega$ (Sheng, 2018, Lean, 2017).

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

$A_{-1}\oplus A_0\to M$0

and structure maps

$A_{-1}\oplus A_0\to M$1

of degrees $A_{-1}\oplus A_0\to M$2, respectively. The bundle of sections $A_{-1}\oplus A_0\to M$3 carries a 2-term $A_{-1}\oplus A_0\to M$4-algebra structure $A_{-1}\oplus A_0\to M$5; $A_{-1}\oplus A_0\to M$6 is a derivation with respect to functions,

$A_{-1}\oplus A_0\to M$7

and $A_{-1}\oplus A_0\to M$8 is $A_{-1}\oplus A_0\to M$9-linear. The anchor compatibilities include

$l_1,l_2,l_3$0

This is the basic form in which Lie 2-algebroids appear in the study of transitive and quadratic cases (Sheng, 2018).

A second, equivalent split presentation uses the graded manifold

$l_1,l_2,l_3$1

A split Lie 2-algebroid in this form consists of an anchored vector bundle $l_1,l_2,l_3$2, another vector bundle $l_1,l_2,l_3$3, a bundle map

$l_1,l_2,l_3$4

a skew-symmetric dull bracket

$l_1,l_2,l_3$5

a $l_1,l_2,l_3$6-connection

$l_1,l_2,l_3$7

and a $l_1,l_2,l_3$8-valued $l_1,l_2,l_3$9-form

$Q[-1]\oplus B^*[-2]$0

subject to five conditions: the dual connection condition on $Q[-1]\oplus B^*[-2]$1, compatibility of $Q[-1]\oplus B^*[-2]$2 with the dull bracket, Jacobi identity up to $Q[-1]\oplus B^*[-2]$3,

$Q[-1]\oplus B^*[-2]$4

a curvature identity for $Q[-1]\oplus B^*[-2]$5 controlled by $Q[-1]\oplus B^*[-2]$6, and the Bianchi identity

$Q[-1]\oplus B^*[-2]$7

The associated homological vector field $Q[-1]\oplus B^*[-2]$8 on $Q[-1]\oplus B^*[-2]$9 satisfies $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$0, and the cited work proves that these split data are exactly equivalent to that homological condition (Lean, 2017).

These two descriptions are not competing definitions. They are equivalent languages for the same object: one emphasizes $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$1-algebroid operations $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$2, the other exposes ordinary bundle-theoretic data $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$3.

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

$(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$4

with core $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$5, together with a Courant algebroid structure on the top bundle $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$6, 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 (Lean, 2017).

Given a VB-Courant algebroid and a chosen Lagrangian splitting

$(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$7

one recovers the split Lie 2-algebroid data as follows. The anchor of a lifted linear section $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$8 defines the $(Q,B,\rho_Q,\ell,[\cdot,\cdot],\nabla,\omega)$9-connection $B^*$0 on $B^*$1. The core anchor

$B^*$2

has dual

$B^*$3

The bracket of two lifted linear sections determines the dull bracket on $B^*$4 and a curvature term $B^*$5, hence a form $B^*$6. Conversely, from split data $B^*$7, one reconstructs a decomposed metric double vector bundle

$B^*$8

with explicit anchor, derivation map, and bracket, and this yields a VB-Courant algebroid (Lean, 2017).

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, $B^*$9-algebroid, or classical double vector bundle language.

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

A Lie 2-algebroid

$3$0

is transitive when the anchor $3$1 is surjective. Writing

$3$2

the restrictions of $3$3 and $3$4 imply that

$3$5

is a graded bundle of Lie 2-algebras over $3$6. In this sense, a transitive Lie 2-algebroid is a nonabelian extension of the tangent Lie algebroid $3$7 by the isotropy Lie 2-algebra bundle $3$8 (Sheng, 2018).

Choosing a splitting of

$3$9

amounts to a section $\omega$0 with $\omega$1, together with a bundle map

$\omega$2

This identifies $\omega$3 and produces the operators

$\omega$4

the curvature $\omega$5-form

$\omega$6

the higher curvature $\omega$7-form

$\omega$8

and mixed tensors

$\omega$9

In terms of these data, the structure maps of the transitive Lie 2-algebroid are written explicitly, with the degree-$L_\infty$0 bracket on $L_\infty$1, the action on $L_\infty$2, and the ternary bracket expressed through $L_\infty$3, $L_\infty$4, $L_\infty$5, and $L_\infty$6 (Sheng, 2018).

The higher Jacobi identities translate into a list of compatibility equations among $L_\infty$7, including

$L_\infty$8

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 $L_\infty$9 and degree $A=A_{-1}\oplus A_0 \longrightarrow M,$0 derivations, forms a strict Lie 2-algebroid $A=A_{-1}\oplus A_0 \longrightarrow M,$1, and then a strict Lie 3-algebroid $A=A_{-1}\oplus A_0 \longrightarrow M,$2. The main structural theorem states that the data of a transitive Lie 2-algebroid are equivalent to a morphism of Lie algebroids

$A=A_{-1}\oplus A_0 \longrightarrow M,$3

with components

$A=A_{-1}\oplus A_0 \longrightarrow M,$4

Conversely, such a morphism reproduces the structure of the transitive Lie 2-algebroid (Sheng, 2018).

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 $A=A_{-1}\oplus A_0 \longrightarrow M,$5 graded symmetric nondegenerate bilinear form

$A=A_{-1}\oplus A_0 \longrightarrow M,$6

satisfying invariance with respect to $A=A_{-1}\oplus A_0 \longrightarrow M,$7, the $A=A_{-1}\oplus A_0 \longrightarrow M,$8-action of $A=A_{-1}\oplus A_0 \longrightarrow M,$9, and the $A_{-1}\oplus A_0\to M$00-bracket $A_{-1}\oplus A_0\to M$01. Nondegeneracy identifies $A_{-1}\oplus A_0\to M$02 with $A_{-1}\oplus A_0\to M$03, so one may write $A_{-1}\oplus A_0\to M$04 and use the standard pairing

$A_{-1}\oplus A_0\to M$05

The resulting fiberwise invariance relations are expressed by

$A_{-1}\oplus A_0\to M$06

$A_{-1}\oplus A_0\to M$07

$A_{-1}\oplus A_0\to M$08

and, after choosing a splitting, by additional compatibility identities involving $A_{-1}\oplus A_0\to M$09 (Sheng, 2018).

From a chosen splitting $A_{-1}\oplus A_0\to M$10, one obtains the curvature $A_{-1}\oplus A_0\to M$11-form

$A_{-1}\oplus A_0\to M$12

and the $A_{-1}\oplus A_0\to M$13-curvature

$A_{-1}\oplus A_0\to M$14

Pairing them by $A_{-1}\oplus A_0\to M$15 gives a $A_{-1}\oplus A_0\to M$16-form

$A_{-1}\oplus A_0\to M$17

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

$A_{-1}\oplus A_0\to M$18

The class is a higher characteristic class: it lives in degree $A_{-1}\oplus A_0\to M$19, not degree $A_{-1}\oplus A_0\to M$20, and pairs a $A_{-1}\oplus A_0\to M$21-curvature with a compatible $A_{-1}\oplus A_0\to M$22-curvature (Sheng, 2018).

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 $A_{-1}\oplus A_0\to M$23 injective, the first Pontryagin class is exactly the obstruction to the existence of a CLWX extension: the extension exists if and only if

$A_{-1}\oplus A_0\to M$24

In this sense, $A_{-1}\oplus A_0\to M$25 plays the role of the obstruction class for lifting a higher infinitesimal structure to a higher Courant-type structure (Sheng, 2018).

The paper also constructs quadratic Lie 2-algebroids from trivial principal $A_{-1}\oplus A_0\to M$26-bundles with $A_{-1}\oplus A_0\to M$27-connection $A_{-1}\oplus A_0\to M$28, where $A_{-1}\oplus A_0\to M$29 is a strict Lie $A_{-1}\oplus A_0\to M$30-group. In the quadratic strict case, the resulting first Pontryagin class vanishes; the corresponding $A_{-1}\oplus A_0\to M$31-cocycle

$A_{-1}\oplus A_0\to M$32

is exact (Sheng, 2018).

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 $A_{-1}\oplus A_0\to M$33-representations. A double Lie algebroid is a double vector bundle

$A_{-1}\oplus A_0\to M$34

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 $A_{-1}\oplus A_0\to M$35-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 $A_{-1}\oplus A_0\to M$36-term representations up to homotopy satisfying compatibility conditions that extend the notion of matched pair of Lie algebroids (Gracia-Saz et al., 2014).

Concretely, if $A_{-1}\oplus A_0\to M$37 is the core, a splitting yields a $A_{-1}\oplus A_0\to M$38-representation of $A_{-1}\oplus A_0\to M$39 on the complex

$A_{-1}\oplus A_0\to M$40

with data $A_{-1}\oplus A_0\to M$41, and a $A_{-1}\oplus A_0\to M$42-representation of $A_{-1}\oplus A_0\to M$43 on

$A_{-1}\oplus A_0\to M$44

with data $A_{-1}\oplus A_0\to M$45. The main theorem identifies the double Lie algebroid condition with explicit matched-pair identities, labeled $A_{-1}\oplus A_0\to M$46, including a common anchor condition, induced Lie algebroid structure on the core, compatibility of the side brackets with $A_{-1}\oplus A_0\to M$47, anchor compatibility, mixed curvature identities, and a coherence equation

$A_{-1}\oplus A_0\to M$48

The tangent double of a Lie algebroid is the canonical example: its two induced $A_{-1}\oplus A_0\to M$49-representations form a matched pair (Gracia-Saz et al., 2014).

The geometric payoff appears in the bicrossproduct construction. For a matched pair of $A_{-1}\oplus A_0\to M$50-representations of Lie algebroids $A_{-1}\oplus A_0\to M$51 and $A_{-1}\oplus A_0\to M$52 on a core $A_{-1}\oplus A_0\to M$53, the cited work constructs a split Lie 2-algebroid on the graded bundle $A_{-1}\oplus A_0\to M$54. The degree-$A_{-1}\oplus A_0\to M$55 bundle is $A_{-1}\oplus A_0\to M$56, its anchor is

$A_{-1}\oplus A_0\to M$57

the lower differential is

$A_{-1}\oplus A_0\to M$58

the dull bracket is

$A_{-1}\oplus A_0\to M$59

the connection on $A_{-1}\oplus A_0\to M$60 is the sum of the two core actions, and the $A_{-1}\oplus A_0\to M$61-form $A_{-1}\oplus A_0\to M$62 is built from $A_{-1}\oplus A_0\to M$63 and $A_{-1}\oplus A_0\to M$64 by the cyclic formula given in the source (Lean, 2017).

This bicrossproduct theorem has a converse: under the conditions that $A_{-1}\oplus A_0\to M$65 and $A_{-1}\oplus A_0\to M$66 are Lie subalgebroids and that $A_{-1}\oplus A_0\to M$67 vanishes on triples from $A_{-1}\oplus A_0\to M$68 and on triples from $A_{-1}\oplus A_0\to M$69, the split Lie 2-algebroid arises from a matched pair of $A_{-1}\oplus A_0\to M$70-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 (Lean, 2017).

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 $A_{-1}\oplus A_0\to M$71-term $A_{-1}\oplus A_0\to M$72-algebra. The big-bracket formalism for weak Lie 2-bialgebras makes this explicit: a weak Lie $A_{-1}\oplus A_0\to M$73-algebra, its dual weak Lie $A_{-1}\oplus A_0\to M$74-coalgebra, and their compatibility are encoded by a master element $A_{-1}\oplus A_0\to M$75 of degree $A_{-1}\oplus A_0\to M$76 satisfying

$A_{-1}\oplus A_0\to M$77

In the strict case, Lie $A_{-1}\oplus A_0\to M$78-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 $A_{-1}\oplus A_0\to M$79-groupoid (Chen et al., 2011).

A complementary limiting picture comes from vector fields on stacks. For a Lie groupoid $A_{-1}\oplus A_0\to M$80, the category $A_{-1}\oplus A_0\to M$81 of multiplicative vector fields and natural transformations is a strict Lie $A_{-1}\oplus A_0\to M$82-algebra. Its degree-$A_{-1}\oplus A_0\to M$83 part is the Lie algebra of multiplicative vector fields, its degree-$A_{-1}\oplus A_0\to M$84 part is canonically $A_{-1}\oplus A_0\to M$85, 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 $A_{-1}\oplus A_0\to M$86-algebra $A_{-1}\oplus A_0\to M$87 of vector fields on any geometric stack $A_{-1}\oplus A_0\to M$88 (Berwick-Evans et al., 2016). The paper is explicit that it remains at the Lie $A_{-1}\oplus A_0\to M$89-algebra level, but it also states that this is exactly the kind of $A_{-1}\oplus A_0\to M$90-term infinitesimal structure one expects in a Lie 2-algebroid theory on stacks.

Another algebraic source of $A_{-1}\oplus A_0\to M$91-term structures is the theory of enhanced Leibniz algebras. An enhanced Leibniz algebra $A_{-1}\oplus A_0\to M$92 induces a semi-strict Lie $A_{-1}\oplus A_0\to M$93-algebra with

$A_{-1}\oplus A_0\to M$94

and a ternary bracket $A_{-1}\oplus A_0\to M$95 measuring Jacobi failure. The cited paper constructs a functor from enhanced Leibniz algebras to Lie $A_{-1}\oplus A_0\to M$96-algebras and emphasizes their role in higher gauge theories with $A_{-1}\oplus A_0\to M$97-form and $A_{-1}\oplus A_0\to M$98-form gauge fields (Strobl et al., 2019). Since Lie $A_{-1}\oplus A_0\to M$99-algebras are Lie 2-algebroids over a point, these constructions supply algebraic models for the local $l_1,l_2,l_3$00-term $l_1,l_2,l_3$01 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-$l_1,l_2,l_3$02 NQ-manifold, a split $l_1,l_2,l_3$03-term $l_1,l_2,l_3$04-algebroid, and—after choosing decompositions—a VB-Courant-type classical object. Its transitive case is controlled by derivation-valued curvature data, its quadratic case carries a degree-$l_1,l_2,l_3$05 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 (Lean, 2017, Sheng, 2018).