Split Courant algebroids as L-structures

https://doi.org/10.1016/j.geomphys.2020.103790Get rights and content

Abstract

We show that split Courant algebroids, i.e., those defined on a Whitney sum AA, are in a one-to-one correspondence with multiplicative curved L-algebras. This one-to-one correspondence extends to Nijenhuis morphisms and behaves well under the operation of twisting by a bivector.

Introduction

Courant algebroids were introduced by Liu, Weinstein and Xu [19] to interpret the bracket defined by Courant to study constraints on Dirac manifolds. In short, a Courant algebroid is a vector bundle EM equipped with a symmetric nondegenerate bilinear form, together with a morphism of vector bundles ρ:ETM and such that the space of sections Γ(E) has the structure of a Leibniz algebra. All these data satisfy some compatibility conditions that we recall in Section 2. This is not the original definition introduced in [19], but an equivalent non-skew-symmetric version that uses what is known as Dorfman bracket instead of the Courant bracket.

There is an alternative way to define Courant algebroids, introduced by Roytenberg [20], which is the one that we consider in this paper. Courant algebroids can be described as degree 2 symplectic graded manifolds together with a degree 3 function Θ satisfying {Θ,Θ}=0, where {,} is the graded Poisson bracket corresponding to the graded symplectic structure. This graded Poisson bracket is called big bracket [14]. The morphism ρ and the Dorfman bracket are recovered as derived brackets (see [20]).

When the Courant structure is defined on the Whitney sum AA of a vector bundle A and its dual, we have what we call a split Courant algebroid. The Courant structure on AA can be the double of a Lie bialgebroid structure on (A,A), the double of a quasi-Lie bialgebroid structure on (A,A) or, more generally, the double of a proto-Lie bialgebroid structure on (A,A) [21].

Besides Courant algebroids, the other relevant structures in this paper are L-algebras, also known as strongly homotopy Lie algebras. They were introduced by Lada and Stasheff [16] and consist of collections of n-ary brackets satisfying higher Jacobi identities. In the original definition of [16], the n-ary brackets are skew-symmetric, but in this paper we consider the equivalent definition where the brackets are graded symmetric. Roytenberg and Weinstein [22] showed that to each Courant algebroid one can associate a Lie 2-algebra and, recently, Lang, Sheng and Xu [17] proved a converse of this result.

In this paper we show that split Courant algebroids AA are in a one-to-one correspondence with multiplicative curved L-algebra structures on Γ(A)[2]. This extends other previous results. In 2002, Roytenberg [21] mentions that each split Courant algebroid which is the double of a quasi-Lie bialgebroid has an associated L-algebra defined on Γ(A)[2] and that the converse holds. No proof is given. In 2015, Frégier and Zambon [9] proved that each split Courant algebroid which is the double of a proto-Lie bialgebroid determines a curved L-algebra structure on Γ(A)[2]. The proof uses the higher derived brackets construction of Voronov [26]. We give an alternative and simpler proof that only uses the properties of the graded Poisson bracket, and we also prove the converse (Theorems Theorem 4.1, Theorem 4.3).

While we were finishing this paper, Cueca and Mehta [8] established an isomorphism between the sheaf of functions on a graded symplectic manifold of degree 2 and the Keller–Waldmann algebra [13] of cochains on a vector bundle EM, equipped with a non-degenerated symmetric bilinear form ,. In this algebra, the 3cochains coincide with the pre-Courant structures on (E,,) and in this case the one-to-one correspondence was already known (see [20]). In the case where E=AA and AA is equipped with the canonical pairing ,, the map M defined by Eqs. (10)–(13) can be recovered from the isomorphism defined in [8].

Having established a one-to-one correspondence between split Courant algebroids and multiplicative curved L-algebras, it seemed interesting to discuss the behaviour of Nijenhuis operators under this correspondence. Nijenhuis morphisms on Courant algebroids were initially considered in [6] and then revisited in [11], under the graded manifold approach to Courant algebroids. Regarding Nijenhuis forms on L-algebras, they were introduced in [4]. This notion also appears in [18], although with a simpler definition which turns out to be a particular case of the one in [4]. In this paper we consider the definition of [4]. Using the Lie 2-algebra associated to each Courant algebroid according to [22], some relations between Nijenhuis morphisms on Courant algebroids and Nijenhuis forms on Lie 2-algebras were already established in [4]. In the current paper the approach is different since split Courant algebroids AA are seen as graded manifolds, which is not the case in [4], and the curved L-algebra structure is defined on Γ(A)[2].

One of the advantages of viewing split Courant algebroids as graded manifolds, besides simpler and more efficient computations, is the relation with Lie algebroid structures on A. Indeed, we have that (AA,Θ=μ) is a Courant algebroid if and only if (A,μ) is a Lie algebroid. Having this is mind, we characterize some known structures on Lie algebroids as Nijenhuis forms on L-algebra structures on Γ(A)[2].

Another type of operation that behave well under the one-to-one correspondence that we established, is the twisting on Courant algebroids and on L-algebras. The twisting of a split Courant algebroid by a bivector was defined in [21], and the same operation can be done on L-algebras. In [10] it is shown that the twisting of a L-algebra by a degree zero element π is an L-algebra provided that π is a Maurer–Cartan element. In the case of a curved L-algebra, we show that π no longer needs to be a Maurer–Cartan element.

The paper is organized as follows. Section 2 contains a brief review of the main notions concerning (pre-)Courant algebroids as well as Nijenhuis morphisms on (pre-)Courant algebroids. In Section 3 we recall the definition of curved L-algebras and of Nijenhuis forms on curved L-algebras. Section 4 contains the main theorem, that establishes a one-to-one correspondence between split Courant algebroids and curved L-algebras. In Section 5 we show that the one-to-one correspondence preserves deformations by Nijenhuis operators. In particular, some Nijenhuis morphisms on Courant algebroids are characterized as Nijenhuis forms on curved L-algebras. Some well known structures on Lie algebroids are viewed as Nijenhuis form on L-algebras. In Section 6 we discuss the twisting of a split Courant algebroid and of a curved L-algebra by πΓ(2A) and we show that the one-to-one correspondence preserves these twisting operations. In Section 7 we combine the one-to-one correspondence with the operations of twisting by π and deformation by a skew-symmetric vector-valued form on Γ(A)[2]. The commutative diagrams included along Sections 4 to 7 can be combined to form a commutative cubic diagram, presented at the end of the paper.

Section snippets

Preliminaries on Courant algebroids and their Nijenhuis morphisms

In this section we recall the definition of Courant algebroid and how it can be seen as a Q-manifold, following the approach of [20], [25]. The notion of Nijenhuis morphism on a (pre)-Courant algebroid is also recalled.

Let EM be a vector bundle equipped with a fibrewise non-degenerate symmetric bilinear form ,.

Definition 2.1

[3]

A pre-Courant structure on (E,,) is a pair (ρ,[,]), where ρ:ETM is a morphism of vector bundles called the anchor, and [,]:Γ(E)×Γ(E)Γ(E) is a R-bilinear bracket, called the

Review on L-algebras and Nijenhuis forms

In this section we recall the definitions of curved (pre-)L-algebra and Nijenhuis form on an L-algebra, following [4]. For the definition of an L-algebra we consider graded symmetric brackets, which is not the case in the original definition introduced in [16]. Both definitions are equivalent, and the equivalence is given by the so-called décalage isomorphism (see [4], [26] for more details).

In what follows, we consider graded vector spaces with all components of finite dimension.

Definition 3.1

A curved

From Courant algebroids to L-algebras and back

In this section we prove a theorem that generalizes a result initially established by Roytenberg [21] in the case of a split Courant algebroid which is the double of a quasi-Lie bialgebroid, and then extended by Frégier and Zambon [9] to the case where the Courant structure is the double of a proto-bialgebroid. A result similar to the one in [9] was obtained by Gualtieri, Matviichuk and Scott [12]. In all cases, given a split Courant algebroid structure, a (curved) L-algebra is constructed.

Nijenhuis on Courant algebroids and on L-algebras

In this section, to each skew-symmetric endomorphism on AA we associate a vector-valued form of degree zero on Γ(A)[2] and we analyse how the induced deformations on pre-Courant algebroids and curved pre-L-algebras are related under the map M. This leads to a relationship between Nijenhuis operators and also enable us to see some structures on Lie algebroids as Nijenhuis forms on L-algebras.

Consider a skew-symmetric endomorphism J:AAAA given as in (5): J=NπωN.Recall that J is

Twisting by a bivector

The purpose of this section is to discuss the twisting of a Courant algebroid and of a curved L-algebra by a bivector.

Given a pre-Courant structure Θ on AA, the notion of twisting Θ by a bivector πΓ(2A) was introduced in [21] as the canonical transformation given by the flow of the Hamiltonian vector field Xπ{π,} associated to π: eπ1+{π,}+12!{π,{π,}}+13!{π,{π,{π,}}}+When applied to Θ=ψ+γ+μ+ϕ yields eπΘ=ψ+{π,γ}+12{π,{π,μ}}+16{π,{π,{π,ϕ}}}+γ+{π,μ}+12{π,{π,ϕ}}+μ+{π,ϕ}+ϕ. Since Xπ{π,}

Twisting and deformation

In this section we combine the operations of twisting and deformation on both (pre-)Courant algebroids and curved (pre-)L-algebras.

Let πΓ(2A) be a bivector. Take NΓ(AA) such that Nπ#=π#N and consider the bivector πNΓ(2A) defined, for all α,βΓ(A), by πN(α,β)=π(Nα,β) or, using the big bracket, by πN=12{π,N}. Mimicking the twisting of a pre-Courant structure by π, we may define the twisting of N by π, and set eπNN+{π,N}. We denote by JN and JπN the skew-symmetric endomorphisms of AA

Acknowledgements

The authors would like to thank Alfonso Tortorella and James Stasheff for several comments and suggestions on a preliminary version of this paper. This work was partially supported by the Centre for Mathematics of the University of Coimbra, Portugal - UIDB/00324/2020, funded by the Portuguese Government, Portugal through FCT/MCTES.

References (26)

  • M. Cueca, R. Mehta, Courant cohomology, Cartan calculus, connections, curvature, characteristic classes,...
  • FrégierY. et al.

    Simultaneous deformations and Poisson geometry

    Compos. Math.

    (2015)
  • GetzlerE.

    Lie theory for nilpotent L-algebras

    Ann. of Math.

    (2009)
  • Cited by (5)

    View full text