Research article Special Issues

An epidemic model with time delays determined by the infectivity and disease durations


  • Received: 17 March 2023 Revised: 19 May 2023 Accepted: 24 May 2023 Published: 02 June 2023
  • We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.

    Citation: Masoud Saade, Samiran Ghosh, Malay Banerjee, Vitaly Volpert. An epidemic model with time delays determined by the infectivity and disease durations[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12864-12888. doi: 10.3934/mbe.2023574

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  • We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.



    In this paper, we study the categorification of VB-Lie algebroids and 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 CLWX 2-algebroids and higher analogues of the string Lie 2-algebra.

    An NQ-manifold is an N-manifold 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 CLWX 2-algebroid (named after Courant-Liu-Weinstein-Xu), which can be viewed as the categorification of a Courant algebroid. Furthermore, 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 CLWX 2-algebroid is a Leibniz 2-algebra, or a Lie 3-algebra. There is also a close relationship between 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 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 VB-Lie algebroid, which is equivalent to Mackenzie's LA-vector bundle ([38]). A 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 VB-algebroid morphisms and representations up to homotopy were studied in [15].

    In his PhD thesis [32], Li-Bland introduced the notion of a VB-Courant algebroid which is the doublization of a Courant algebroid [35], and established abstract correspondence between NQ-manifolds of degree 2 and 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 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]), VB-Lie algebroids ([7,19]) and VB-Lie groupoids ([7,20]) became more and more important recently and are widely studied. In particular, the Lie theory relating VB-Lie algebroids and 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 VB-Lie 2-algebroid, which can be viewed as the categorification of a VB-Lie algebroid, or doublization of a Lie 2-algebroid:

    We show that the tangent prolongation of a Lie 2-algebroid is a VB-Lie 2-algebroid and the graded fat bundle associated to a 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 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 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 VB-CLWX 2-algebroid, which can be viewed as both the doublization of a CLWX 2-algebroid and the categorification of a VB-Courant algebroid. More importantly, we show that after choosing a splitting, there is a one-to-one correspondence between VB-CLWX 2-algebroids and split Lie 3-algebroids (NQ-manifolds of degree 3). The tangent prolongation of a CLWX 2-algebroid is a VB-CLWX 2-algebroid naturally. We go on defining E-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 VB-Courant algebroid, there is an E-Courant algebroid [24,31], we show that on the graded fat bundle associated to a VB-CLWX 2-algebroid, there is an E-CLWX 2-algebroid structure naturally. Similar to the case of a CLWX 2-algebroid, an E-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, VB-Lie algebroids and VB-Courant algebroids. In Section 3, we introduce the notion of a 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 VB-Lie 2-algebroid naturally. In Section 4, first we construct a strict Lie 3-algebroid End(E)=(End2(E),End1(E),D(E),p,d,[,]C) from a 3-term complex of vector bundles E:E2πE1πE0 and then we define a flat superconnection of a Lie 2-algebroid A=(A1,A0,a,l1,l2,l3) on this 3-term complex of vector bundles to be a morphism from A to End(E). We show that after choosing a splitting, 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 VB-CLWX 2-algebroid and show that after choosing a splitting, there is a one-to-one correspondence between VB-CLWX 2-algebroids and Lie 3-algebroids. In Section 6, we introduce the notion of an E-CLWX 2-algebroid and show that the graded fat bundle associated to a VB-CLWX 2-algebroid is an E-CLWX 2-algebroid naturally. In particular, the graded jet bundle of a CLWX 2-algebroid, which is the graded fat bundle of the tangent prolongation of this CLWX 2-algebroid, is a TM-CLWX 2-algebroid. We can also obtain a Lie 3-algebra from an E-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

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

    A morphism of double vector bundles

    (φ;fA,fB;fM):(D;A,B;M)(D;A,B;M)

    consists of maps φ: DD, fA:AA, fB:BB, fM:MM, such that each of (φ,fB), (φ,fA), (fA,fM) and (fB,fM) is a morphism of the relevant vector bundles.

    The space of sections ΓB(D) of the vector bundle DB is generated as a C(B)-module by core sections ΓcB(D) and linear sections ΓlB(D). See [41] for more details. For a section c:MC, the corresponding core section c:BD is defined as

    c(bm)=˜0bm+A¯c(m),mM,bmBm,

    where ˉ means the inclusion CD. A section ξ:BD is called linear if it is a bundle morphism from BM to DA over a section XΓ(A). We will view BC both as Hom(B,C) and Hom(C,B) depending on what it acts. Given ψΓ(BC), there is a linear section ˜ψ:BD over the zero section 0A:MA given by

    ˜ψ(bm)=˜0bm+A¯ψ(bm).

    Note that ΓlB(D) is locally free as a C(M)-module. Therefore, ΓlB(D) is equal to Γ(ˆA) for some vector bundle ˆAM. The vector bundle ˆA 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

    0BCˆAprA0. (2.1)

    Definition 2.1. ([19,Definition 3.4]) A VB-Lie algebroid is a double vector bundle (D;A,B;M) equipped with a Lie algebroid structure (DB,a,[,]D) such that the anchor a:DTB is linear, i.e. a:(D;A,B;M)(TB;TM,B;M) is a morphism of double vector bundles, and the Lie bracket [,]D is linear:

    [ΓlB(D),ΓlB(D)]DΓlB(D),[ΓlB(D),ΓcB(D)]DΓcB(D),[ΓcB(D),ΓcB(D)]D=0.

    The vector bundle AM is then also a Lie algebroid, with the anchor a and the bracket [,]A defined as follows: if ξ1,ξ2 are linear over X1,X2Γ(A), then the bracket [ξ1,ξ2]D is linear over [X1,X2]A.

    Definition 2.2. ([32,Definition 3.1.1]) A VB-Courant algebroid is a metric double vector bundle (D;A,B;M) such that (DB,S, [[,]],ρ) is a Courant algebroid and the following conditions are satisfied:

    (i) The anchor map ρ:DTB is linear;

    (ii) The Courant bracket is linear. That is

    [[ΓlB(D),ΓlB(D)]]ΓlB(D),   [[ΓlB(D),ΓcB(D)]]ΓcB(D),[[ΓcB(D),ΓcB(D)]]=0.

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

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

    Definition 3.1. ([52,Definition 2.1]) A split Lie n-algebroid is a non-positively graded vector bundle A=A0A1An+1 over a manifold M equipped with a bundle map a:A0TM (called the anchor), and n+1 many brackets li:Γ(iA)Γ(A) with degree 2i for 1in+1, such that

    1. Γ(A) is an n-term L-algebra:

    i+j=k+1(1)i(j1)σSh1i,kisgn(σ)Ksgn(σ)lj(li(Xσ(1),,Xσ(i)),Xσ(i+1),,Xσ(k))=0,

    where the summation is taken over all (i,ki)-unshuffles Sh1i,ki with i1 and "Ksgn(σ)" is the Koszul sign for a permutation σSk, i.e.

    X1Xk=Ksgn(σ)Xσ(1)Xσ(k).

    2. l2 satisfies the Leibniz rule with respect to the anchor a:

    l2(X0,fX)=fl2(X0,X)+a(X0)(f)X,X0Γ(A0),fC(M),XΓ(A).

    3. For i2, li's are C(M)-linear.

    Denote a split Lie n-algebroid by (An+1,,A0,a,l1,,ln+1), or simply by A. We will only use a split Lie 2-algebroid (A1,A0,a,l1,l2,l3) and a split Lie 3-algebroid (A2,A1,A0,a,l1,l2,l3,l4). For a split Lie n-algebroid, we have a generalized Chevalley-Eilenberg complex (Γ(Symm(A[1])),δ). See [5,52] for more details. Then A[1] is an NQ-manifold of degree n. A split Lie n-algebroid morphism AA can be defined to be a graded vector bundle morphism f:Symm(A[1])Symm(A[1]) such that the induced pull-back map f:C(A[1])C(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 (l3=0,l4=0) and this is what we will use in this paper to define flat superconnections.

    Definition 3.2. Let A=(A1,A0,a,l1,l2,l3) be a split Lie 2-algebroid and A=(A2,A1,A0,a,l1,l2) a strict split Lie 3-algebroid. A morphism F from A to A consists of:

    a bundle map F0:A0A0,

    a bundle map F1:A1A1,

    a bundle map F20:2A0A1,

    a bundle map F21:A0A1A2,

    a bundle map F3:3A0A2,

    such that for all X0,Y0,Z0,X0iΓ(A0), i=1,2,3,4, X1,Y1Γ(A1), we have

    aF0=a,l1F1=F0l1,F0l2(X0,Y0)l2(F0(X0),F0(Y0))=l1F20(X0,Y0),F1l2(X0,Y1)l2(F0(X0),F1(Y1))=F20(X0,l1(Y1))l1F21(X0,Y1),l2(F1(X1),F1(Y1))=F21(l1(X1),Y1)F21(X1,l1(Y1)),l2(F0(X0),F2(Y0,Z0))F20(l2(X0,Y0),Z0)+c.p.=F1(l3(X0,Y0,Z0))+l1F3(X0,Y0,Z0),l2(F0(X0),F21(Y0,Z1))+l2(F0(Y0),F21(Z1,X0))+l2(F1(Z1),F20(X0,Y0))=F21(l2(X0,Y0),Z1)+c.p.+F3(X0,Y0,l1(Z1)),

    and

    4i=1(1)i+1(F21(X0i,l3(X01,,^X0i,X04))+l2(F0(X0i),F3(X01,,^X0i,X04)))+i<j(1)i+j(F3(l2(X0i,X0j),X0k,X0l)+c.p.12l2(F20(X0i,X0j),F20(X0k,X0l)))=0,

    where k<l and {k,l}{i,j}=.

    Let (A1,A0,a,l1,l2,l3) be a split Lie 2-algebroid. Then for all X0,Y0Γ(A0) and X1Γ(A1), Lie derivatives L0X0:Γ(Ai)Γ(Ai), i=0,1, L1X1:Γ(A1)Γ(A0) and L3X0,Y0:Γ(A1)Γ(A0) are defined by

    {L0X0α0,Y0=ρ(X0)Y0,α0α0,l2(X0,Y0),L0X0α1,Y1=ρ(X0)Y1,α1α1,l2(X0,Y1),L1X1α1,Y0=α1,l2(X1,Y0),L3X0,Y0α1,Z0=α1,l3(X0,Y0,Z0), (3.1)

    for all α0Γ(A0),α1Γ(A1),Y1Γ(A1),Z0Γ(A0). If (A[1],a,l1,l2,l3) is also a split Lie 2-algebroid, we denote by L0,L1,L3,δ 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 (D1;A1,B1;M1D0;A0,B0;M0). Morphisms between graded double vector bundles can be defined in an obvious way. We will denote by D and A the graded vector bundles DB0DB1 and A0A1 respectively. Now we are ready to introduce the main object in this section.

    Definition 3.3. A VB-Lie 2-algebroid is a graded double vector bundle

    (D1;A1,B;MD0;A0,B;M)

    equipped with a Lie 2-algebroid structure (DB1,DB0,a,l1,l2,l3) on D such that

    (i) The anchor a:D0TB is linear, i.e. we have a bundle map a:A0TM such that (a;a,idB;idM) is a double vector bundle morphism (see Diagram (i));

    (ii) l1 is linear, i.e. we have a bundle map l1:A1A0 such that (l1;l1,idB;idM) is a double vector bundle morphism (see Diagram (ii));

    (iii) l2 is linear, i.e.

    l2(ΓlB(D0),ΓlB(D0))ΓlB(D0),l2(ΓlB(D0),ΓcB(D0))ΓcB(D0),l2(ΓlB(D0),ΓlB(D1))ΓlB(D1),l2(ΓlB(D0),ΓcB(D1))ΓcB(D1),l2(ΓcB(D0),ΓlB(D1))ΓcB(D1),l2(ΓcB(D0),ΓcB(D1))=0;l2(ΓcB(D0),ΓcB(D0))=0.

    (iv) l3 is linear, i.e.

    l3(ΓlB(D0),ΓlB(D0),ΓlB(D0))ΓlB(D1),l3(ΓlB(D0),ΓlB(D0),ΓcB(D0))ΓcB(D1),l3(ΓcB(D0),ΓcB(D0),)=0.

    Since Lie 2-algebroids are the categorification of Lie algebroids, VB-Lie 2-algebroids can be viewed as the categorification of VB-Lie algebroids.

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

    Theorem 3.4. Let (D1;A1,B;MD0;A0,B;M) be a VB-Lie 2-algebroid. Then

    (A1,A0,a,l1,l2,l3)

    is a split Lie 2-algebroid, where l2 is defined by the property that if ξ01,ξ02,ξ0ΓlB(D0) are linear sections over X01,X02,X0Γ(A0), and ξ1ΓlB(D1) is a linear section over X1Γ(A1), then l2(ξ01,ξ02)ΓlB(D0) is a linear section over l2(X01,X02)Γ(A0) and l2(ξ0,ξ1)ΓlB(D1) is a linear section over l2(X0,X1)Γ(A1). Similarly, l3 is defined by the property that if ξ01,ξ02,ξ03ΓlB(D0) are linear sections over X01,X02,X03Γ(A0), then l3(ξ01,ξ02,ξ03)ΓlB(D1) is a linear section over l3(X01,X02,X03)Γ(A1).

    Proof. Since l2 is linear, for any ξiΓlB(Di) satisfying πAi(ξi)=0, we have

    πA(i+j)(l2(ξi,ηj))=0,ηjΓlB(Dj).

    This implies that l2 is well-defined. Similarly, l3 is also well-defined.

    By the fact that l1:D1D0 is a double vector bundle morphism over l1:A1A0, we can deduce that (Γ(A1),Γ(A0),l1,l2,l3) is a Lie 2-algebra. We only give a proof of the property

    l1(l2(X0,X1))=l2(X0,l1(X1)),X0Γ(A0),X1Γ(A1). (3.2)

    The other conditions in the definition of a Lie 2-algebra can be proved similarly. In fact, let ξ0ΓlB(D0),ξ1ΓlB(D1) be linear sections over X0,X1 respectively, then by the equality l1(l2(ξ0,ξ1))=l2(ξ0,l1(ξ1)), we have

    πA0l1(l2(ξ0,ξ1))=πA0l2(ξ0,l1(ξ1)).

    Since l1:D1D0 is a double vector bundle morphism over l1:A1A0, the left hand side is equal to

    πA0l1(l2(ξ0,ξ1))=l1πA1l2(ξ0,ξ1)=l1l2(X0,X1),

    and the right hand side is equal to

    πA0l2(ξ0,l1(ξ1))=l2(πA0(ξ0),πA0(l1(ξ1)))=l2(X0,l1(X1)).

    Thus, we deduce that (3.2) holds.

    Finally, for all X0Γ(A0), YiΓ(Ai) and fC(M), let ξ0ΓlB(D0) and ηiΓlB(Di),i=0,1 be linear sections over X0 and Yi. Then qB(f)ηi is a linear section over fYi. By the fact that a is a double vector bundle morphism over a, we have

    l2(X0,fYi)=πAil2(ξ0,qB(f)ηi)=πAi(qB(f)l2(ξ0,ηi)+a(ξ0)(qB(f))ηi)=fl2(X0,Yi)+a(X0)(f)Yi.

    Therefore, (A1,A0,a,l1,l2,l3) is a Lie 2-algebroid.

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

    Consider the associated graded fat bundle ˆA1ˆA0, obviously we have

    Proposition 1. Let (D1;A1,B;MD0;A0,B;M) be a VB-Lie 2-algebroid. Then (ˆA1,ˆA0,ˆa,ˆl1,ˆl2,ˆl3) is a split Lie 2-algebroid, where ˆa=apr and ˆl1,ˆl2,ˆl3 are the restriction of l1,l2,l3 on linear sections respectively.

    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 BC1BC0. Since l1 is linear, it induces a bundle map lC1:C1C0. The restriction of ˆl1 on BC1 is given by

    ˆl1(ϕ1)=lC1ϕ1,ϕ1Γ(BC1)=Γ(Hom(B,C1)). (3.4)

    Since the anchor a:D0TB is a double vector bundle morphism, it induces a bundle map ϱ:C0B via

    ϱ(c0),ξ=a(c0)(ξ),c0Γ(C0),ξΓ(B). (3.5)

    Then by the Leibniz rule, we deduce that the restriction of ˆl2 on Γ(BC1BC0) is given by

    ˆl2(ϕ0,ψ0)=ϕ0ϱψ0ψ0ϱϕ0, (3.6)
    ˆl2(ϕ0,ψ1)=ˆl2(ψ1,ϕ0)=ψ1ϱϕ0, (3.7)

    for all ϕ0,ψ0Γ(BC0)=Γ(Hom(B,C0)), ψ1Γ(BC1)=Γ(Hom(B,C1)). Since l3 is linear, the restriction of l3 on BC1BC0 vanishes. Obviously, the anchor is trivial. Thus, the split Lie 2-algebroid structure on BC1BC0 is exactly given by (3.4), (3.6) and (3.7). Therefore, BC1BC0 is a graded bundle of strict Lie 2-algebras.

    An important example of 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 AM, TA is a Lie algebroid over TM. A section σ:MA gives rise to a linear section σTdσ:TMTA and a core section σC:TMTA by contraction. Any section of TA over TM is generated by such sections. A function fC(M) induces two types of functions on TM by

    fC=qf,fT=df,

    where q:TMM is the projection. We have the following relations about the module structure:

    (fσ)C=fCσC,(fσ)T=fTσC+fCσT. (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, (3.9)

    for all XX(M). See [32,Example 2.5.4] and [40] for more details.

    Now for split Lie 2-algebroids, we have

    Proposition 2. Let A=(A1,A0,a,l1,l2,l3) be a split Lie 2-algebroid. Then

    (TA1,TA0,a,l1,l2,l3)

    is a split Lie 2-algebroid over TM, where a:TA0TTM is given by

    a(σ0T)=a(σ0)T,a(σ0C)=a(σ0)C, (3.10)

    l1:ΓTM(TA1)ΓTM(TA0) is given by

    l1(σ1T)=l1(σ1)T,l1(σ1C)=l1(σ1)C, (3.11)

    l2:ΓTM(TAi)×ΓTM(TAj)ΓTM(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,

    and l3:3ΓTM(TA0)ΓTM(TA1) is given by

    l3(σ0T,τ0T,ς0T)=l3(σ0,τ0,ς0)T,l3(σ0T,τ0T,ς0C)=l3(σ0,τ0,ς0)C, (3.12)

    and l3(σ0T,τ0C,ς0C)=0, for all σ0,τ0,ς0Γ(A0) and σ1,τ1Γ(A1). Moreover, we have the following VB-Lie 2-algebroid:

    Proof. By the fact that A=(A1,A0,a,l1,l2,l3) is a split Lie 2-algebroid, it is straightforward to deduce that (TA1,TA0,a,l1,l2,l3) is a split Lie 2-algebroid over TM. Moreover, a,l1,l2,l3 are all linear, which implies that it is a VB-Lie 2-algebroid.

    The associated fat bundles of double vector bundles (TA1;A1,TM;M) and (TA0;A0,TM;M) are the jet bundles JA1 and JA0 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 (A1,A0,a,l1,l2,l3) be a split Lie 2-algebroid. Then we obtain that (JA1,JA0,ˆa,ˆl1,ˆl2,ˆl3) is a split Lie 2-algebroid, where ˆa,ˆl1,ˆl2,ˆl3 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,

    for all σ0,τ0,ζ0Γ(A0) and τ1Γ(A1).

    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 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 E2πE1πE0 by E. Sections of the covariant differential operator bundle D(E) are of the form d=(d0,d1,d2), where di:Γ(Ei)Γ(Ei) are R-linear maps such that there exists XX(M) satisfying

    di(fei)=fdi(ei)+X(f)ei,fC(M),eiΓ(Ei).

    Equivalently, D(E)=D(E0)×TMD(E1)×TMD(E2). Define p:D(E)TM by

    p(d0,d1,d2)=X. (4.1)

    Then the covariant differential operator bundle D(E) fits the following exact sequence:

    0End(E0)End(E1)End(E2)D(E)TM0. (4.2)

    Denote by End1(E)=Hom(E0,E1)Hom(E1,E2). Denote by End2(E)=Hom(E0,E2). Define d:End2(E)End1(E) by

    d(θ2)=πθ2θ2π,θ2Γ(Hom(E0,E2)), (4.3)

    and define d:End1(E)D(E) by

    d(θ1)=πθ1+θ1π,θ1Γ(Hom(E0,E1)Hom(E1,E2)). (4.4)

    Then we define a degree 0 graded symmetric bracket operation [,]C on the section space of the graded bundle End2(E)End1(E)D(E) by

    [d,t]C=dttd,d,tΓ(D(E)), (4.5)
    [d,θi]C=dθiθid,dΓ(D(E)),θiΓ(Endi(E)), (4.6)
    [θ1,ϑ1]C=θ1ϑ1+ϑ1θ1,θ1,ϑ1Γ(End1(E)). (4.7)

    Denote by Dπ(E)D(E) the subbundle of D(E) whose section dΓ(Dπ(E)) satisfying πd=dπ, or in term of components,

    d0π=πd1,d1π=πd2.

    It is obvious that Γ(Dπ(E)) is closed under the bracket operation [,]C and

    d(End1(E))Dπ(E).

    Then it is straightforward to verify that

    Theorem 4.1. Let E2πE1πE0 be a 3-term complex of vector bundles over M. Then (End2(E),End1(E),Dπ(E),p,d,[,]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 (A1,A0,a,l1,l2,l3) on a 3-term complex of vector bundles E2πE1πE0 consists of:

    a bundle morphism F0:A0Dπ(E),

    a bundle morphism F1:A1End1(E),

    a bundle morphism F20:2A0End1(E),

    a bundle morphism F21:A0A1End2(E),

    a bundle morphism F3:3A0End2(E).

    A superconnection is called flat if (F0,F1,F20,F21,F3) is a Lie n-algebroid morphism from the split Lie 2-algebroid (A1,A0,a,l1,l2,l3) to the strict split Lie 3-algebroid (End2(E),End1(E),Dπ(E),p,d,[,]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 E1πE0, a superconnection will only consists of

    a bundle morphism F0=(F00,F01):ADπ(E),

    a bundle morphism F20:2A0Hom(E0,E1).

    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 VB-Lie 2-algebroids. Let (DB1,DB0,a,l1,l2,l3) be a VB-Lie 2-algebroid structure on the graded double vector bundle (D1;A1,B;MD0;A0,B;M). Recall from Theorem 3.4 and Proposition 1 that both (A1,A0,a,l1,l2,l3) and (ˆA1,ˆA0,ˆa,ˆl1,ˆl2,ˆl3) are split Lie 2-algebroids.

    Choose a horizontal lift s=(s0,s1):A0A1ˆA0ˆA1 of the short exact sequence of split Lie 2-algebroids (3.3). Define B:A0D(B) by

    BX0b,ξ=a(X0)ξ,bb,ˆa(s0(X0))(ξ),X0Γ(A0),bΓ(B),ξΓ(B).

    Since for all ϕ0Γ(BC0), we have ˆa(ϕ0)=0, it follows that B is well-defined.

    We define 0:A0D(C0) and 1:A0D(C1) by

    0X0c0=l2(s0(X0),c0),1X0c1=l2(s0(X0),c1), (4.8)

    for all X0Γ(A0),c0Γ(C0),c1Γ(C1). Define Υ1:A1Hom(B,C0) and Υ2:A1Hom(C0,C1) by

    Υ1X1=s0(l1(X1))ˆl1(s1(X1)),Υ2X1c0=l2(s1(X1),c0), (4.9)

    for all X1Γ(A1),c0Γ(C0). Since l2 is linear, 0, 1 and Υ are well-defined.

    Define R0:2Γ(A0)Γ(Hom(B,C0)), Λ:2Γ(A0)Γ(Hom(C0,C1)) and R1:Γ(A0)Γ(A1)Γ(Hom(B,C1)) by

    R0(X0,Y0)=s0l2(X0,Y0)ˆl2(s0(X0),s0(Y0)), (4.10)
    Λ(X0,Y0)(c0)=l3(s0(X0),s0(Y0),c0), (4.11)
    R1(X0,Y1)=s1l2(X0,Y1)ˆl2(s0(X0),s1(Y1)), (4.12)

    for all X0,Y0Γ(A0) and Y1Γ(A1) Finally, define Ξ:3Γ(A0)Hom(B,C1) by

    Ξ(X0,Y0,Z0))=s1l3(X0,Y0,Z0)ˆl3(s0(X0),s0(Y0),s0(Z0)). (4.13)

    By the equality l1l2(s0(X0),c1)=l2(s0(X0),lC1(c1)), we obtain

    lC11X0=0X0lC1. (4.14)

    By the fact that a:D0TB preserves the bracket operation, we obtain

    BX0ϱ(c0),ξ=a(X0)ϱ(c0),ξϱ(c0),a(s0(X0))(ξ)=[a(s0(X0)),a(c0)]TB(ξ)=a(l2(s0(X0),c0))(ξ)=ϱ0X0c0,ξ,

    which implies that

    BX0ϱ=ϱ0X0. (4.15)

    By (4.14) and (4.15), we deduce that (BX0,0X0,1X0)D(E), where E is the 3-term complex of vector bundles C1lC1C0ϱB. Then we obtain a superconnection (F0,F1,F20,F21,F3) of the Lie 2-algebroid (A1,A0,a,l1,l2,l3) on the 3-term complex of vector bundles C1lC1C0ϱB, where

    F0=(B,0,1),F1=(Υ1,Υ2),F20=(R0,Λ),F21=R1,F3=Ξ.

    Theorem 4.3. There is a one-to-one correspondence between VB-Lie 2-algebroids (D1;A1,B;MD0;A0,B;M) and flat superconnections (F0,F1,F20,F21,F3) of the split Lie 2-algebroid (A1,A0,a,l1,l2,l3) on the 3-term complex of vector bundles C1lC1C0ϱB by choosing a horizontal lift s=(s0,s1):A0A1ˆA0ˆA1.

    Proof. First it is obvious that

    pF0=a. (4.16)

    Using equalities al1=0 and al1=0, we have

    Bl1X1b,ξ=a(l1(X1))b,ξb,a(s0(l1(X1)))(ξ)=b,a(Υ1X1)(ξ),

    which implies that

    Bl1X1=ϱΥ1X1. (4.17)

    For 0, we can obtain

    0l1(X1)=l2(s0l1(X1),)|C0=l2(l1(s1(X1))+Υ1X1,)|C0=lC1Υ2X1+Υ1X1ϱ. (4.18)

    For 1, we have

    1l1(X1)=l2(s0l1(X1),)|C1=l2(l1(s1(X1))+Υ1X1,)|C1=Υ2X1lC1. (4.19)

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

    F0l1=dF1. (4.20)

    By straightforward computation, we have

    Bl2(X0,Y0)bBX0BY0b+BY0BX0b,ξ=b,a(ˆl2(s0(X0),s0(Y0))s0l2(X0,Y0))(ξ)=b,a(R0(X0,Y0))(ξ),

    which implies that

    Bl2(X0,Y0)BX0BY0+BY0BX0=ϱR0(X0,Y0). (4.21)

    Similarly, we have

    0l2(X0,Y0)c00X00Y0c0+0Y00X0c0=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),

    which implies that

    0l2(X0,Y0)0X00Y0+0Y00X0=lC1Λ(X0,Y0)+R0(X0,Y0)ϱ, (4.22)

    and

    1l2(X0,Y0)c11X01Y0c1+1Y01X0c1=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),

    which implies that

    1l2(X0,Y0)1X01Y0+1Y01X0=Λ(X0,Y0)lC1. (4.23)

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

    F0(l2(X0,Y0))[F0(X0),F0(Y0)]C=dF20(X0,Y0). (4.24)

    By the equality

    l2(s0(X0),l2(s1(Y1),c0))+c.p.=ˆl3(s0(X0),l1(s1(Y1)),c0),

    we obtain

    [F0(X0),Υ2Y1]CΥ2l2(X0,Y1)=Λ(X0,l1(Y1))R1(X0,Y1)ϱ. (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))lC1R1(X0,Y1)=s0l2(X0,l1(Y1))ˆl2(s0(X0),s0l1(Y1)Υ1Y1)lC1R1(X0,Y1)=[F0(X0),Υ1Y1]C+R0(X0,l1(Y1))lC1R1(X0,Y1). (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). (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. (4.28)

    By the equality

    ˆl2(s0(X0),ˆl2(s0(Y0),s0(Z0)))+c.p.=ˆl1ˆl3(s0(X0),s0(Y0),s0(Z0)),

    we deduce that

    [F0(X0),R0(Y0,Z0)]C+R0(X0,l2(Y0,Z0))+c.p.=Υ1l3(X0,Y0,Z0)+lC1Ξ(X0,Y0,Z0). (4.29)

    By the equality

    l2(s0(X0),l3(s0(Y0),s0(Z0),c0))l3(l2(s0(X0),s0(Y0)),s0(Z0),c0)+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. (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). (4.31)

    Then by the equality

    ˆl2(s0(X0),ˆl2(s0(Y0),s1(Z1)))+c.p.=ˆl3(s0(X0),s0(Y0),ˆl1(s1(Z1))),

    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. (4.32)

    Finally, by the equality

    4i=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,

    we deduce that

    4i=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. (4.33)

    By (4.16), (4.20), (4.24), (4.27), (4.28), (4.31)-(4.33), we deduce that (F0,F1,F20,F21,F3) is a morphism from the split Lie 2-algebroid (A1,A0,a,l1,l2,l3) to the strict split Lie 3-algebroid

    (End2(E),End1(E),Dπ(E),p,d,[,]C).

    Conversely, let (A1,A0,a,l1,l2,l3) be a split Lie 2-algebroid and (F0,F1,F20,F21,F3) a flat superconnection on the 3-term complex C1lC1C0ϱB. Then we can obtain a VB-Lie 2-algebroid structure on the split graded double vector bundle (A1BC1;A1,B;MA0BC0;A0,B;M). We leave the details to readers. The proof is finished.

    In this section, first we recall the notion of a CLWX 2-algebroid. Then we explore what is a metric graded double vector bundle, and introduce the notion of a VB-CLWX 2-algebroid, which can be viewed as the categorification of a 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-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-algebras are equivalent. Due to this reason, a 2-term Lod-algebra will be called a Leibniz 2-algebra directly in the sequel.

    Definition 5.1. ([34]) A CLWX 2-algebroid is a graded vector bundle E=E1E0 over M equipped with a non-degenerate graded symmetric bilinear form S on E, a bilinear operation :Γ(Ei)×Γ(Ej)Γ(E(i+j)), 0i+j1, which is skewsymmetric on Γ(E0)×Γ(E0), an E1-valued 3-form Ω on E0, two bundle maps :E1E0 and ρ:E0TM, such that E1 and E0 are isotropic and the following axioms are satisfied:

    (i) (Γ(E1),Γ(E0),,,Ω) is a Leibniz 2-algebra;

    (ii) for all eΓ(E), ee=12DS(e,e), where D:C(M)Γ(E1) is defined by

    S(Df,e0)=ρ(e0)(f),fC(M),e0Γ(E0); (5.1)

    (iii) for all e11,e12Γ(E1), S((e11),e12)=S(e11,(e12));

    (iv) for all e1,e2,e3Γ(E), ρ(e1)S(e2,e3)=S(e1e2,e3)+S(e2,e1e3);

    (v) for all e01,e02,e03,e04Γ(E0), S(Ω(e01,e02,e03),e04)=S(e03,Ω(e01,e02,e04)).

    Denote a CLWX 2-algebroid by (E1,E0,,ρ,S,,Ω), or simply by E. Since the section space of a 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 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],Θ) be a symplectic NQ manifold of degree 3, where A is an ordinary vector bundle and Θ is a degree 4 function on T[3]A[2] satisfying {Θ,Θ}=0. Here {,} is the canonical Poisson bracket on T[3]A[2]. Then (A[1],A,,ρ,S,,Ω) is a CLWX 2-algebroid, where the bilinear form S is given by

    S(X+α,Y+β)=X,β+Y,α,X,YΓ(A),α,βΓ(A),

    and , ρ, and Ω are given by derived brackets. More precisely, we have

    α={α,Θ},αΓ(A),ρ(X)(f)={f,{X,Θ}},XΓ(A),fC(M),XY={Y,{X,Θ}},X,YΓ(A),Xα={α,{X,Θ}},XΓ(A),αΓ(A),αX={X,{α,Θ}},XΓ(A),αΓ(A),Ω(X,Y,Z)={Z,{Y,{X,Θ}}},X,Y,ZΓ(A).

    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 (D1;A1,B;MD0;A0,B;M) equipped with a degree 1 nondegenerate graded symmetric bilinear form S on the graded bundle DB1DB0 such that it induces an isomorphism between graded double vector bundles

    where B means dual over B.

    Given a metric graded double vector bundle, we have

    C0A1,C1A0.

    In the sequel, we will always identify C0 with A1, C1 with A0. Thus, a metric graded double vector bundle is of the following form:

    Now we are ready to put a CLWX 2-algebroid structure on a graded double vector bundle.

    Definition 5.4. A VB-CLWX 2-algebroid is a metric graded double vector bundle

    ((D1;A1,B;MD0;A0,B;M),S),

    equipped with a CLWX 2-algebroid structure (DB1,DB0,,ρ,S,,Ω) such that

    (i) is linear, i.e. there exists a unique bundle map ¯:A1A0 such that :D1D0 is a double vector bundle morphism over ¯:A1A0 (see Diagram (iii));

    (ii) the anchor ρ is a linear, i.e. there exists a unique bundle map ¯ρ:A0TM such that ρ:D0TB is a double vector bundle morphism over ¯ρ:A0TM (see Diagram (iv));

    (iii) the operation is linear;

    (iv) Ω is linear.

    Since a CLWX 2-algebroid can be viewed as the categorification of a Courant algebroid, we can view a VB-CLWX 2-algebroid as the categorification of a VB-Courant algebroid.

    Example 1. Let (A1,A0,a,l1,l2,l3) be a Lie 2-algebroid. Let E0=A0A1, E1=A1A0 and E=E0E1. Then (E1,E0,,ρ,S,,Ω) is a CLWX 2-algebroid, where :E1E0 is given by

    (X1+α0)=l1(X1)+l1(α0),X1Γ(A1),α0Γ(A0),

    ρ:E0TM is given by

    ρ(X0+α1)=a(X0),X0Γ(A0),α1Γ(A1),

    the symmetric bilinear form S=(,)+ is given by

    (X0+α1+X1+α0,Y0+β1+Y1+β0)+=X0,β0+Y0,α0+X1,β1+Y1,α1,

    the operation is given by

    {(X0+α1)(Y0+β1)=l2(X0,Y0)+L0X0β1L0Y0α1,(X0+α1)(X1+α0)=l2(X0,X1)+L0X0α0+ιX1δ(α1),(X1+α0)(X0+α1)=l2(X1,X0)+L1X1α1ιX0δ(α0), (5.2)

    and the E1-valued 3-form Ω is defined by

    Ω(X0+α1,Y0+β1,Z0+ζ1)=l3(X0,Y0,Z0)+L3X0,Y0ζ1+L3Z0,X0β1+L3Y0,Z0α1,

    where L0,L1,L3 are given by (3.1). It is straightforward to see that this CLWX 2-algebroid gives rise to a VB-CLWX 2-algebroid:

    Example 2. For any manifold M, (T[1]M,TM,=0,ρ=id,S,,Ω=0) is a CLWX 2-algebroid, where S is the natural symmetric pairing between TM and TM, and is the standard Dorfman bracket given by

    (X+α)(Y+β)=[X,Y]+LXβιYdα,X,YX(M),α,βΩ1(M). (5.3)

    See [34,Remark 3.4] for more details. In particular, for any vector bundle E, (TE,TE,=0,ρ=id,S,,Ω=0) is a CLWX 2-algebroid, which gives rise to a VB-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 VB-CLWX 2-algebroids.

    Proof. Let A=(A2,A1,A0,a,l1,l2,l3,l4) be a split Lie 3-algebroid. Then T[3]A[1] is a symplectic NQ manifold of degree 3. Note that

    T[3]A[1]=T[3](A0×MA1×MA2)[1],

    where A0×MA1×MA2 is viewed as a vector bundle over the base A2 and A1×MA0×MA2 is its dual bundle. Denote by (xi,μj,ξk,θl,pi,μj,ξk,θl) a canonical (Darboux) coordinate on T[3](A0×MA1×MA2)[1], where xi is a smooth coordinate on M, μjΓ(A2) is a fibre coordinate on A2, ξkΓ(A0) is a fibre coordinate on A0, θlΓ(A1) is a fibre coordinate on A1 and (pi,μj,ξk,θl) are the momentum coordinates for (xi,μj,ξk,θl). About their degrees, we have

    (xiμjξkθlpiμjξkθl00113322)

    The symplectic structure is given by

    ω=dxidpi+dμjdμj+dξkdξk+dθldθl,

    which is degree 3. The Lie 3-algebroid structure gives rise to a degree 4 function Θ satisfying {Θ,Θ}=0. By Theorem 5.2, we obtain a CLWX 2-algebroid (D1,D0,,ρ,S,,Ω), where D1=A1×MA0×MA2 and D0=A0×MA1×MA2 are vector bundles over A2. Obviously, they give the graded double vector bundle

    (A1×MA0×MA2;A1,A2;MA0×MA1×MA2;A0,A2;M).

    The section space ΓA2(D0) are generated by Γ(A1) (the space of core sections) and Γ(A2A1)Γ(A0) (the space of linear sections) as C(A2)-module. Similarly, The section space ΓA2(D1) are generated by Γ(A0) and Γ(A2A0)Γ(A1) as C(A2)-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),

    for all e0=X0+ψ1+α1ΓA2(D0) and e1=X1+ψ0+α0ΓA2(D1), where XiΓ(Ai), ψiΓ(A2Ai) and αiΓ(Ai). Then it is obvious that

    ((A1×MA0×MA2;A1,A2;MA0×MA1×MA2;A0,A2;M),S)

    is a metric graded double vector bundle.

    The bundle map :D1D0 is given by

    (X1+ψ0+α0)=l1(X1)+l2(X1,)|A1+ψ0l1+l1(α0).

    Thus, :D1D0 is a double vector bundle morphism over l1:A1A0.

    Note that functions on A2 are generated by fibrewise constant functions C(M) and fibrewise linear functions Γ(A2). For all fC(M) and X2Γ(A2), the anchor ρ:D0TA2 is given by

    ρ(X0+ψ1+α1)(f+X2)=a(X0)(f)+α1,l1(X2)+l2(X0,X2)+ψ1(l1(X2)).

    Therefore, for a linear section X0+ψ1ΓlA2(D0), the image ρ(X0+ψ1) is a linear vector field and for a core section α1Γ(A1), the image ρ(α1) is a constant vector field. Thus, ρ is linear.

    The bracket operation is given by

    (X0+ψ1+α1)(Y0+ϕ1+β1)=l2(X0,Y0)+l3(X0,Y0,)|A1+l2(X0,ϕ1())ϕ1l2(X0,)|A1+L0X0β1+ψ1l2(Y0,)|A1l2(Y0,ψ1())+ψ1l1ϕ1ϕ1l1ψ1β1l1ψ1L0Y0α1+α1l1ϕ1,(X0+ψ1+α1)(Y1+ϕ0+β0)=l2(X0,Y1)+l3(X0,,Y1)|A0+l2(X0,ϕ0())ϕ0l2(X0,)|A0+L0X0β0ψ1l2(,Y1)|A0+δ(ψ1(Y1))+ψ1l1ϕ0+ιY1δα1+α1l1ϕ0,(Y1+ϕ0+β0)(X0+ψ1+α1)=l2(Y1,X0)l3(X0,,Y1)|A0l2(X0,ϕ0())+ϕ0l2(X0,)|A0+δ(ϕ0(X0))ιX0δβ0+ψ1l2(,Y1)|A0ψ1l1ϕ0+L1Y1α1α1l1ϕ0.

    Then it is straightforward to see that the operation is linear.

    Finally, Ω is given by

    Ω(X0+ψ1+α1,Y0+ϕ1+β1,Z0+φ1+γ1)=l3(X0,Y0,Z0)+l4(X0,Y0,Z0,)φ1l3(X0,Y0,)|A0ϕ1l3(Z0,X0,)|A0ψ1l3(Y0,Z0,)|A0+L3X0,Y0γ1+L3Y0,Z0α1+L3Z0,X0β1,

    which implies that Ω is also linear.

    Thus, a split Lie 3-algebroid gives rise to a split VB-CLWX 2-algebroid:

    Conversely, given a split VB-CLWX 2-algebroid:

    where D1=A1×MA0×MB and D0=A0×MA1×MB, then we can deduce that the corresponding symplectic NQ-manifold of degree 3 is T[3]A[1], where A=A0A1B 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 A. We omit details.

    Remark 3. Since every double vector bundle is splitable, every VB-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 VB-CLWX 2-algebroids. We omit such details.

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

    Proposition 3. Let (E1,E0,,ρ,S,,Ω) be a CLWX 2-algebroid. Then we obtain that (TE1,TE0,˜,˜ρ,˜S,˜,˜Ω) is a CLWX 2-algebroid over TM, where the bundle map ˜:TE1TE0 is given by

    ˜(σ1T)=(σ1)T,˜(σ1C)=(σ1)C,

    the bundle map ˜ρ:TE0TTM is given by

    ˜ρ(σ0T)=ρ(σ0)T,˜ρ(σ0C)=ρ(σ0)C,

    the degree 1 bilinear form ˜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,

    the bilinear operation ˜ is given by

    σ0T˜τ0T=(σ0τ0)T,σ0T˜τ0C=τ0C˜σ0T=(σ0τ0)C,σ0C˜τ0C=0,σ0T˜τ1T=(σ0τ1)T,σ0T˜τ1C=σ0C˜τ1T=(σ0τ1)C,σ0C˜τ1C=0,τ1T˜σ0T=(τ1σ0)T,τ1C˜σ0T=τ1T˜σ0C=(τ1σ0)C,τ1C˜σ0C=0,

    and ˜Ω:3TE0TE1 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,

    for all σ0,τ0,ς0Γ(E0) and σ1,τ1Γ(E1).

    Moreover, we have the following VB-CLWX 2-algebroid:

    Proof. Since (E1,E0,,ρ,S,,Ω) is a CLWX 2-algebroid, it is straightforward to deduce that (TE1,TE0,˜,˜ρ,˜S,˜,˜Ω) is a CLWX 2-algebroid over TM. Moveover, it is obvious that ˜,˜ρ,˜S,˜,˜Ω are all linear, which implies that we have a VB-CLWX 2-algebroid.

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

    There is an E-valued pairing ,E between the jet bundle JE and the first order covariant differential operator bundle DE defined by

    μ,dEd(u),d(DE)m,μ(JE)m,uΓ(E)statisfyingμ=[u]m.

    Definition 6.1. Let E be a vector bundle. An E-CLWX 2-algebroid is a 6-tuple (K,,ρ,S,,Ω), where K=K1K0 is a graded vector bundle over M and

    :K1K0 is a bundle map;

    S:KKE is a surjective graded symmetric nondegenerate E-valued pairing of degree 1, which induces an embedding: KHom(K,E);

    ρ:K0DE is a bundle map, called the anchor, such that ρ(JE)K1, i.e.

    S(ρ(μ),e0)=μ,ρ(e0)E,μΓ(JE),e0Γ(K0);

    :Γ(Ki)×Γ(Kj)Γ(K(i+j)),0i+j1 is an R-bilinear operation;

    Ω:3K0K1 is a bundle map,

    such that the following properties hold:

    (E1) (Γ(K),,,Ω) is a Leibniz 2-algebra;

    (E2) for all eΓ(K), ee=12DS(e,e), where D:Γ(E)Γ(K1) is defined by

    S(Du,e0)=ρ(e0)(u),uΓ(E),e0Γ(K0); (6.1)

    (E3) for all e11,e12Γ(K1), S((e11),e12)=S(e11,(e12));

    (E4) for all e1,e2,e3Γ(K), ρ(e1)S(e2,e3)=S(e1e2,e3)+S(e2,e1e3);

    (E5) for all e01,e02,e03,e04Γ(K0), S(Ω(e01,e02,e03),e04)=S(e03,Ω(e01,e02,e04));

    (E6) for all e01,e02Γ(K0), ρ(e01e02)=[ρ(e01),ρ(e02)]D, where [,]D is the commutator bracket on Γ(DE).

    A CLWX 2-algebroid can give rise to a Lie 3-algebra ([34,Theorem 3.11]). Similarly, an E-CLWX 2-algebroid can also give rise to a Lie 3-algebra. Consider the graded vector space e=e2e1e0, where e2=Γ(E), e1=Γ(K1) and e0=Γ(K0). We introduce a skew-symmetric bracket on Γ(K),

    [[e1,e2]]=12(e1e2e2e1),e1,e2Γ(K), (6.2)

    which is the skew-symmetrization of .

    Theorem 6.2. An E-CLWX 2-algebroid (K,,ρ,S,,Ω) gives rise to a Lie 3-algebra (e,l1,l2,l3,l4), where li are given by

    l1(u)=D(u),uΓ(E),l1(e1)=(e1),e1Γ(K1),l2(e01,e02)=[[e01,e02]],e01,e02Γ(K0),l2(e0,e1)=[[e0,e1]],e0Γ(K0),e1Γ(K1),l2(e0,f)=12S(e0,Df),e0Γ(K0),fΓ(E),l2(e11,e12)=0,e11,e12Γ(K1),l3(e01,e02,e03)=Ω(e01,e02,e03),e01,e02,e03Γ(K0),l3(e01,e02,e1)=T(e01,e02,e1),e01,e02Γ(K0),e1Γ(K1),l4(e01,e02,e03,e04)=¯Ω(e01,e02,e03,e04),e01,e02,e03,e04Γ(K0),

    where the totally skew-symmetric T:Γ(K0)×Γ(K0)×Γ(K1)Γ(E) is given by

    T(e01,e02,e1)=16(S(e01,[[e02,e1]])+S(e1,[[e01,e02]])+S(e02,[[e1,e01]])), (6.3)

    and ¯Ω:4Γ(K0)Γ(E) is given by

    ¯Ω(e01,e02,e03,e04)=S(Ω(e01,e02,e03),e04).

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

    Let (DB1,DB0,,ρ,S,,Ω) be a VB-CLWX 2-algebroid on the graded double vector bundle (D1;A1,B;MD0;A0,B;M). Then we have the associated graded fat bundles ˆA1ˆA0, which fit the exact sequences:

    0BA0ˆA1A10,0BA1ˆA0A00.

    Since the bundle map is linear, it induces a bundle map ˆ:ˆA1ˆA0. Since the anchor ρ is linear, it induces a bundle map ˆρ:ˆA0DB, where sections of DB 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 ˆS on the graded fat bundle ˆA1ˆA0. Since the operation is linear, it induces an operation ˆ:ˆAi׈AjˆA(i+j), 0i+j1. Finally, since Ω is linear, it induces an ˆΩ:Γ(3^A0)ˆA1. Then we obtain:

    Theorem 6.3. A VB-CLWX 2-algebroid gives rise to a B-CLWX 2-algebroid structure on the corresponding graded fat bundle. More precisely, let (DB1,DB0,,ρ,S,,Ω) be a VB-CLWX 2-algebroid on the graded double vector bundle (D1;A1,B;MD0;A0,B;M) with the associated graded fat bundle ˆA1ˆA0. Then (ˆA1,ˆA0,ˆ,ˆρ,ˆS,ˆ,ˆΩ) is a B-CLWX 2-algebroid.

    Proof. Since all the structures defined on the graded fat bundle ˆA1ˆA0 are the restriction of the structures in the VB-CLWX 2-algebroid, it is straightforward to see that all the axioms in Definition 6.1 hold.

    Example 3. Consider the VB-CLWX 2-algebroid given in Example 2, the corresponding E-CLWX 2-algebroid is ((JE)[1],DE,=0,ρ=id,S=(,)E,,Ω=0), where the graded symmetric nondegenerate E-valued pairing (,)E is given by

    (d+μ,t+ν)E=μ,tE+ν,dE,d+μ,t+νDEJE,

    and is given by

    See [10] for more details.

    Example 4. Consider the VB-CLWX 2-algebroid given in Proposition 3. The graded fat bundle is JE1JE0. It follows that the graded jet bundle associated to a CLWX 2-algebroid is a TM-CLWX 2-algebroid. This is the higher analogue of the result that the jet bundle of a Courant algebroid is TM-Courant algebroid given in [11]. See also [24] for more details. $

    As applications of E-CLWX 2-algebroids introduced in the last section, we construct Lie 3-algebras from Lie 3-algebras in this section. Let (g2,g1,g0,l1,l2,l3,l4) be a Lie 3-algebra. By Theorem 5.5, the corresponding VB-CLWX 2-algebroid is given by

    where D1=g1g0g2 and D0=g0g1g2.

    By Theorem 6.3, we obtain:

    Proposition 4. Let (g2,g1,g0,l1,l2,l3,l4) be a Lie 3-algebra. Then there is an E-CLWX 2-algebroid (Hom(g0,g2)g1,Hom(g1,g2)g0,,ρ,S,,Ω), where for all xi,yi,zigi, ϕi,ψi,φiHom(gi,g2), :Hom(g0,g2)g1Hom(g1,g2)g0 is given by

    (ϕ0+x1)=ϕ0l1+l2(x1,)|g1+l1(x1), (7.1)

    ρ:Hom(g1,g2)g0gl(g2) is given by

    ρ(ϕ1+x0)=ϕ1l1+l2(x0,)|g2, (7.2)

    the g2-valued pairing S is given by

    S(ϕ1+x0,ψ0+y1)=ϕ1(y1)+ψ0(x0), (7.3)

    the operation is given by

    {(x0+ψ1)(y0+ϕ1)=l2(x0,y0)+l3(x0,y0,)|g1+l2(x0,ϕ1())ϕ1l1ψ1ϕ1l2(x0,)|g1+ψ1l2(y0,)|g1l2(y0,ψ1())+ψ1l1ϕ1,(x0+ψ1)(y1+ϕ0)=l2(x0,y1)+l3(x0,,y1)|g0+l2(x0,ϕ0())ϕ0l2(x0,)|g0ψ1l2(,y1)|g0+δ(ψ1(y1))+ψ1l1ϕ0,(y1+ϕ0)(x0+ψ1)=l2(y1,x0)l3(x0,,y1)|g0l2(x0,ϕ0())+ϕ0l2(x0,)|g0+δ(ϕ0(x0))+ψ1l2(,y1)|g0ψ1l1ϕ0, (7.4)

    and Ω is given by

    Ω(ϕ1+x0,ψ1+y0+φ1+z0)=l3(x0,y0,z0)+l4(x0,y0,z0,)φ1l3(x0,y0,)|g0ϕ1l3(z0,x0,)|g0ψ1l3(y0,z0,)|g0. (7.5)

    By (7.2), it is straightforward to deduce that the corresponding D:g2Hom(g0,g2)g1 is given by

    D(x2)=l2(,x2)+l1(x2) (7.6)

    Then by Theorem 6.2, we obtain:

    Proposition 5. Let (g2,g1,g0,l1,l2,l3,l4) be a Lie 3-algebra. Then there is a Lie 3-algebra (¯g2,¯g1,¯g0,l1,l2,l3,l4), where ¯g2=g2, ¯g1=Hom(g0,g2)g1, ¯g0=Hom(g1,g2)g0, and li are given by

    l1(x2)=D(x2),x2g2,l1(ϕ0+x1)=ϕ0l1+l2(x1,)|g1+l1(x1),ϕ0+x1¯g1,l2(e01,e02)=e01e02,e01,e02¯g0,l2(e0,e1)=12(e0e1e1e0),e0¯g0,e1¯g1,l2(e0,x2)=12S(e0,Dx2),e0¯g0,x2g2,l2(e11,e12)=0,e11,e12¯g1,l3(e01,e02,e03)=Ω(e01,e02,e03),e01,e02,e03¯g0,l3(e01,e02,e1)=T(e01,e02,e1),e01,e02¯g0,e1¯g1,l4(e01,e02,e03,e04)=¯Ω(e01,e02,e03,e04),e01,e02,e03,e04¯g0,

    where the operation D, , Ω are given by (7.6), (7.4), (7.5) respectively, T:¯g0ׯg0ׯg1g2 is given by

    T(e01,e02,e1)=16(S(e01,l2(e02,e1))+S(e1,l2(e01,e02))+S(e02,l2(e1,e01))),

    and ¯Ω:4¯g0g2 is given by

    ¯Ω(e01,e02,e03,e04)=S(Ω(e01,e02,e03),e04).

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

    Example 5. We view a 3-term complex of vector spaces V2l1V1l1V0 as an abelian Lie 3-algebra. By Proposition 5, we obtain the Lie 3-algebra

    (V2,Hom(V0,V2)V1,Hom(V1,V2)V0,l1,l2,l3,l4=0),

    where li,i=1,2,3 are given by

    l1(x2)=l1(x2),l1(ϕ0+y1)=ϕ0l1+l1(y1),l2(ψ1+x0,ϕ1+y0)=ψ1l1ϕ1ϕ1l1ψ1,l2(ψ1+x0,ϕ0+y1)=12l1(ψ1(y1)ϕ0(x0))+ψ1l1ϕ0,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(ψ1l1ϕ1(z1)ϕ1l1ψ1(z1)ψ1l1φ0(y0)+ϕ1l1φ0(x0)),

    for all x2V2,ψ0+x1,ϕ0+y1,φ0+z1Hom(V0,V2)V1,ψ1+x0,ϕ1+y0,φ1+z0Hom(V1,V2)V0.

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

    A Lie 2-algebra (g1,g0,~l1,~l2,~l3) gives rise to a Lie 3-algebra (R,g1,g0,l1,l2,l3,l4=0) naturally, where li, 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,

    for all x0,y0,z0g0, x1,y1,z1g1, and r,sR. By Proposition 5, we obtain the Lie 3-algebra (R,g1g0,g0g1,l1,l2,l3,l4), where li, i=1,2,3,4 are given by

    l1(r)=0,l1(x1+α0)=l1(x1)+l1(α0),l2(x0+α1,y0+β1)=l2(x0,y0)+ad0x0β1ad0y0α1,l2(x0+α1,y1+β0)=l2(x0,y1)+ad0x0β0ad1y1α1,l2(x1+α0,y1+β0)=0,l2(x0+α1,r)=0,l3(x0+α1,y0+β1,z0+ζ1)=l3(x0,y0,z0)+ad3x0,y0ζ1+ad3y0,z0α1+ad3z0,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)

    for all x0,y0,z0,u0g0, x1,y1,z1g1, α1,β1,ζ1,γ1g1, α0,β0g0, where ad0x0:gigi, ad1x1:g1g0 and ad3x0,y0:g1g0 are defined respectively by

    ad0x0α1,x1=α1,l2(x0,x1),ad0x0α0,y0=α0,l2(x0,y0),ad1x1α1,y0=α1,l2(x1,y0),ad3x0,y0α1,z0=α1,l3(x0,y0,z0).

    Remark 4. For any Lie algebra (h,[,]h), we have the semidirect product Lie algebra (hadh,[,]ad), which is a quadratic Lie algebra naturally. Consequently, one can construct the corresponding Lie 2-algebra (R,hadh,l1=0,l2=[,]ad,l3), where l3 is given by

    l3(x+α,y+β,z+γ)=γ,[x,y]h+β,[z,x]h+α,[y,z]h,x,y,zh,α,β,γh.

    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 (g1,g0,~l1,~l2,~l3), there is a naturally a quadratic Lie 2-algebra structure on (g1g0)(g0g1) ([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 (g1,g0,l1,l2,l3) equipped with a degree 1 graded symmetric nondegenerate bilinear form S which induces an isomorphism between g1 and g0, such that the following invariant conditions hold:

    S(l1(x1),y1)=S(l1(y1),x1), (7.7)
    S(l2(x0,y0),z1)=S(l2(x0,z1),y0), (7.8)
    S(l3(x0,y0,z0),u0)=S(l3(x0,y0,u0),z0), (7.9)

    for all x0,y0,z0,u0g0, x1,y1g1.

    Let (g1,g0,l1,l2,l3,S) be a quadratic Lie 2-algebra. On the 3-term complex of vector spaces Rg1g0, where R is of degree 2, we define li, 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), (7.10)

    for all x0,y0,z0,u0g0, x1,y1,z1g1 and rR.

    Theorem 7.1. With above notations, (R,g1,g0,l1,l2,l3,l4) 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 l3 and l4 using the invariant conditions (7.7)-(7.9). We omit details.



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