Abstract
All gauge-natural bilinear operators \(A: \Gamma ^l_E(TE\oplus T^*E)\times \Gamma ^l_E(TE\oplus T^*E)\rightarrow \Gamma ^l_E(TE\oplus T^*E)\) transforming pairs of linear sections of the “doubled” tangent bundle \(TE\oplus T^*E\) of a vector bundle E into linear sections of \(TE\oplus T^*E\) are completely described. Then, all such A with the Jacobi identity in Leibniz form are extracted.
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1 Introduction
All manifolds considered in the paper are assumed to be Hausdorff, second countable, finite dimensional, without boundary, and smooth (of class \({\mathcal {C}}^\infty \)). Maps between manifolds are assumed to be \({\mathcal {C}}^\infty \).
In [3, 8], the authors completely described bilinear operators on sections of \(TN\oplus T^*N\rightarrow N\) (for N a smooth manifold), which are \({\mathcal {M}} f_m\)-natural, i.e., invariant under the morphisms in the category \({\mathcal {M}} f_m\) of m-dimensional manifolds and their submersions. The principal result of [3] is precisely the full classification of such operators which also, like the Courant bracket, satisfy the Jacobi identity in Leibniz form. The Courant bracket, defined in [2], is of particular interest, because it involves in the concept of Dirac structures and in the concept of generalised complex structures on N, see [2, 4, 5].
This article classifies bilinear operators on the linear sections of the double vector bundles (TE; E, TM; M) and \((T^*E;E,E^*;M)\) (for \(E\rightarrow M\) a smooth vector bundle), which are gauge-natural, i.e., invariant under the morphisms in the category \(\mathcal {VB}_{m,n}\) of rank-n vector bundles over m-dimensional bases and their vector bundle isomorphisms onto images. These double vector bundles are of particular interest, because their direct sum \((TE\oplus T^*E;E,TM\oplus E^*;M)\) is the standard VB-Courant algebroid. The Dorfman–Courant bracket is part of this structure and an example of a \(\mathcal {VB}_{m,n}\)-gauge natural operator \(\Gamma ^l_E(TE\oplus T^*E)\times \Gamma ^l_E(TE\oplus T^*E)\rightarrow \Gamma ^l_E(TE\oplus T^*E)\), where \(\Gamma ^l_E(TE\oplus T^*E)\) denotes the space of linear sections of \(TE\oplus T^*E\rightarrow E\). The Dorfman–Courant bracket is the restriction of the Courant bracket to linear sections of \(TE\oplus T^*E\rightarrow E\), see [6]. (It can be also interpreted as the bracket of the Omni–Lie algebroid \(Der(E^*)\oplus J^1(E^*)\), studied in [1].)
The principal result of the paper is precisely the full classification of such operators which also, like the Dorfman–Courant bracket, satisfy the Jacobi identity in Leibniz form. The article first establishes the general form of \(\mathcal {VB}_{m,n}\)-gauge natural bilinear operators \(A:\Gamma ^l_E(TE\oplus T^*E)\times \Gamma ^l_E(TE\oplus T^*E)\rightarrow \Gamma ^l_E(TE\oplus T^*E)\), while its later half is dedicated to establishing which of these operators A satisfy the Jacobi identity in Leibniz form. Thus, the main result of the paper is the following.
Theorem 1.1
Let \(m\ge 2\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma _E^l(TE\oplus T^*E)\times \Gamma _E^l(TE\oplus T^*E)\rightarrow \Gamma _E^l(TE\oplus T^*E)\) is of the form
for arbitrary (uniquely determined by A) real numbers \(a, b_1,b_2,b_3,b_4,b_5,b_6\), where \([-,-]\) is the usual bracket on vector fields, \({\mathcal {L}}\) denotes the Lie derivative, d denotes the exterior derivative, i denotes the insertion derivative and L denotes the Euler vector field.
Moreover, such A satisfies the Jacobi identity in Leibniz form (i.e.
for any \(\nu ^i\in \Gamma _E^l(TE\oplus T^*E)\) for \(i=1,2,3\) ) if and only if \((a,b_1,b_2,b_3,b_4,b_5,b_6)\) is from the following list of 7-tuples:
where \(c, \lambda , \mu \) are arbitrary real numbers with \(c\not =0\).
It seems that the case \(m=1\) and \(n\ge 1\) is more complicated. It remains open.
Most proofs in the paper hinge the application of the following multilinear Peetre theorem in the same manner: This implies that any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(\Gamma ^l_E(TE\oplus T^*E)\times \Gamma ^l_E(TE\oplus T^*E)\rightarrow \Gamma ^l_E(TE\oplus T^*E)\) is of finite order.
Multilinear Peetre Theorem
(Theorem 19.9 in [7]). Let \(L_1,...,L_k\) be vector bundles over the same base M, \(L\rightarrow N\) be another vector bundle and let \(\pi :N\rightarrow M\) be continuous and locally non-constant. If \(D:C^\infty (L_1)\times ...\times C^\infty (L_k)\rightarrow C^\infty (L)\) is a k-linear \(\pi \)-local operator, then for every compact set \(K\subset N\), there is a natural number r such that for every \(x\in \pi (K)\) and all sections \(s,q\in C^\infty (L_1\oplus ...\oplus L_k)\), the condition \(j^rs(x)=j^rq(x)\) implies \(Ds\vert (\pi ^{-1}(x)\cap K)=Dq\vert (\pi ^{-1}(x)\cap K)\).
From now on, let \({\mathbf {R}}^{m,n}\) be the trivial vector bundle over \({\mathbf {R}}^m\) with the standard fibre \({\mathbf {R}}^n\) and let \(x^1,...,x^m,y^1,...,y^n\) be the usual coordinates on \({\mathbf {R}}^{m,n}\).
2 The Gauge-Natural Bilinear Operators Similar to the Dorfman–Courant Bracket
Let \(E=(E\rightarrow M)\) be a vector bundle.
Applying the tangent and the cotangent functors to \(E\rightarrow M\), we obtain double vector bundles (TE; E, TM; M) and \((T^*E;E,E^*;M)\).
A vector field X on E is called linear if it is a vector bundle map \(X:E\rightarrow TE\) between \(E\rightarrow M\) and \(TE\rightarrow TM\). Equivalently, a vector field X on E is linear iff it has expression
in any local vector bundle trivialization on E. The Euler vector field L on E is an example of a linear vector field on E. (We recall that the coordinate expression of L is \(L=\sum _{j=1}^ny^j{\partial \over \partial y^j}\).)
A 1-form \(\omega \) on E is called linear if it is a vector bundle map \(\omega :E\rightarrow T^*E\) between \(E\rightarrow M\) and \(T^*E\rightarrow E^*\). Equivalently, a 1-form \(\omega \) on E is linear iff it has expression
in any local vector bundle trivialization on E.
We need the following definition being respective modification of the general one from the fundamental monograph [7].
Definition 2.1
A \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) is a \(\mathcal {VB}_{m,n}\)-invariant family of \({\mathbf {R}}\)-bilinear operators
for all \(\mathcal {VB}_{m,n}\)-objects E, where \(\Gamma _E^l(TE\oplus T^*E)\) is the vector space of linear sections of \(TE\oplus T^*E\) (i.e. couples \(X\oplus \omega \) of linear vector fields X and linear 1-forms \(\omega \) on E). The \(\mathcal {VB}_{m,n} \)-invariance of A means that if \((X^1\oplus \omega ^1, X^2\oplus \omega ^2)\in \Gamma ^l_E(TE\oplus T^*E)\times \Gamma _E^l(TE\oplus T^*E)\) and \(({\overline{X}}^1\oplus \overline{\omega }^1,{\overline{X}}^2\oplus {\overline{\omega }}^2)\in \Gamma ^l_{{\overline{E}}}(T{\overline{E}}\oplus T^*{\overline{E}})\times \Gamma _{{\overline{E}}}^l(T {\overline{E}}\oplus T^*{\overline{E}}))\) are \(\varphi \)-related by an \(\mathcal {VB}_{m,n}\)-map \(\varphi :E\rightarrow {\overline{E}}\) (i.e. \({\overline{X}}^i\circ \varphi =T\varphi \circ X^i\) and \(\overline{\omega }^i\circ \varphi =T^*\varphi \circ \omega ^i\) for \(i=1,2\)), then so are \(A(X^1\oplus \omega ^1,X^2\oplus \omega ^2)\) and \(A(\overline{X}^1\oplus {\overline{\omega }}^1,{\overline{X}}^2\oplus {\overline{\omega }}^2)\).
Remark 2.2
Quite similarly, we can define \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\), \(\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\), etc. For example, a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) is a \(\mathcal {VB}_{m,n}\)-invariant family of \({\mathbf {R}}\)-bilinear operators \(A:\Gamma _E^l(TE)\times \Gamma _E^l(T^*E)\rightarrow \Gamma _E^l(TE)\) for all \(\mathcal {VB}_{m,n}\)-objects E, where \(\Gamma _E^l(TE)\) is the space of linear vector fields on E and \(\Gamma _E^l(T^*E)\) is the space of linear 1-forms on E.
Example 2.3
The usual bracket [X, Y] of (linear) vector fields defines a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \([-,-]:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\).
Example 2.4
The Lie derivative \({\mathcal {L}}_X\omega \) of linear 1-forms \(\omega \) with respect to linear vector fields X defines a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \({\mathcal {L}}:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\).
Example 2.5
Let \(\omega \) be a linear 1-form and X be a linear vector field on a vector bundle E. Then, we have linear 1-form \(i_Xd\omega \), and we have the corresponding \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\), where d denotes the exterior derivative and i denotes the insertion derivative.
Example 2.6
The Dorfman–Courant bracket \([[X^1\oplus \omega ^1,X^2\oplus \omega ^2]]:=[X^1,X^2]\oplus ({\mathcal {L}}_{X^1}\omega ^2-i_{X^2}d\omega ^1)\) gives \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \([[-,-]]:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\).
Example 2.7
Let \(\omega \) be a linear 1-form and X be a linear vector field on a vector bundle E and let L denotes the Euler vector field on E. Then, we have linear 1-form \({\mathcal {L}}_{X}di_L\omega \), and we have the corresponding \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\).
Lemma 2.8
Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator A (in question) is of finite order. It means that there is a finite number r (depending on A) such that
for any \(\mathcal {VB}_{m,n}\)-object \(E\rightarrow M\), any linear sections \(\nu _1\) and \(\nu _2\) on E and any \(x\in M\), where \(E_x\) is the fibre of \(E\rightarrow M\) over \(x\in M\), and \(j^r_x\nu _1=j^r_x{\overline{\nu }}_1\) means that \(j^r_v\nu _1=j^r_v{\overline{\nu }}_1\) for any \(v\in E_x\) (or equivalently for any v from the basis of \(E_x\)).
Proof
The space \(\Gamma ^l_{E}(TE\oplus T^*E)\) is a locally free \({\mathcal {C}}^\infty (M)\)-module. Hence, there is a vector bundle \(\hat{E}\) over M such that \(\Gamma ^l_E(TE\oplus T^*E)\) is isomorphic to \(\Gamma \hat{E}\) as \({\mathcal {C}}^\infty (M)\)-module. So, we can treat A as bilinear local operator \(A:\Gamma \hat{E}\times \Gamma \hat{E}\rightarrow \Gamma \hat{E}\). Then, the multi-linear Peetre theorem (cited in Introduction) for \(k=2\), \(M=N={\mathbf {R}}^m\), \(\pi =\mathrm {id}\), \(K=\{0\}\), \(L_1=L_2=L=\hat{{\mathbf {R}}}^{m,n}\) and \(D=A\) implies that there is a natural number r such that for every pairs \((\nu _1,\nu _2)\) and \(({\overline{\nu }}_1,{\overline{\nu }}_2)\) of linear sections, the condition \(j^r_0(\nu _1,\nu _2)=j_0^r({\overline{\nu }}_1,{\overline{\nu }}_2)\) implies \(A(\nu _1,\nu _2)=A({\overline{\nu }}_1,{\overline{\nu }}_2)\) at 0. Then, using the invariance of A, we complete the proof.
For the other operators mentioned in Remark 2.2, the proofs are similar.
\(\square \)
A linear vector field X on \({\mathbf {R}}^{m,n}\) is monomial if it is of the form \(x^\alpha {\partial \over \partial x^i}\) or \(x^\alpha y^j{\partial \over \partial y^k}\), where \(\alpha =(\alpha ^1,...,\alpha ^m)\) is a m-tuple of non-negative integers and \(i=1,...,m\), \(j,k=1,...,n\), Of course, \(x^\alpha :=(x^1)^{\alpha ^1}\cdot ...\cdot (x^m)^{\alpha ^m}\).
Similarly, a linear 1-form \(\omega \) on \({\mathbf {R}}^{m,n}\) is monomial if it is of the form \(x^\alpha dy^j\) or \(x^\alpha y^jdx^i\).
A linear section \(X\oplus \omega \) is called monomial if (X is monomial and \(\omega =0\)) or (\(X=0\) and \(\omega \) is monomial).
Lemma 2.9
Let A be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator (in question) such that \(A(\nu _1,\nu _2)=0\) over \(0\in {\mathbf {R}}^m\) for all monomial linear sections \(\nu _1\) and \(\nu _2\) on \({\mathbf {R}}^{m,n}\). Then \(A=0\).
Proof
Because of the invariance of A with respect to local trivialization, it suffices to show that \(A(\nu _1,\nu _2)=0\) over \(0\in {\mathbf {R}}^m\) for any linear sections \(\nu _1\) and \(\nu _2\) on \({\mathbf {R}}^{m,n}\). Since A is of finite order r (because of Lemma 2.8), we may assume that \(\nu _1\) and \(\nu _2\) are polynomial of degree not more than r. Then, since A is bilinear, we may assume that \(\nu _1\) and \(\nu _2\) are monomial. \(\square \)
Lemma 2.10
Let \(A:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\) (or \(A:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T^*)\)) be a \(\mathcal {V B}_{m,n}\)-gauge-natural bilinear operator. Assume that \(m\ge 2\) and \(A({\partial \over \partial x^1}, (x^1)^q {\partial \over \partial x^2})=0\) over \(0\in {\mathbf {R}}^m\) for all \(q=0,1,...\) . Then \(A=0\).
Proof
Because of the invariance of A with respect to local trivialization, it suffices to show that \(A(X,Y)=0\) over \(0\in {\mathbf {R}}^m\) for any linear vector fields X and Y on \({\mathbf {R}}^{m,n}\). We can assume X is not vertical over 0. Then by the Frobenius theorem and the invariance of A, we can assume \(X={\partial \over \partial x^1}\). Next, by the similar arguments as in the proof of Lemma 2.9, we may assume that Y is monomial.
So, let \(\beta =(\beta _1,\beta _2,...,\beta _m) \in ({\mathbf {N}}\cup \{0\})^{m}\) and \(j,k=1,...,n\) and \(i=1,...,m\).
There exists a \(\mathcal {VB}_{m,n}\)-map \(\psi :{\mathbf {R}}^{m,n}\rightarrow {\mathbf {R}}^{m,n}\) preserving \(x^1\) and \({\partial \over \partial x^1}\) and sending the germ at \(0\in {\mathbf {R}}^m\) of \({\partial \over \partial x^2}\) into the germ at \(0\in {\mathbf {R}}^m\) of \({\partial \over \partial x^2}+(x^2)^{\beta _2}\cdot ...\cdot (x^m)^{\beta _m}y^j{\partial \over \partial y^k}\). Then, by the invariance of A with respect to \(\psi \), from assumption \(A({\partial \over \partial x^1}, (x^1)^{\beta _1} {\partial \over \partial x^2})=0\) over \(0\in {\mathbf {R}}^m\), we get \(A({\partial \over \partial x^1}, (x^1)^{\beta _1}{\partial \over \partial x^2}+x^{\beta }y^j{\partial \over \partial y^k})=0\) over \(0\in {\mathbf {R}}^m\). Then \(A({\partial \over \partial x^1}, x^{\beta }y^j{\partial \over \partial y^k})=0\) over \(0\in {\mathbf {R}}^m\).
If \(i=2,...,m\), there exists a \(\mathcal {V B}_{m,n}\)-map \(\varphi :{\mathbf {R}}^{m,n}\rightarrow {\mathbf {R}}^{m,n}\) preserving \(x^1\) and \({\partial \over \partial x^1}\) and sending the germ at \(0\in {\mathbf {R}}^m\) of \({\partial \over \partial x^2}\) into the germ at \(0\in {\mathbf {R}}^m\) of \({\partial \over \partial x^2}+(x^2)^{\beta _2}\cdot ...\cdot (x^m)^{\beta _m}{\partial \over \partial x^i}\). Then, similarly as above (using the invariance with respect to \(\varphi \)), we get \(A({\partial \over \partial x^1}, x^{\beta }{\partial \over \partial x^i})=0\) over \(0\in {\mathbf {R}}^m\).
Then, using the invariance of A with respect to \((x^1+\tau x^2, x^2,...,x^m,y^1,...,y^n)\), we get \(A({\partial \over \partial x^1},(x^1-\tau x^2)^{\beta _1}(x^2)^{\beta _2}\cdot ...\cdot (x^m)^{\beta _m}({\partial \over \partial x^2}+\tau {\partial \over \partial x^1}))(e)=0\) for any \(\tau \in {\mathbf {R}}\) and any e in the fibre of \({\mathbf {R}}^{m,n}\) over \(0\in {\mathbf {R}}^m\). Considering the coefficient on \(\tau \), we get \(A({\partial \over \partial x^1},x^{\beta }{\partial \over \partial x^1})(e)-\beta _1A({\partial \over \partial x^1},(x^1)^{\beta _1-1}(x^2)^{\beta _2+1}(x^3)^{\beta _3}\cdot ... \cdot (x^m)^{\beta _m}{\partial \over \partial x^2})(e)=0\). Then \(A({\partial \over \partial x^1},x^{\beta }{\partial \over \partial x^1})=0\) over \(0\in {\mathbf {R}}^m\).
The lemma is complete \(\square \)
Lemma 2.11
Let \(m\ge 1\) and \(n\ge 1\). Let \(A:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator. Assume that \(A({\partial \over \partial x^1}, \omega )=0\) over \(0\in {\mathbf {R}}^m\) for all monomial linear 1-forms \(\omega \) on \({\mathbf {R}}^{m,n}\). Then \(A=0\).
Proof
It suffices to show that \(A(X,\omega )=0\) over \(0\in {\mathbf {R}}^m\) for any linear vector field X and any linear 1-form \(\omega \) on \({\mathbf {R}}^{m,n}\). We can assume \(X={\partial \over \partial x^1}\) and \(\omega \) is monomial. \(\square \)
Lemma 2.12
Let \(m\ge 1\) and \(n\ge 1\). Let \(A:\Gamma ^l(T^*)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) (or \(A:\Gamma ^l(T^*)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\)) be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator. Assume that \(A(\omega ^1,\omega ^2)=0\) over \(0\in {\mathbf {R}}^m\) for all monomial linear 1-forms \(\omega ^1,\omega ^2\) on \({\mathbf {R}}^{m,n}\). Then \(A=0\).
Proof
It is the particular case of Lemma 2.9. \(\square \)
Lemma 2.13
Let \(A:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) be a \(\mathcal {V B}_{m,n}\)-gauge-natural bilinear operator. Assume that \(m\ge 2\) and
for all \(k=0,1,...\) , where \(e_1=(1,0,...,0)\in {\mathbf {R}}^n\) is the element in the fibre over \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\). Then \(A=0\).
Proof
It suffices to show that \(A({\partial \over \partial x^1},\omega )=0\) over \(0\in {\mathbf {R}}^m\) for any monomial linear 1-form \(\omega \) on \({\mathbf {R}}^{m,n}\).
(I) At first, we prove that \(A({\partial \over \partial x^1}, x^{\beta } dy^j)=0\) over \(0\in {\mathbf {R}}^m\) for all m-tuples \(\beta =(\beta _1,...,\beta _m)\) of non-negative integers and all \(j=1,...,n\).
Consider a m-tuple \(\beta =(\beta _1,...,\beta _m)\) of non-negative integers. Let \(j=1,...,n\) and \(e=(\xi _1,...,\xi _n)\) be a point from the fibre over \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\). We may assume \(\xi _j\not =0\). Let \(k=\vert \beta \vert \). Using the invariance of A with respect to \(\mathcal {VB}_{m,n}\)-maps
for \(\tau _1\not =0, \tau _2,...,\tau _m\) (sending \({\partial \over \partial x^1}\) into \({1\over \tau _1}{\partial \over \partial x^1}\) and preserving \(e_1\)), from the assumption of the lemma, we get
Then, considering the coefficients on \((\tau _1)^{\beta _1}\cdot ...\cdot (\tau _m)^{\beta _m}\) of these polynomials in \(\tau _1,...,\tau _m\), we get \(A({\partial \over \partial x^1}, x^{\beta } dy^1)(e_1)=0\). Since \(\xi _j\not =0\), there is a linear isomorphism \(\varphi :{\mathbf {R}}^n\rightarrow {\mathbf {R}}^n\) sending \(y^1\) into \({1\over \xi _j}y^j\) and \(e_1\) into e. Then, using the invariance of A with respect to \((x^1,...,x^m,\varphi (y^1,..,y^n))\), we get
i.e. \(A({\partial \over \partial x^1}, x^{\beta } dy^j)(e)=0\).
(II) Now, we prove \(A({\partial \over \partial x^1}, x^{\beta } y^j dx^i)=0\) over \(0\in {\mathbf {R}}^m\) for all m-tuples \(\beta =(\beta _1,...,\beta _m)\) of non-negative integers and \(j=1,...,n\) and \(i=1,...,m\).
Let \(\beta =(\beta _1,...,\beta _m)\) be an m-tuple of non-negative integers, \(j=1,...,n\), \(i=1,...,m\) and e be from the fibre over \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\). If \(\beta _2+...+\beta _m\ge 1\), using the invariance of A with respect to \(\mathcal {VB}_{m,n}\)-map
(preserving \({\partial \over \partial x^1}\) and \(e_1\)), from the assumption \(A({\partial \over \partial x^1}, (x^1)^{\beta _1}y^1dx^1)(e_1)=0\) we get
Consequently, \(A({\partial \over \partial x^1}, x^{\beta } y^1 dx^1)(e_1)=0\) (for \(\beta _2+...+\beta _m=0\), too). So, if \(i=2,...,m\), then, by the invariance of A with respect to \((x^1+\tau x^i, x^2,...,x^m,y^1,y^2...,y^n)^{-1}\) (preserving \({\partial \over \partial x^1}\) and \(e_1\)), we get
Then, considering the coefficient on \(\tau \), we get
where \(B:= A({\partial \over \partial x^1}, (x^1)^{\beta _1-1}(x^2)^{\beta _2}\cdot ...\cdot (x^i)^{\beta _i+1}\cdot ...\cdot (x^m)^{\beta _m}y^1dx^1)(e_1)\). If \(\beta _1\not =0\), \(B=0\) (it is proved above). If \(\beta _1=0\), the term \(\beta _1 B\) does not occur. Consequently, \(A({\partial \over \partial x^1}, x^{\beta } y^1 dx^i)(e_1)=0\) (for \(i=1\), too). Then (using similar arguments to the one of the end of the part (I) of the proof) \(A({\partial \over \partial x^1}, x^{\beta } y^j dx^i)(e)=0\). \(\square \)
Lemma 2.14
The collection of \(\mathcal {VB}_{m,n}\)-gauge-natural operators \(A^i:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) for \(i=1,2,3\) given by \(A^1(X,\omega )={\mathcal {L}}_X\omega \), \(A^2(X,\omega )=i_Xd\omega \) and \(A^3(X,\omega )={\mathcal {L}}_Xdi_L\omega \) is \({\mathbf {R}}\)-linearly independent.
Proof
We know that \(L=\sum _{j=1}^ny^j{\partial \over \partial y^j}\) end \(e_1=(1,0...,0)\in {\mathbf {R}}^n\) is the element over the fibre over \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\). Then, it is easy to compute that \(A^1({\partial \over \partial x^1},y^1dx^1)(e_1)=0\), \(A^2({\partial \over \partial x^1},y^1dx^1)(e_1)=-d_{e_1}y^1\), \(A^3({\partial \over \partial x^1},y^1dx^1)(e_1)=0\), \(A^1({\partial \over \partial x^1},x^1dy^1)(e_1)=d_{e_1}y^1\), \(A^2({\partial \over \partial x^1},x^1dy^1)(e_1)=d_{e_1}y^1\), \(A^3({\partial \over \partial x^1},x^1dy^1)(e_1)=d_{e_1}y^1\), \(A^1({\partial \over \partial x^1},(x^1)^2dy^1)(e_1)=0\), \(A^2({\partial \over \partial x^1},(x^1)^2dy^1)(e_1)= 0\), \(A^3({\partial \over \partial x^1},(x^1)^2dy^1)(e_1)= 2d_{e_1}x^1\). Now, the lemma is clear. \(\square \)
Proposition 2.15
Let \(m\ge 2\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\) is the constant multiple of the usual bracket \([-,-]\) on (linear) vector fields.
Proof
Let k be a non-negative integer. We can write
for any \(e=(e^1,...,e^n)\) from the fibre \({\mathbf {R}}^n\) at \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\), where \(f^{[k,i]}\) and \(g^{[k,l]}_j\) are the real numbers (independent of e).
Using the invariance of A with respect to \((x^1,tx^2, x^3,...,x^m,y^1,...,y^n)\) for \(t>0\), we get the conditions \(tf^{[k,i]}=f^{[k,i]}\) for \(i=1,3,...,m\) (i.e. for \(i\not =2\)) and \(tg^{[k,l]}_j=g^{[k,l]}_j\) for \(j,l=1,...,n\). Then \(f^{[k,i]}=0\) for \(i=1,3,...,m\) and \(g^{[k,l]}_j=0\) for \(j,l=1,...,n\).
Similarly, by the invariance of A with respect to \(({1\over t}x^1,x^2,...,x^m,y^1,...,y^n)\), we get \(t^{k-1}f^{[k,2]} =f^{[k,2]}\), i.e. \(f^{[k,2]}=0\) if \(k\not =1\).
Then by Lemma 2.10, A is determined by the value \(f^{[1,2]}\in {\mathbf {R}}\). Consequently, the vector space of all such A is of dimension not more than 1. Then, the dimension argument completes the proof of the proposition. \(\square \)
Proposition 2.16
Let \(m\ge 1\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) is zero.
Proof
Let \(\alpha =(\alpha ^1,...,\alpha ^m)\in ({\mathbf {N}}\cup \{0\})^m\) be an m-tuple of non-negative integers and let \(i_o\in \{1,...,m\}\) and \(j_o\in \{1,...,n\}\).
We can write
for any \(e=(e^1,...,e^n)\) from the fibre \({\mathbf {R}}^n\) at \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\), where \(f^i\) and \(g^l_j\) are the real numbers (depending on \(\alpha \), \( i_o\) and \(j_o\) and independent of e).
Using the invariance of A with respect to \((x^1,...,x^m,{1\over t}y^1,...,{1\over t}y^n)\) for \(t>0\), we get
Then \(A({\partial \over \partial x^1}, x^\alpha y^{j_o} dx^{i_o})=0\) over \(0\in {\mathbf {R}}^m\) for \(i_o=1,...,m\), \(j_o=1,...,n\), \(\alpha \in ({\mathbf {N}}\cup \{0\})^m\). By the same argument (replacing \(x^\alpha y^{j_o}dx^{i_o}\) by \(x^\alpha dy^{j_o}\)), we derive \(A({\partial \over \partial x^1}, x^\alpha dy^{j_o})=0\) over \(0\in {\mathbf {R}}^m\) for \(j_o=1,...,n\) and \(\alpha \in ({\mathbf {N}}\cup \{0\})^m\). Consequently, \(A({\partial \over \partial x^1},\omega )=0\) over \(0\in {\mathbf {R}}^m\) for any linear 1-form \(\omega \) on \({\mathbf {R}}^{m,n}\).
Now, applying Lemma 2.11, we complete the proof. \(\square \)
Proposition 2.17
Let \(m\ge 1\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T^*)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) is zero.
Proof
We proceed quite similar as for Proposition 2.16. Let \(\omega ^1=x^\beta y^{j_1}dx^{i_1}\) or \(\omega ^1=x^\beta dy^{j_1}\) and let \(\omega ^2= x^\alpha y^{j_o}dx^{i_o}\) or \(\omega ^2 =x^\alpha dy^{j_o}\). We can write
for any \(e=(e^1,...,e^n)\) from the fibre \({\mathbf {R}}^n\) at \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\), where \(f^i\) and \(g^l_j\) are the real numbers (depending on \(\omega ^1\) and \(\omega ^2\) and independent of e). Using the invariance of A with respect to \((x^1,...,x^m,{1\over t}y^1,...,{1\over t}y^n)\) for \(t>0\), we get
Then \(A(\omega ^1,\omega ^2)=0\) over 0. \({\mathbf {R}}^{m,n}\). Then \(A=0\) because of Lemma 2.12. \(\square \).
Proposition 2.18
Let \(m\ge 2\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T^*)\) is zero.
Proof
Let k be a non-negative integer. We can write
for any \(e=(e^1,...,e^n)\) from the fibre \({\mathbf {R}}^n\) at \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\), where \(f^{[k]}_j\) and \(g^{[k]}_{il}\) are the real numbers (independent of e). By the invariance of A with respect to \((x^1,tx^2,...,x^m, y^1,...,y^n)\) for \(t>0\), we get
(the Kronecker delta). Then \(A({\partial \over \partial x^1}, (x^1)^k {\partial \over \partial x^2})=0\) over 0 for \(k=0,1,...\). Then \(A=0\) because of Lemma 2.10. \(\square \)
Proposition 2.19
Let \(m\ge 1\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T^*)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) is the zero one.
Proof
Let \(\omega ^1=x^\beta y^{j_1}dx^{i_1}\) or \(\omega ^1=x^\beta dy^{j_1}\) and let \(\omega = x^\alpha y^{j_o}dx^{i_o}\) or \(\omega =x^\alpha dy^{j_o}\). We can write
for any \(e=(e^1,...,e^n)\) from the fibre \({\mathbf {R}}^n\) at \(0\in {\mathbf {R}}^m\) of \({\mathbf {R}}^{m,n}\), where \(f_j\) and \(g_{il}\) are the real numbers (independent of e). Using the invariance of A with respect to \((x^1,...,x^m,{1\over t}y^1,...,{1\over t}y^n)\), we get
Then \(A(\omega ^1,\omega ^2)=0\) over 0. Then \(A=0\) because of Lemma 2.12. \(\square \)
Proposition 2.20
Let \(m\ge 2\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) is of the form
for the (uniquely determined by A) real numbers \(c_1,c_2,c_3\).
Proof
Let k be a non-negative integer and \(e_1=(1,0,...,0)\in {\mathbf {R}}^n\). We can write
where \(f^{(k)}_j\) and \(g^{(k)}_{i}\) are the real numbers. Using the invariance of A with respect to \(({1\over t}x^1,...,{1\over t} x^m,y^1,...,y^n)\), we get \(t^{k-1}f^{(k)}_j=f^{(k)}_j\ \text {and}\ t^{k-1}g^{(k)}_{i}=g^{(k)}_{i}t.\) Then \(f^{(k)}_j=0\) for \(j=1,...,n\) if \(k\not =1\), and if \(k\not =2\) then \(g^{(k)}_{i}=0\) for \(i=1,...,m\). Hence,
\(A({\partial \over \partial x^1},x^1dy^1)(e_1)= \sum _{j=1}^n f^{(1)}_jd_{e_1}y^j\ \text {and}\ A({\partial \over \partial x^1},(x^1)^2dy^1)(e_1)=\sum _{i=1}^mg^{(2)}_id_{e_1}x^i .\) Now, using the invariance of A with respect to \((x^1,tx^2,...,tx^m, y^1,ty^2,...,ty^n)\) for \(t>0\), we deduce that
Similarly, we can write
where \(\tilde{f}^{(k)}_j\) and \({\tilde{g}}^{(k)}_{i}\) are the real numbers. Then quite similarly as above, we get
and \(A({\partial \over \partial x^1},(x^1)^ky^1dx^1)(e_1)=0 \ \text {if} \ k=2,3,4,5,... \)
We prove that
We know (see, above) that
Consequently, by the invariance of A with respect to \((x^1,...,x^m,y^1+\tau x^1y^1,y^2,...,y^n)\) preserving \(e_1\) and sending \({\partial \over \partial x^1}\) into \({\partial \over \partial x^1}+{\tau \over 1+\tau x^1}y^1{\partial \over \partial y^1}\) and \(y^1\) into \(y^1-\tau x^1y^1+\tau ^2(x^1)^2y^1-....\), we get
where \(r=\max (2,{\tilde{r}})\) and \({\tilde{r}}\) is the finite order of A. Considering the coefficients on \(\tau \), we get
But \(A({\partial \over \partial x^1}, d((x^1)^2y^1))(e_1)= 2A({\partial \over \partial x^1}, x^1y^1 dx^1)(e_1)+A({\partial \over \partial x^1}, (x^1)^2dy^1)(e_1)\). Then
Further, we have observed above that \(A({\partial \over \partial x^1},dy^1)(e_1)=0\). Then, using the invariance of A with respect to \((x^1,...,x^m,y^1+\tau x^1y^1,y^2,...,y^n)\), we deduce that
Then, considering the coefficients on \(\tau \), we get
Then, using the invariance of A with respect to \((x^1,...,x^m,y^1+\tau x^1y^1,y^2,...,y^n)\) preserving \(y^1{\partial \over \partial y^1}\) (as it is the Euler vector field in \(\mathcal {VB}_{1,1}\) and then it is \(\mathcal {VB}_{1,1}\)-invariant), we deduce that
So, considering the coefficients on \(\tau \), we get
That is why, \( 2\tilde{g}^{(1)}_1+ g^{(2)}_1=2(f^{(1)}_1 + \tilde{f}^{(0)}_1) \ .\)
So, by Lemma 2.13, A is determined by the real numbers \(f^{(1)}_1, g^{(2)}_1 \) and \({\tilde{f}}^{(0)}_1\). Then the dimension of vector space of all A in question is not more than 3. So, the dimension argument (Lemma 2.14) completes the proposition. \(\square \)
Now, we are in position to obtain the following theorem corresponding to the first part of Theorem 1.1.
Theorem 2.21
Let \(m\ge 2\) and \(n\ge 1\). Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) is of the form
for arbitrary (uniquely determined by A) real numbers \(a, b_1,b_2,b_3,b_4,b_5,b_6\).
Proof
Let \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator. Let \(X^1\oplus \omega ^1, X^2\oplus \omega ^2\in \Gamma _E^l(TE\oplus T^*E)\). We can write
where \(\hat{A}(X^1\oplus \omega ^1,X^2\oplus \omega ^2)\in \Gamma ^l_E(TE)\) and \(\check{A}(X^1\oplus \omega ^1,X^2\oplus \omega ^2)\in \Gamma ^l_E(T^*E)\). Next,
where \(\hat{A}^{(1)}(X^1,X^2)=\hat{A}(X^1\oplus 0, X^2\oplus 0) \ , \ \hat{A}^{(2)}(X^1,\omega ^2)=\hat{A}(X^1\oplus 0,0\oplus \omega ^2) ,\ etc. \ , \) and similarly for \(\check{A}\) instead of \(\hat{A}\). Hence, A defines (is determined by) eight \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(\hat{A}^{(1)}:\Gamma ^l(T)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\), \(\hat{A}^{(2)}:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\), ... , \(\check{A}^{(4)}:\Gamma ^l(T^*)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\). Further, the \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(B:\Gamma ^l(T^*)\times \Gamma ^l(T)\rightsquigarrow \Gamma ^l(T)\) are in bijection with the \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(B^{op}:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T)\) by \(B^{op}(X,\omega )=B(\omega ,X)\), etc. So, our theorem is a immediate consequence of Propositions 2.15–2.20 and the expression \({\mathcal {L}}_X=i_Xd+di_X\). \(\square \)
We end this section by the following two lemmas.
Lemma 2.22
For any linear vector field X and any linear 1-form \(\omega \) on a vector bundle E (with the basis of dimension \(\ge 2\)), we have
where L is the Euler vector field on E.
Proof
We have three \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(A^1, A^2, A^3:\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) given by
It remains to show that \(A^1=A^2=A^3\).
By Lemma 2.13, it is sufficient to verify that
and \(A^1({\partial \over \partial x^1},(x^1)^ky^1dx^1)= A^2({\partial \over \partial x^1},(x^1)^ky^1dx^1)= A^3({\partial \over \partial x^1},(x^1)^ky^1dx^1)=0 .\)
\(\square \)
Lemma 2.23
For any linear vector field X and any linear 1-form \(\omega \) on a vector bundle E (with the basis of dimension \(\ge 2\)), we have
Proof
We have two \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operators \(A^1, A^2 :\Gamma ^l(T)\times \Gamma ^l(T^*)\rightsquigarrow \Gamma ^l(T^*)\) given by \(A^1(X,\omega )=di_Ldi_X\omega \ , \ A^2(X,\omega )=di_X\omega \ .\) It remains to show that \(A^1=A^2\).
By Lemma 2.13, it is sufficient to see that \(A^1({\partial \over \partial x^1},(x^1)^kdy^1)= A^2({\partial \over \partial x^1},(x^1)^kdy^1) =0\); and \(A^1({\partial \over \partial x^1},(x^1)^ky^1dx^1)\)\(= A^2({\partial \over \partial x^1},(x^1)^ky^1dx^1) =k(x^1)^{k-1}y^1dx^1+(x^1)^kdy^1.\)\(\square \)
3 The Complete Description of All \(\mathcal {VB}_{m,n}\)-Gauge-Natural Operators \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l (T\oplus T^*)\) Satisfying the Jacobi Identity in Leibniz Form
Let \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l (T\oplus T^*)\) be a \(\mathcal {V B}_{m,n}\)-gauge-natural bilinear operator in the sense of Definition 2.1.
Definition 3.1
We say that A satisfies the Jacobi identity in Leibniz form if
for any linear sections \(\nu ^i=X^i\oplus \omega ^i\in \Gamma _E^l(TE\oplus T^*E)\) for \(i=1,2,3\) and any \(\mathcal {VB}_{m,n}\)-object E.
By Theorem 2.21, A is of the form (1) for (uniquely determined by A) real numbers \(a, b_1,b_2,...,b_6\). We are going to obtain some conditions on the numbers \(a,b_1,b_2,b_3,b_4,b_5,b_6\) equivalent to the Jacobi identity in Leibniz form of A.
Lemma 3.2
For any linear vector fields \(X^1,X^2,X^3\) on \({\mathbf {R}}^{m,n}\) and any linear 1-forms \(\omega ^1,\omega ^2,\omega ^3\) on \({\mathbf {R}}^{m,n}\), we can write
where
The Jacobi identity in Leibniz form of A is equivalent to
Proof
The lemma is obvious. \(\square \)
Lemma 3.3
The Jacobi identity in Leibniz form of A is equivalent to the system of equalities (6), (7) and (8) for all linear vector fields \(X^1,X^2,X^3\) and all linear 1-forms \(\omega ^1, \omega ^2, \omega ^3\) on \({\mathbf {R}}^{m,n}\), where
Proof
If we put \(\omega ^1=\omega ^2=0\) (in 5), we get (6). Similarly, if we put \(\omega ^1=\omega ^3=0\), we get (7). Similarly, if we put \(\omega ^2=\omega ^3=0\), we get (8). Conversely, adding the above equalities (6)–(8), we get (5). The lemma is complete. \(\square \)
Proposition 3.4
The Jacobi identity in Leibniz form of A is equivalent to the system consisting of conditions (9) and (10)–(12) for all linear vector fields \(X^1,X^2, X^3\) and all linear 1-forms \(\omega ^1,\omega ^2,\omega ^3\) on \({\mathbf {R}}^{m,n}\), where
Proof
Applying the differential d to both sides of the equalities (6)–(8) and applying the formulas \(d^2=0\) and \(d{\mathcal {L}}_X={\mathcal {L}}_Xd\), we immediately obtain
for all linear \(X^1,X^2,\omega ^3\), and
for all linear \(X^1,X^3,\omega ^2\), and
for all linear \(X^2,X^3,\omega ^1\).
Then by the formula \({\mathcal {L}}_{[X,Y]}={\mathcal {L}}_X{\mathcal {L}}_Y-{\mathcal {L}}_Y{\mathcal {L}}_X\), we get
for all linear \(X^1,X^2,\omega ^3\), and
for all linear \(X^1,X^3,\omega ^2\), and
for all linear \(X^2,X^3,\omega ^1\).
Considering linear \(X^1,X^2,\omega ^3\) such that \({\mathcal {L}}_{[X^1,X^2]}d\omega ^3\not =0\) (for example \(X^1={\partial \over \partial x^1}\) and \(X^2=x^1{\partial \over \partial x^1}\) and \(\omega ^3=(x^1)^2 dy^1\)), from (16) we get
Similarly, considering linear \(X^1,X^3,\omega ^2\) with \({\mathcal {L}}_{[X^1,X^3]}d\omega ^2\not =0\) (for example, \(X^1={\partial \over \partial x^1}\) and \(X^3=x^1{\partial \over \partial x^1}\) and \(\omega ^2=(x^1)^2 dy^1\)), from (17) we get
Similarly, considering linear \(X^2,X^3,\omega ^1\) with \({\mathcal {L}}_{X^3}{\mathcal {L}}_{X^2}d\omega ^1\not =0\) (for example, \(X^3={\partial \over \partial x^1}\) and \(X^2=x^1{\partial \over \partial x^1}\) and \(\omega ^1=(x^1)^2 dy^1 \)), from (18) and (20) we get
Consequently, we obtain (9).
Conversely, if \(b_1\), \(b_2\) and a satisfy (9), then using the formula \({\mathcal {L}}_X{\mathcal {L}}_{Y}\omega -\mathcal { L}_Y{\mathcal {L}}_X\omega ={\mathcal {L}}_{[X,Y]}\omega \), we get
for all linear \(X^1,X^2,\omega ^3\), and
for all linear \(X^1,X^3,\omega ^2\), and
for all linear \(X^2,X^3,\omega ^1\).
Now, we can easily see that the proposition is a simple consequence of Lemma 3.3. \(\square \)
We prove the following theorem corresponding to the second part of Theorem 1.1.
Theorem 3.5
Let \(m\ge 2\) and \(n\ge 1\) be natural numbers. Any \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l (T\oplus T^*)\) of the form (1) satisfies the Jacobi identity in Leibniz form if and only if \((a,b_1,b_2,b_3,b_4,b_5,b_6)\) is from the following list of 7-tuples:
where \(c, \lambda , \mu \) are arbitrary real numbers with \(c\not =0\).
Proof
At first we prove the part “ \(\Rightarrow \) ” of the theorem.
Let \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l (T\oplus T^*)\) be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator of the form (1). Assume that A satisfies the Jacobi identity in Leibniz form.
If we put linear vector fields \(X^1={\partial \over \partial x^1}\) and \(X^2={\partial \over \partial x^2}\) and linear 1-form \(\omega ^3=x^2y^1dx^1\) into (10), we get
Then \(b_5b_3dy^1+b_3^2dy^1=0\). Then
If we put linear vector fields \(X^1={\partial \over \partial x^1}\) and \(X^2=x^1{\partial \over \partial x^1}\) and linear 1-form \(\omega ^3=y^1dx^1\) into (10), we get
Then \(b_1b_3+b_3b_1+b^2_3+b_5b_3=b_3a\), i.e. \(b_3(2b_1+b_3+b_5-a)=0\). Then
If we put linear vector fields \(X^1={\partial \over \partial x^1}\) and \(X^2=x^1{\partial \over \partial x^1}\) and linear 1-form \(\omega ^3=x^1dy^1\) into (10), we get
Then, \(b_1b_5+b_3b_5+b_5b_1+b^2_5=b_5a\), i.e. \(b_5(2b_1+b_3+b_5-a)=0\). Then,
If we put linear vector fields \(X^2={\partial \over \partial x^1}\) and \(X^3=x^1{\partial \over \partial x^1}\) and linear 1-form \(\omega ^1=y^1dx^1\) into (12), we get
Then \(b_4a=b_1b_4+b_3b_2+b_3b_4+b_5b_4\). Then
If we put linear vector fields \(X^2={\partial \over \partial x^1}\) and \(X^3=x^1{\partial \over \partial x^1}\) and linear 1-form \(\omega ^1=x^1dy^1\) into (12), we get
Then \(b_6a=b_1b_6+b_3b_6+b_5b_2+b_5b_6\). Then
If we put linear vector fields \(X^2={\partial \over \partial x^1}\) and \(X^3={\partial \over \partial x^2}\) and linear 1-form \(\omega ^1=x^2y^1dx^1\) into (12), we get
Then \(0=b_2b_4+b^2_4+b_6b_4+b_3b_2\). Then
If we put linear vector fields \(X^2=x^1{\partial \over \partial x^1}\) and \(X^3={\partial \over \partial x^1}\) and linear 1-form \(\omega ^1=x^1dy^1\) into (12), we get
Then \(-b_6a=b_2b_6+ b_4b_6+b_6b_2+b_6^2\). Then \(b_6(2b_2+b_4+b_6+a)=0\), i.e.
It remains to consider two cases consisting of several subcases and sub-subcases.
Case I. \(a\not =0\).
If \(b_3\not =0\), then by (26) \(b_3+b_5=0\), and then by (27), \(2b_1=a\). So, using (9), we get \(a=0\). Contradiction. So, in our case
We consider two subcases.
Subcase I.1. \(b_5\not =0\).
By (9), we have three sub-subcases.
Sub-subcase I.1.1. \((b_1,b_2)=(0,0)\).
Since \(b_1=0\) and \(b_3=0\) and \(b_5\not =0\), then by (28) we have \(b_5=a\).
Since \(b_2=0\), then by (31), \(b_4(b_4+b_6)=0\), i.e. \(b_4=0\) or \(b_4+b_6=0\).
If \(b_4=0\), then (since \(b_2=0\)) by (32), \(b_6=0\) or \(b_6=-a\).
If \(b_4+b_6=0\), then (since \(b_2=0\)) by (32), \(b_6=0\) (as \(a\not =0\)), and then \(b_4=-b_6=0\).
Summing up, in our sub-subcase, we have
Sub-subcase I.1.2. \((b_1,b_2)=(a,0)\).
By (28), since \(b_5\not =0\) and \(b_1=a\) and \(b_3=0\) (see (33)), \(2a+0+b_5-a=0\), i.e. \(b_5=-a\).
Since \(b_2=0\), then by (31), \(b_4(b_4+b_6)=0\), i.e. \(b_4=0\) or \(b_4+b_6=0\).
If \(b_4=0\), then by (32) since \(b_2=0\), we get \(b_6=0\) or \(b_6=-a\). So, since \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,0,0,0,-a,-a)\) do not satisfy (30), then \(b_6=0\).
If \(b_4+b_6=0\), then by (32) and \(b_2=0\), we get \(b_6=0\) (as \(a\not =0\)), and then and \(b_4=-b_6=0\).
Summing up, in our sub-subcase, we have
Sub-case I.1.3. \((b_1,b_2)=(a,-a)\).
By (28), since \(b_5\not =0\) and \(b_1=a\) and \(b_3=0\) (see (33)), \(2a+0+b_5-a=0\), i.e. \(b_5=-a\).
Then, by (30), we have \(b_6(a-a-0)=(-a)(-a+b_6)\), i.e. \(0=-a(-a+b_6)\). Then \(b_6=a\).
Moreover, by (29), we have \(b_4(a-a-(-a))=0\), i.e. \(b_4a=0\). Then \(b_4=0\).
Summing up, in our sub-subcase, we have
Subcase I. 2. \(b_5=0\).
By (9) we have three sub-subcases.
Sub-subcase I.2.1. \((b_1,b_2)=(0,0)\).
By (30), we have \(b_6(a-0-0)=0\), i.e. \(b_6=0\). Then by (31) we have \(b_4(b_4+0)=0\), i.e. \(b_4=0\).
Summing up, in our sub-subcase, we have
Sub-subcase I.2.2. \((b_1,b_2)=(a,0)\).
Suppose \(b_4\not =0\). By (31), \(b_4(b_4+b_6) =0\). Then \(b_4+b_6=0\). Then by (32), \(b_6=0\), and then and \(b_4=-b_6=0\). Contradiction. So, \(b_4=0\).
Then by (32), \(b_6=0\) or \(b_6=-a\).
Summing up, in our sub-subcase, we have
Sub-subcase I.2.3. \((b_1,b_2)=(a,-a)\).
By (31), we have \(b_4^2+ b_4b_6=ab_4\), i.e. \(b_4=0\) or \(b_4+b_6=a\).
If \(b_4=0\) then by (32), \(b_6=0\) or \(b_6=a\).
Summing up, in our sub-subcase, we have
For \(\lambda =a\) we realise the case with \(b_4=0\) and \(b_6=a\). That is why we do not expose separately this in above.
Case II. \(a=0\).
Then by (9), \(b_1=b_2=0\). So, if \(b_3\not =0\) or \(b_5\not =0\), then by (26) and (28) we have \(b_3+b_5=0\). If \(b_3=b_5=0\), then we also have \(b_3+b_5=0\). Similarly, if \(b_4\not =0\) or \(b_6\not =0\), then by (31) and (32) we have \(b_4+b_6=0\). If \(b_4=b_6=0\) then of course \(b_4+b_6=0\).
Summing up, in our case, we have
The part “ \(\Rightarrow \) ” of the theorem is complete.
Now, we are going to prove the part “ \(\Leftarrow \) ” of the theorem.
Let \((a,b_1,b_2,b_3,b_4,b_5,b_6)\) be a arbitrary 7-tuple from the list (25). By Proposition 3.4, it is sufficient to show that \((a,b_1,b_2,b_3,b_4,b_5,b_6)\) satisfies conditions (9)–(12) for all linear vector fields \(X^1,X^2, X^3\) and all linear 1-forms \(\omega ^1,\omega ^2, \omega ^3\) on \({\mathbf {R}}^{m,n}\).
We consider two cases and several subcases.
Case 1. \(a\not =0\) .
Subcase 1.1. \((b_1,b_2,b_3,b_4,b_5,b_6)=(0,0,0,0,0,0)\).
The condition (9) holds as \((b_2,b_1)=(0,0)\). The equalities (10)–(12) are \(0=0\).
Subcase 1.2. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,0,0,0,0,0)\).
The condition (9) holds as \((b_2,b_1)=(0,a)\). The equalities (10)–(12) are \(0=0\).
Subcase 1.3. \((b_1,b_2,b_3,b_4,b_5,b_6)=(0,0,0,0,a,0)\).
The condition (9) holds as \((b_2,b_1)=(0,0)\). The equalities (11) and (12) are \(0=0\). Using (2), the equality (10) can be written as
It is satisfied (by \((b_1,b_2,b_3,b_4,b_5, b_6)\)) because \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \).
Subcase 1.4. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,-a,0,0,0, 0)\).
The condition (9) holds as \((b_2,b_1)=(-a,a)\). The equalities (10)–(12) are \(0=0\).
Subcase 1.5. \((b_1,b_2,b_3,b_4,b_5,b_6)=(0,0,0,0,a,-a)\).
The condition (9) holds as \((b_2,b_1)=(0,0)\). Using (2), the equalities (10)–(12) can be written as
They are satisfied because \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \).
Subcase 1.6. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,0,0,0,0,-a)\).
The condition (9) holds as \((b_2,b_1)=(0,a)\). The equality (10) is \(0=0\). Using (2), equalities (11) and (12) can be written as
They are satisfied because \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \).
Subcase 1.7. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,0,0,0,-a,0)\).
The condition (9) holds as \((b_2,b_1)=(0,a)\). Equalities (11) and (12) are \(0=0\). Using (2), equality (10) can be written as
or (after reduction of similar terms) as
It is satisfied because \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \).
Subcase 1.8. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,-a,0,0,-a,a)\).
The condition (9) holds as \((b_2,b_1)=(-a,a)\). Using (2), equality (10) can be written as
or (after reduction of similar terms) as
Similarly, (11) can be written as
or (after reduction of similar terms) as
Similarly, (12) can be written as
or (after reduction of similar terms) as
So, (10)–(12) are satisfied because of \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \).
Subcase 1.9. \((b_1,b_2,b_3,b_4,b_5,b_6)=(a,-a,0,a-\lambda ,0, \lambda )\).
The condition (9) holds as \((b_2,b_1)=(-a,a)\). Condition (10) is \(0=0\). Using (2), (11) can be written as
Then, using formulas \({\mathcal {L}}_{[X,Y]}\omega ={\mathcal {L}}_X{\mathcal {L}}_Y\omega -{\mathcal {L}}_Y{\mathcal {L}}_X\omega \) and \(d{\mathcal {L}}_X={\mathcal {L}}_Xd\), condition (11) can be written as
So, (11) is satisfied because \({\mathcal {L}}_X i_Y- i_Y{\mathcal {L}}_X=i_{[X,Y]}\).
Using (2) and \(d {\mathcal {L}}_X={\mathcal {L}}_X d\), (12) is
So, to prove that (12) is satisfied, it is sufficient to show that the coefficients on \(\lambda ^0\) of both sides of (41) are equal, and the coefficients on \(\lambda ^1\) of both sides of (41) are equal, and the coefficients on \(\lambda ^2\) of both sides of (41) are equal.
Comparing the coefficients on \(\lambda ^0\) in (41), we obtain
This condition is satisfied because
as \(di_{X^3}di_{X^2}=d(di_{X^3}+i_{X^3}d)i_{X^2}=d{\mathcal {L}}_{X^3}i_{X^2}\).
Comparing the coefficients on \(\lambda \) in (41) and using \(d{\mathcal {L}}_X={\mathcal {L}}_Xd\), we obtain
Using the formulas \({\mathcal {L}}_{[X^2,X^3]}={\mathcal {L}}_{X^2}{\mathcal {L}}_{X^3}-{\mathcal {L}}_{X^3}{\mathcal {L}}_{X^2}\) and \(i_{[X^2,X^3]}={\mathcal {L}}_{X^2}i_{X^3}-i_{X^3}{\mathcal {L}}_{X^2}\) we can short equivalently (42) to
By (3), we have \(di_Ldi_{X^2}\omega ^1=di_{X^2}\omega ^1\). Then \({\mathcal {L}}_{X^3}di_Ldi_{X^2}\omega ^1={\mathcal {L}}_{X^3}di_{X^2}\omega ^1\). Moreover, by the formulas \({\mathcal {L}}_X=i_Xd+di_X\) and \(d^2=0\) and \({\mathcal {L}}_X d= d{\mathcal {L}}_X\), we have
Also \(di_{X^3}{\mathcal {L}}_{X^2}di_L\omega ^1 = (di_{X^3}+i_{X^3}d)d{\mathcal {L}}_{X^2}i_L\omega ^1= {\mathcal {L}}_{X^3}{\mathcal {L}}_{X^2}di_L\omega ^1\), i.e.
So, our equality (43) can be equivalently rewritten as
i.e. as \(0=0\). So, (42) holds.
Comparing the coefficients on \(\lambda ^2\) in (41), we get
This condition holds because of (44) and (2) and (45) it can be rewritten as
Case 2. \(a=0\) and \((b_1,b_2,b_3,b_4,b_5,b_6)=(0,0,\lambda ,\mu ,-\lambda , -\mu )\).
The condition (9) holds as \((b_2,b_1)=(0,0)\).
Condition (10) is
It is satisfied because by (44) and (45) and (2) and (3) it can be rewritten as
So, it can be reduced to \(0=0\).
Condition (11) is
It is satisfied because by (44) and (45) and (2) and (3) it can be rewritten as
i.e as \(0=0\).
Condition (12) is
It is satisfied because by (44) and (45) and (2) and (3) it can be rewritten as
i.e. \(0=0\).
The theorem is complete. \(\square \)
Remark 3.6
The space \(\Gamma ^l_{E}(TE\oplus T^*E)\) is a locally free \({\mathcal {C}}^\infty (M)\)-module. Hence, there is a vector bundle \(\hat{E}\) over M such that \(\Gamma ^l_E(TE\oplus T^*E)\) is isomorphic to \(\Gamma \hat{E}\) as \({\mathcal {C}}^\infty (M)\)-modules. The vector bundle \(\hat{E}\) is called the fat vector bundle. It is isomorphic to the Omni–Lie algebroid \({\mathcal {A}}(E):=Der(E^*)\oplus J^1(E^*)\), studied in [1], where \(Der(E^*)\) is the bundle of derivations on \(E^*\), and \(J^1(E^*)\) the first jet prolongation bundle, see [6]. Denote \({\mathcal {A}}(E):=\hat{E}\). Any \(\mathcal {VB}_{m,n}\)-map \(f:E\rightarrow E_1\) with the base map \(f:M\rightarrow M_1\) induces in obvious (functor) way the vector bundle map \({\mathcal {A}}(f):{\mathcal {A}}(E)\rightarrow {\mathcal {A}}(E_1)\) covering \({\underline{f}}\). In other words, we have a so-called vector gauge bundle functor \({\mathcal {A}}:\mathcal {VB}_{m,n}\rightarrow \mathcal {VB}\). Thus a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) is a (usual) \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator \(A:{\mathcal {A}}\times {\mathcal {A}}\rightsquigarrow {\mathcal {A}}\) (in the sense of [7]). Thus, Theorem 3.5 gives the full description of all \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear brackets satisfying the Jacobi identity in Leibniz form on sections of the Omni–Lie algebroid of E.
Definition 3.7
A natural Lie bracket on \(\Gamma _E^l(TE\oplus T^*E)\) is a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear skew-symmetric operator \(A: \Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) satisfying the Jacobi identity in Leibniz form.
We have the following immediate consequence of Theorems 2.21 and 3.5.
Corollary 3.8
Let \(m\ge 2\) and \(n\ge 1\) be natural numbers. Let \(A:\Gamma ^l(T\oplus T^*)\times \Gamma ^l(T\oplus T^*)\rightsquigarrow \Gamma ^l(T\oplus T^*)\) be a \(\mathcal {VB}_{m,n}\)-gauge-natural bilinear operator. Then, A is skew-symmetric if and only if it is of the form (1) for arbitrary (uniquely determined by A) real numbers \(a,b_1,b_2,b_3,b_4,b_5,b_6\) satisfying
Moreover, such A is a Lie bracket if and only if \((a,b_1,b_2,b_3,b_4,b_5,b_6)\) is from the following list of 7-tuples:
where \(c, \lambda \) are arbitrary real numbers with \(c\not =0\).
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Mikulski, W.M. The Gauge-Natural Bilinear Operators Similar to the Dorfman–Courant Bracket. Mediterr. J. Math. 17, 40 (2020). https://doi.org/10.1007/s00009-020-1472-1
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DOI: https://doi.org/10.1007/s00009-020-1472-1
Keywords
- Natural operator
- linear vector field
- linear 1-form
- Dorfman–Courant bracket
- Jacobi identity in Leibniz form