The conference proceedings for Geometry of Jets and Fields in honour of Professor Janusz Grabowski are now published: you can find an online version here.

I have a contribution with Janusz Grabowski, Katarzyna Grabowska and Paweł Urbański entitled New developments in geometric mechanics.

Gennadi Sardanashvily – passed away on the September 1, 2016 – also has a contribution in the proceedings. I did not know Sardanashvily well, but our few interactions told me he was a nice guy. I am sure the community will miss him.

In better news, my wife Gemma had a portrait of Janusz Grabowski published in the proceedings!

Our paper, Polarisation of Graded Bundles, with Janusz Grabowski and Mikołaj Rotkiewicz has now been published in SIGMA [1].

In the paper we show that Graded bundles (cf. [2]), which are a particular kind of graded manifold (cf. [3]), can be `fully linearised’ or `polarised’. That is, given any graded bundle of degree k, we can associate with it in a functorial way a k-fold vector bundle – we call this the full linearisation functor. In the paper [1], we fully characterise this functor. Hopefully, this notion will prove fruitful in applications as k-fold vector bundles are nice objects that that various equivalent ways of describing them.

Graded Bundles
Graded bundles are particular examples of polynomial bundles: that is we have a fibre bundle whose are \(\mathbb{R}^{N}\) and the admissible changes of local coordinates are polynomial. A little more specifically, a graded bundle $F$, is a polynomial bundle for which the base coordinates are assigned a weight of zero, while the fibre coordinates are assigned a weight in \(\mathbb{N} \setminus 0\). Moreover we require that admissible changes of local coordinates respect the weight. The degree of a graded bundle is the highest weight that we assign to the fibre coordinates.

Any graded bundle admits a series of affine fibrations
\(F = F_k \rightarrow F_{k-1} \rightarrow \cdots \rightarrow F_{1} \rightarrow F_{0} =M\),
which is locally given by projecting out the higher weight coordinates.

For example, a graded bundle of degree 2 admits local coordinates \((x, y ,z)\) of weight 0,1, and 2 respectively. Changes of coordinates are then, `symbolically’
\(x’ = x'(x)\),
\(y’ = y T(x)\),
\(z’ = z G(x) + \frac{1}{2} y y H(x)\),
which clearly preserve the weight.

We then have a series of fibrations
\(F_2 \rightarrow F_1 \rightarrow M\),
given (locally) by
\((x,y,z) \mapsto (x,y) \mapsto (x)\).

Linearisation
The basic idea of the full linearisation is quite simple – I won’t go into details here. Recall the notion of polarisation of a homogeneous polynomial. The idea is that one adjoins new variables in order to produce a multi-linear form from a homogeneous polynomial. The original polynomial can be recovered by examining the diagonal.

As graded bundles are polynomial bundles, and the changes of local coordinates respect the weight, we too can apply this idea to fully linearise a graded bundle. That is, we can enlarge the manifold by including more and more coordinates in the correct way as to linearise the changes of coordinates. In this way we obtain a k-fold vector bundle, and the original graded bundle, which we take to be of degree k.

So, how do we decide on these extra coordinates? The method is to differentiate, reduce and project. That is we should apply the tangent functor as many times as is needed and then look for a substructure thereof. So, let us look at the degree 2 case, which is simple enough to see what is going on. In particular we only need to differentiate once, but you can quickly convince yourself that for higher degrees we just repeat the procedure.

The tangent bundle \( T F_2\) – which we consider the tangent bundle as a double graded bundle – admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)} \; \underbrace{\dot{x}}_{(0,1)}, \; \underbrace{\dot{y}}_{(1,1)} ,\; \underbrace{\dot{z}}_{(2,1)})\)

The changes of coordinates for the ‘dotted’ coordinates are inherited from the changes of coordinates on \(F_2\),
\(\dot{x}’ = \dot{x}\frac{\partial x’}{\partial x}\),
\( \dot{y}’ = \dot{y}T(x) + y \dot{x} \frac{\partial T}{\partial x}\),
\(\dot{z}’ = \dot{z}G(x) + z \dot{x}\frac{\partial G}{\partial x} + y \dot{y}H(x) + \frac{1}{2}y y \dot{x}\frac{\partial H}{\partial x}\).
Thus we have differentiated.

Clearly we can restrict to the vertical bundle while still respecting the assignment of weights – one inherited from \(F_2\) and the other comes from the vector bundle structure of a tangent bundle. In fact, what we need to do is shift the first weight by minus the second weight. Technically, this means that we no longer are dealing with graded bundles, the coordinate \(\dot{x}\) will be of bi-weight (-1,1). However, the amazing thing here is that we can set this coordinate to zero – as we should do when looking at the vertical bundle – and remain in the category of graded bundles. That is, not only is setting \(\dot{x}=0\) well-defined, you see this from the coordinate transformations; but also this keeps us in the right category. We have preformed a reduction of the (shifted) tangent bundle.

Thus we arrive at a double graded bundle \(VF_2\) which admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
and the obvious admissible changes thereof.

Now, observe that we have the degree of \(z\) as (2,0), which is the coordinate with the highest first component of the bi-weight. Thus, as we have the structure of a graded bundle, we can project to a graded bundle of one lower degree \(\pi : VF_2 \rightarrow l(F_2)\). The resulting double vector bundle is what we will call the linearisation of \(F_2\).

So we have constructed a manifold with coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
with changes of coordinates
\(x’ = x'(x)\),
\(y’ = y T(x)\)
\( \dot{y}’ = \dot{y}T(x)\),
\(\dot{z}’ = \dot{z}G(x) + y \dot{y}H(x)\).

Then, by comparison with the changes of local coordinates on \(F_2\) you see that we have a canonical embedding of the original graded bundle in its linearisation as a ‘diagonal’
\(\iota : F_2 \rightarrow l(F_2)\),
by setting \(\dot{y} = y\) and \(\dot{z} = 2 z\).

References
[1] Andrew James Bruce, Janusz Grabowski and Mikołaj Rotkiewicz, Polarisation of Graded Bundles, SIGMA 12 (2016), 106, 30 pages.

[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36.

[3] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168.

Janusz Grabowski and I have placed a prepint on the arXiv with the title Pre-Courant Algebroids.

In the `classical language’, a Courant algebroid is a vector bundle, whose sections come equipped with a bracket – bilinear map – together with an anchor map and a nondegenerate symmetric bilinear form that satisfy some compatibility conditions. The bracket on the space of sections is not a Lie bracket, but rather a non-skewsymmetric bracket that satisfies the Jacobi identity in Loday-Leibniz form. This bracket is usually called the Courant–Dorfman bracket.

A pre-Courant algebroid can be thought of as a Courant algebroid but without the Jacobi identity on the Courant–Dorfman pre-bracket.

It has long be known, due to Roytenberg [1], that Courant algebroids are `really’ symplectic Lie 2-algebroids. That is, we have an N-manifold of degree 2 (a supermanifold with a particular additional grading), equipped with a nondegenerate Poisson bracket of degree -2 and a homological vector field of degree 1 that is Hamiltonian. The brackets of Courant algebroid can then be recovered using the derived bracket formalism and the bilinear form is encoded in the symplectic structure.

Pre-Courant algebroids in the superlanguage
So, do we have a similar understanding of pre-Courant algebroids? The answer is yes…

First back to Courant algebroids. As stated above, they can be encoded in a Hamiltonian vector field – and so they can be encoded in a Grassmann odd Hamiltonian of degree/weight 3, which we denote as \( \Theta\). The fact that the Hamiltonian vector field is homological (Grassmann odd and squares to zero) is equivalent to

\( \{ \Theta, \Theta \} =0 \).

This condition encodes all the compatibility conditions between the bracket and the anchor map (a particular vector bundle map to the tangent bundle). More than that, this condition also encodes the Jacobi identity for the bracket. Thus, we need a weaker condition that is not too weak – we only want to lose the Jacobi identity and keep the other conditions. It turns out that we require

\( \{\{ \Theta, \Theta \}, f\} =0 \),

for all weight zero functions f, if we want to encode a pre-Courant algebroid in exactly the same way as we do a Courant algebroid. In the preprint we define what we call symplectic almost Lie 2-algebroids in this way and show how they correspond to pre-Courant algebroids.

Does this help any?
This change in starting position simplifies many basic facts about pre-Courant algebroids – just as it does with Courant algebroids. In particular, the notion a Dirac structures as a particular Lagrangian submanifolds is quite clear.

In the preprint was also show that including a compatible N-grading is quite simple when one uses the language of homogeneity structures [2]. One should also consult [3,4] where the notion of weighted Lie groupoids and weighted Lie algebroids are explored. As an example VB-Courant algebroids – Courant algebroids with a compatible vector bundle structure – are natural examples of weighted (pre-)Courant algebroids. This change of postion to `graded super bundles’ with some additional structures allows for a very neat understanding of weighted Dirac structure and in particular VB-Dirac structures. This framework simplifes the understanding of many thing.

Conclusion
The bottom line seems to be that Courant algebroids are `really’ sympelectic Lie 2-algebroids and pre-Courant algebroids are really symplectic almost Lie 2-algebroids.

References
[1] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in: Quantization, Poisson brackets and beyond (Manchester, 2001), 169–185, Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002.

[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21–36.

[3] A.J. Bruce, K. Grabowska & J. Grabowski, Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090.

[4] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101
(2016), 71–99.

My paper with Alfonso Tortorell on higher versions of Kirillov’s local Lie algebras has now been published in Diffrential Geometry and Applications [1]. If you have access to this journal you can follow this link.

In this paper we take the point of view that Jacobi geometry is best understood as homogeneous Poisson geometry – that is Poisson geometry on principle \(\mathbb{R}^{\times}\)-bundles. Every line bundle over a manifold can be understood in terms of such a principle bundle.

The same holds try when we pass to supermanifolds. With this in mind Alfonso and I more-or-less just replace Poisson with higher or homotopy Poisson. This allows us to neatly define an \(L_{\infty}\)-algebra on the space of sections of an even line bundle in the categeory of supermanifolds. This algebra is the higher/homotopy generalisation of Kirillov’s local Lie algebra on the space of sections of a line bundle.

We show that the basic theorems from Kirillov’s local Lie algebras or Jacobi bundles all passes to this higher case.

Refrences
[1] Andrew James Bruce & Alfonso Giuseppe Tortorella, Kirillov structures up to homotopy, Differential Geometry and its Applications Volume 48, October 2016, Pages 72–86.