Category Archives: Research work

Functional analytic questions and products of higher graded supermanifolds

board In two preprints Functional Analytic Issues in \(\mathbb{Z}^n_2\)-geometry and Products in the category of \(\mathbb{Z}^n_2\)-manifolds Norbert Poncin and I explore in some detail the Fréchet algebra structure on the structure sheaf of a \(\mathbb{Z}^n_2\)-manifold and use this to deduce several important results including the fact that the category of \(\mathbb{Z}^n_2\)-manifolds admits (finite) products.

Loosley, \(\mathbb{Z}^n_2\)-manifolds are manifold-like objects for which we have local coordinates that are assigned a grading in \(\mathbb{Z}^n_2 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \mathbb{Z}_2\) (n-times) and the coordinates are \(\mathbb{Z}^n_2\)-commutative with the sign factor being given by the standard scalar product on \(\mathbb{Z}^n_2\). Note that this means that the sign factors are not determined by the parity, i.e., the sum of the components of the \(\mathbb{Z}_2^n\)-degree. In particular, we may have coordinates that anticommute but are none the less non-nilpotent. This is in stark contrast to the standard case of supermanifolds. The upshot is that we have non-nilpotent formal coordinates and must use power series and not polynomials in the formal coordinates when defining the structure sheaf. This can lead to many subtleties when developing the theory. The basic theory using locally ringed spaces is quite new [1,2] and many basic questions remain.

In the two preprints, we address some foundational issues anchored in functional analysis. Alongside other results, we have shown the following:

  • The structure sheaf of a \(\mathbb{Z}^n_2\)-manifold is a nuclear Fréchet sheaf of \(\mathbb{Z}^n_2\)-graded \(\mathbb{Z}^n_2\)-commutative algebras;
  • Morphisms of \(\mathbb{Z}^n_2\)-manifolds are continous with respect to the local convex topolgies on spaces of local sections;
  • All the information about a \(\mathbb{Z}^n_2\)-manifold is completely encoded in the algebra of global sections of the structure sheaf – we have a reconstruction theorem and an embedding of the category of \(\mathbb{Z}^n_2\)-manifold into the (opposite) category of unital \(\mathbb{Z}^n_2\)-graded \(\mathbb{Z}^n_2\)-commutative algebras;
  • The cartesian product of \(\mathbb{Z}^n_2\)-manifolds is well defined and satisfies the required universal properties to be a categorical product. Thus, the category \(\mathbb{Z}^n_2\)-manifold admits products.

While none of the above results are very surprising given that the same statements can be made for smooth manifolds and indeed supermanifolds, the non-trivial problems arise due to the fact that we are forced to deal with algebras of formal power series. Some of the proof are minor modifications of the proofs for supermanifolds (the n=1 case), while others really required a lot of work in checking things carefully.

At every stage, it seems that while \(\mathbb{Z}^n_2\)-manifolds are a non-trivial extension of supermanifolds, they do provide a nice workable example of noncommutative geometries in which one can keep a large part of one’s classical thinking – with some care. So far, the basic theory of smooth manifolds extends to the theory of \(\mathbb{Z}^n_2\)-manifolds. The exception here seems to be the theory of integration, which is already more complicated for supermanifolds as compared with classical manifolds. The interested reader may consult [3] for a review of the current state of affairs.

Now, with these results in place, it seems the right time to look for further applications of \(\mathbb{Z}^n_2\)-manifolds… watch this space!

References
[1] Covolo, Tiffany; Grabowski, Janusz; Poncin, Norbert The category of \(\mathbb{Z}^n_2\)-supermanifolds, J. Math. Phys. 57 (2016), no. 7, 073503, 16 pp.

[2] Covolo, Tiffany; Grabowski, Janusz; Poncin, Norbert Splitting theorem for \(\mathbb{Z}^n_2\)-supermanifolds, J. Geom. Phys. 110 (2016), 393–401.

[3] Poncin, Norbert Towards integration on colored supermanifolds. Geometry of jets and fields, 201–217, Banach Center Publ., 110, Polish Acad. Sci. Inst. Math., Warsaw, 2016.

Mixed symmetry tensors and their graded description

board In a preprint `The Graded Differential Geometry of Mixed Symmetry Tensors , Eduardo Ibarguengoytia and I describe how one use the recently developed theory of \(\mathbb{Z}^n_2\)-manifolds [1].

Background
Differential forms are covariant tensor fields that are completely antisymmetric in their indices and it is well-known that supermanifolds offer a neat way to encode such tensors. Mixed symmetry tensor fields are covariant tensors fields are a natural generalisation of differential forms in which the tensors are neither fully symmetric nor antisymmetric. In physics, such tensor fields appear in the context of higher spin fields and dual gravitons. In particular, the particle spectrum of string theory contains beyond the massless particles of the effective supergravity theory, an infinite tower of massive particles of ever higher spin. Thus, if one wants to consider the effective theory beyond the effective supergravity theory, one is forced to contend with mixed symmetry tensors. The first study of mixed symmetry tensors field from a physics perspective was Curtright [2] who developed a generalised version of gauge theory using higher rank tensors. It was Hull [3] who suggested that such fields, in particular, the dual gravition and double dual gravition, maybe useful in probing various aspects of M-theory.

Recently, Chatzistavrakidis, Khoo, Roest, & Schupp [4] used a “generalised supermanifold” in which we have two sets of anticommuting coordinates which mutually commute in order to describe certain mixed symmetry tensors. It turns out that they are unknowingly using particular \(\mathbb{Z}^2_2\)-manifolds!

Our contribution
In our short note (6 pages), we highlight the use of \(\mathbb{Z}^2_2\)-manifolds to describe mixed symmetry tensors with two blocks of antisymmetric indices. We show that many of the known expressions involving Curtright’s dual gravition in five dimensions can be neatly expressed using these higher graded manifolds. We briefly discuss the flat space-time situation and the case of curved space-times where we really do see some differences as compared with the theory of standard differential forms. We hope that this observation could be useful to others working in string theory and related topics.

References
[1] Covolo, T., Grabowski, J. & Poncin, N., The category of \(\mathbb{Z}^n_2\)-supermanifolds, J. Math. Phys. 57 (2016), no. 7, 073503, 16 pp.

[2] Curtright, T., Generalized gauge fields, Physics Letters B. 165 (1985), 304–308.

[3] Hull, C.M., Strongly coupled gravity and duality, Nuclear Phys. B 583 (2000), no. 1-2, 237–259.

[4] Chatzistavrakidis, A., Khoo, F.S., Roest, D. & Schupp, P., Tensor Galileons and gravity, J. High Energy Phys.(2017), no.3, 070.

Almost commutative versions of Lie algebroids?

board In a preprint `Almost Commutative Q-algebras and Derived brackets , I describe how one can in part generalise the notion of Lie algebroid using Vaintrob’s understanding interms of Q-manifolds [1].

A question that I posed to myself a while ago was if the `super-understanding’ of Lie algebroids in terms of a graded supermanifold equipped with a homological vector field can be generalised to the noncommutative world. Lie–Rinehart pairs have long been understood as the algebraic counterpart to Lie algebroids and offer a direct route to the noncommutative world. However, the idea is to start with Vaintrob’s picture of Lie algebroids. The full problem in the setting of noncommutative geometry seems not to be so tractable. However, the problem in the context of almost commutative geometry (see [2]) has now been tackled.

It turns out that almost commutative algebras, loosely algebras in which elements `almost’ commute, i.e., ab = k ba for some number k, one can mimic the classical case closely. In particular, almost commutativity is close enough to commutativity or supercommutativity (things commute up to signs), that one can make sense of non-negatively graded almost commutative algebras. Philosophically, such algebras are thought of as the total spaces of some `almost commutative vector bundles’ following the ethos of Grabowski & Rotkiewicz [3] (and Th. Voronov in several of his papers). We can make sense of homological derivations of weight one and push the derive bracket formalism of Kosmann-Schwarzbach [4] through and construct a kind of Lie bracket and anchor map. In short, with a little care, all the basic ideas of describing Lie algebroids in terms of supergeometry can be generalised to almost commutative geometry.

While the results are essentially the expected ones, this shows that ideas from graded and supergeometry, including derived brackets, can be applied to specific versions of noncommutative geometries. We hope to further explore this in the near future.

Thanks
I thank Prof. Tomasz Brzezinski and Prof. Richard Szabo for their advice with parts of this preprint.

References
[1] Vaĭntrob, A. Yu. Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), no. 2, 428–429

[2] Bongaarts, P. J. M. & Pijls, H. G. J. Almost commutative algebra and differential calculus on the quantum hyperplane, J. Math. Phys. 35 (1994), no. 2, 959–970.

[3] Grabowski, J. & Rotkiewicz, M. Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36.

[4] Kosmann-Schwarzbach, Y. Derived brackets, Lett. Math. Phys. 69 (2004), 61–87.

Filtered bundles

board The paper `On the concept of a filtered bundle ‘ with Katarzyna Grabowska and Janusz Grabowski to appear in International Journal of Geometric Methods in Modern Physics is now `online ready’ and available for free for the rest of this October!

In the paper we generalise the notion of a graded bundle – a particularly nice kind on non-negatively graded manifold – allow for coordinate changes that do not strictly preserve the grading `on the nose’, but instead include lower degree terms. The coordinate changes are thus filtered. It turns out that many nice things from the theory for graded bundles can naturally be generalised to this filtered setting. One of the nice results is that any filtered bundle is non-canonically isomorphic to a graded bundle, and so furthermore any filtered bundle is non-canonically isomorphic to a Whitney sum of vector bundles. We also show that the linearisation process as given in [1] also carries over to this filtered setting.

Many examples of the polynomial bundles found in geometric mechanics and geometric formulations of field theories are not graded bundles, but rather they have a filtered structure. In the paper we take an abstraction of some of the basic structure of jet bundles and similar with an eye for future applications. Hopefully some of these ideas will be useful.

References
[1] 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.

Paper on supermechanics

board The paper `On a geometric framework for Lagrangian supermechanics‘ with Katarzyna Grabowska and Giovanni Moreno now been published in The Journal of Geometric Mechanics (JGM).

In the paper we re-examine Lagrangian mechanics on supermanifolds (loosely `manifolds’ with both commuting and anticommuting coordinates) in the geometric framework of Tulczyjew [2]. The only real deviation from the classical setting is that one now needs to understand curves in a more categorical framework as maps \(S \times \mathbb R \rightarrow M\), where \(M\) is the supermanifold understudy and \(S\) is some arbitrary supermanifold [1]. Thus one needs to think of families of curves parameterised by arbitrary supermanifolds and thus we have families of Lagrangians similarly parameterised. In our opinion, although the super-mechanics is now an old subject, none of the existing literature really explains what a solution to the dynamics is. We manage to describe the phase dynamics in this super-setting and give real meaning to solutions thereof, albeit one needs to think more categorically than the classical case.

Our philosophy is that time is the only true meaningful parameter describing dynamics on supermanifolds. The auxiliary parameterisations by \(S\) are necessary in order to property `track out’ paths on the supermanifold, but they must play no fundamental role in the theory other than that. Mathematically, this is described in terms of category theory as all the constructions are natural in \(S\). This basically means that if we change \(S\) then the theory is well behaved and that nothing fundamentally depends on our choice of \(S\). The theory holds for all \(S\), and to fully determine the dynamics one needs to consider all supermanifolds as `probes’ – in more categorical language we are constructing certain functors. This seems to take us away from the more familiar setting of classical mechanics, but it seems rather unavoidable. Supermanifolds are examples of `mild’ noncommutative spaces and as such we cannot expect there to be a very simple and universal notion of a curve – this is tied to the localisation problem in noncommutative geometry. Specifically, supermanifolds are not just collections of points together with a topology (or something similar).

The bottom line is that the classical framework of geometric mechanics following Tulczyjew generalises to the super-case upon taking some care with this `extended’ notion of a curve.

References
[1] A. J. Bruce, On curves and jets of curves on supermanifolds, Arch. Math. (Brno), 50 (2014), 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.

Remarks on Contact and Jacobi Geometry

board The paper `Remarks on Contact and Jacobi Geometry‘ with Katarzyna Grabowska and Janusz Grabowski now been published in SIGMA [1].

In the paper we present a rather general formalism to define and study Jacobi and Kirllov structures using principle \(\mathbb{R}^\times\)-bundles equipped with homogeneous Poisson structures. This approach was first described by Grabowski [2]. This set-up allows for a rather economical description of contact/Jacobi groupoids and related structures. Importantly, by using homogeneous Poisson structures we simplify the overall picture of contact/Jacobi/Kirillov geometry and show that many technical proofs of various statements in the theory are drastically simplified. We think that this approach gives new insight into the existing theory and hopefully the ideas will be useful to others.

Contact Geometry
Contact geometry is motivated by the formalism of classical mechanics, and in particular looking at constant energy surfaces in phase space. Jacobi geometry is the `degenerate brother’ of contact geometry, and Kirillov geometry is the `twisted sister’ of Jacobi geometry – for those that know think of the relation between symplectic and Poisson geometry, and then trivial and non-trivial line bundles. Contact geometry clearly from its conception has broad applications in physics, ranging from classical mechanics, geometric optics and thermodynamics. There are also some mathematical applications such as knot invariants and invariants in low dimensional topology.

Lie groupoids
Another facet of this paper are Lie groupoids, which should be through of as a wider setting to discuss symmetries than groups. Very loosely, a groupoid is a `many object’ group, and a Lie groupoid is a `geometric’ version of a groupoid. Associated with any Lie group is a Lie algebra, which describes infinitesimal (so `very small’) symmetries of geometric entities. Likewise, associated with any Lie groupoid is a Lie algebroid. Without any details, a Lie algebroid should be considered as describing `very small’ symmetries associated with a Lie groupoid. However, unlike Lie groups and algebras, not every Lie algebroid comes from a Lie groupoid!

Why study contact/Jacobi/Kirillov Groupoids?
Bringing contact and groupoids together is, in the standard setting, not so easy. Our formalism makes this much clearer and allows for direct generalisations to Jacobi geometry. But why bring them together in the first place?

Alan Weinstein [3] introduced the notion of a symplectic groupoid with the intention of extending methods from geometric quantisation to Poisson manifolds. Very loosely, the geometry of Lie groupoids is needed in geometric approaches that allow a passage from classical mechanics to quantum mechanics. In a sense, one can think of symplectic and Poisson groupoids as the Lie groupoid versions of the phase spaces found in classical mechanics, i.e., the spaces formed by position and momentum.

Since the initial work of Weinstein the topic of symplectic and Poisson groupoids has exploded, largely motivated by the geometry of classical mechanics – not that all practitioners see this!

Similarly, given the role of contact geometry in physics, it is natural to think about groupoid versions of contact and Jacobi geometry. More than this, it turns out that the integrating objects of Jacobi/Kirillov structures are precisely contact groupoids (as we define them). That is, as soon as one thinks about the `degenerate brother’ and `twisted sister’ of contact geometry one encounters contact groupoids as the `finite versions’.

For me, all this is strongly motivated by the basic questions of the geometry of classical mechanics. It is rather amazing that we are pushed rather quickly into more and more difficult ideas in geometry. And this is before we get into the quantum world!

Acknowledgments
I personally thank the anonymous referees for their effort in reading the paper and providing many helpful comments and suggestions. For sure the paper would not be what it is today without them.

References
[1] Andrew James Bruce, Katarzyna Grabowska and Janusz Grabowski, Remarks on Contact and Jacobi Geometry, SIGMA 13 (2017), 059, 22 pages.

[2] Janusz Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013), 27-58.

[3] Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101-104.

Modular classes of Q-manifolds

board Q-manifolds are supermanifolds equipped with a Grassmann odd vector field that `squares to zero’, which is known as a homological vector field. Such things can be found behind the AKSZ-BV formalism in mathematical physics and in differential geometry they encode Lie algebroids and Courant algebroids amongst other things. The notion of the modular class of a Q-manifold is known to experts but there is not much in the literature to date.

In the preprint entitled “Modular classes of Q-manifolds: a review and some applications”, I review the notion of the modular class of a Q-manifold – which is understood as the obstruction to the existence of a Berezin volume that is invariant under the action of the homological vector field. The modular class is naturally defined in terms of the divergence of a chosen Berezin volume, but is independent of this choice. The notion directly generalises the notion of the modular class of a Poisson manifold (Koszul [1] and Weinstein [2]) and that of a Lie algebroid (Evans & Weinstein [3]).

I discuss the basic constructs and immediate consequences, all of which are probably known to the handful of experts. Maybe more interesting is that fact that I then apply this to double Lie algebroids ([4,5,6] ) and higher Poisson manifolds [7]. Along the way I make several observations which I believe maybe genuinely new. Either way, having these ideas written clearly in one place is beneficial to the community.

The basic idea
A Q-manifold is a pair \((M,Q)\), where \(M\) is a supermanifold and \(Q \in Vect(M)\) is an odd vector field that ‘self commutes’

\(Q^2 = \frac{1}{2} [Q,Q] =\frac{1}{2} \left( Q \circ Q – (-1)^{1} Q \circ Q \right)\),

note the extra minus sign as compared with the classical case of vector fields on a manifold. This means that `squaring to zero’ is a non-trivial condition. Moreover, as we have an odd vector field that squares to zero we have a differential and so a cohomology theory. In particular, \((C^{\infty}(M), Q )\) is a cochain complex and the related cohomology we refer to as the standard cohomology.

Given any Berezin volume \(\mathbf{\rho} = D[x] \rho(x)\), we can define the divergence of \(Q\) with respect to this volume:

\(L_{Q} \mathbf{\rho} = \mathbf{\rho} {Div}_{\rho}(Q). \)

Note that \({Div}_{\rho}(Q)\) is then a Grassmann odd function on \(M\) and it is \(Q\)-closed. Moreover, it turns out that under change of the Berezin volume the divergence of \(Q\) changes by a \(Q\)-exact term. Thus, we can define the modular class as the standard cohomology class of the divergence of the homological vector field and this does not depend on any chosen Berezin volume

\(Mod(Q) = [Div_{\mathbf{\rho}}(Q)]_{St}. \)

In local coordinates \(Q = Q^{a}(x)\frac{\partial}{\partial x^a}\) and so the modular class has a local characteristic representative

\(\phi_{Q}(x) = \frac{\partial Q^{a}}{\partial{x^a}}(x),\)

which corresponds to picking the standard coordinate volume (we simply drop the \(Q\)-exact term in the definition of the divergence). Moreover, we do not have a Poincare lemma here and so thinking of local representatives of cohomology classes makes sense in general.

In this way we associate to any Q-manifold a characteristic class in its standard cohomology. The modular class is one of the simplest such classes one can imagine on a Q-manifold. There are more complicated things, see [8].

Thanks
I thank prof. Janauzs Grabowski for giving me the opportunity to present some of the ideas in this preprint at a Geometric Methods in Physics seminar in Warsaw on April 26th 2017. I also thank Florian Schatz for reading an earlier draft of this preprint.

References
[1] Koszul, J., Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Elie Cartan (Lyon, 1984), Asterisque 1985, Numero Hors Serie, 257–271.

[2] Weinstein A., The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379–394.

[3] Evens, S., Lu, J.H., Weinstein, A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser. 2 50 (1999), 417–436.

[4] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, I., Adv. Math. 94 (1992), no. 2, 180–239.

[5] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, II., Adv. Math. 154 (2000), no. 1, 46–75.

[6] Voronov, Th., Q-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), no. 2, 279–310.

[7] Voronov, Th., Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133–153.

[8] Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A., Characteristic classes of Q-manifolds: classification and applications, J. Geom. Phys. 60 (2010), no. 5, 729–759.

Representations theory of Lie algebroids and weighted Lie algebroids

board Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds for weighted Lie algebroids as these are a wide generalization fo VB-algebroids.

In a preprint entitled “Graded differential geometry and the representation theory of Lie algebroids” with Janusz Grabowski and Luca Vitagliano, we look at the relation between weighted Lie algebroids [1], Lie algebroid modules [2] and representations up to homotopy of Lie algebroids [3]. We show that associated with any weighted Lie algebroid is a series of canonical Lie algebroid modules over the underlying weight zero Lie algebroid. Moreover, we know, due to Mehta [4], that a Lie algebroid module is (up to isomorphisms classes) equivalent to a representation up to homotopy of the Lie algebroid.

Weighted Lie groupoids were first defined and studied in [5] and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem [6] holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond [7].

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

[2] Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428–429.

[3] Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew.Math, 663 (2012), 91–126.

[4] Mehta R.A., Lie algebroid modules and representations up to homotopy. Indag. Math. (N.S.) 25 (2014), no. 5, 1122–1134.

[5] Bruce A.J., Grabowska K., Grabowski J., Graded Bundles in the Category of Lie Groupoids, SIGMA 11 (2015), 090, 25 pages.

[6] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv, 78 (2003), 681–72.

[7] Cabrera A., Drummond T., Van Est isomorphism for homogeneous cochains, Pacific J. Math. 287 (2017), 297–336

Geometry of Jets and Fields in honour of Professor Janusz Grabowski

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!

Linearising graded manifolds

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.