How can you superise a graded manifold?

The question J. Grabowski, M. Rotkiewicz and I asked was ‘how can we superise a (purely even) graded manifold?’ We propose an interesting solution in our latest preprint Superisation of graded manifolds.

We start with the problem of passing from a particular ‘species’ of graded manifold, known as graded bundles [1]. Graded bundles are non-negatively graded (purely even) manifolds for which the grading is associated with a smooth action of the multiplicative monoid of reals. Such graded manifolds have a well defined structure, nice topological properties and a well defined differential calculus. For these reason we decided that this special class of graded manifold should be the starting place.

Moreover, any vector bundle structure can be encoded in a regular action of the monoid of multiplicative reals. A graded bundle is a ‘vector bundle’ for which we relax the condition of being regular. As everyone knows, the parity reversion functor takes a vector bundle (the total space of) and produces a linearly fibred supermanifold. This functor just declares the fibre coordinates of the vector bundle (in the category of smooth manifolds) to be Grassmann odd. Importantly, one can ‘undo’ this superisation by once again shifting the Grassmann parity of the fibre coordinates. Thus, the parity reversion functor acting on purely even vector bundles is an inconvertible functor and we establish a categorical equivalence between vector bundles and linearly fibred supermanifolds.

Passing to graded bundles
However, such a direct functor cannot exist for graded bundles. Graded bundles are not ‘linear objects’, the changes of non-zero weight local coordinates are polynomial. Simply declaring some coordinates to be Grassmann odd is not going to produce an invertible functor: we have nilpotent elements and now terms that are skew-symmetric which by contraction with symmetric terms in the transformation laws will vanish. In short, some information about the changes of local coordinates is going to be lost when we superise by brute force. We do obtain a functor that takes a graded bundle and produces a supermanifold, but we cannot go back!

Any meaningful ‘superisation’ of a graded bundle must be in terms of an invertible functor and allow us to establish a categorical equivalence between the category of graded bundles and some subcategory of the category of supermanifolds (or some other ‘super-objects’).

Our solution to this conundrum is a two stage plan of attack: first fully linearise and then superise.

Full linearisation
First we fully linearise a graded bundle by repeated application of the linearisation functor [2]. In this way we get a functor that takes a graded bundle of degree k and produces a k-fold vector bundle. In the paper we characterise this functor and make several interesting observations, especially in relation to the degree two case.

The basic idea of the full linearisation is that we polarise the polynomial changes of local coordinates. That is, we add more and more local coordinates in such a way as to fully linearise the changes of coordinates. We do this by repeated application of the tangent functor and substructures thereof. We also have an inverse procedure of diagonalisation, which allows us to ‘undo’ the full linearsation.

As a k-fold vector bundle is ‘multi-linear’ we can superise it!

Standard superisation
Following Voronov [3], we can apply the standard parity reversion functor to a k-fold vector bundle in each ‘direction’ and obtain a supermanifold. Thus, by fully linearising a graded bundle and then application of the parity reversion functor in each ‘direction’ we obtain a supermanifold.

However, this procedure is not really unique: one obtains different functors depending on which order each parity reversion functor is applied. These different functor are of course related by a natural transformation, so there is no deep problem here. However, when we consider just vector bundles the parity reversion functor works perfectly and we have no ambiguities in our choice of functor. This suggest that we can do something better for k-fold vector bundles and our superisation of graded bundles.

Higher supermanifolds
Instead of using standard supermanifolds we can employ \(\mathbb{Z}_{2}^{k}\)-supermanifolds [4]. It is known from [4] that these ‘higher supermanifolds’ offer a neat way to superise k-fold vector bundles without any ambiguities. Thus, in our paper we apply this higher superisation to the lineariastion of a graded bundle.

In short, we can in a functorial and invertible way associate a \(\mathbb{Z}_{2}^{k}\)-supermanifold with a graded bundle answering our opening question.

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

[2] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).

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

[4] T. Covolo, J. Grabowski & N. Poncin, \(\mathbb{Z}_{2}^{n}\)-Supergeometry I: Manifolds and Morphisms, arXiv:1408.2755[math.DG], (2014).

Paper on weighted Groupoids publsihed in SIGMA

Our paper ‘Graded bundles in the Category of Lie Groupoids‘, written with K. Grabowska and J. Grabowski, has now been published in the journal Symmetry, Integrability and Geometry: Methods and Applications (SIGMA).

In this paper we define weighed Lie groupoids as Lie groupoids with a compatible action of the multiplicative monoid of reals. Such actions are known as homogeneity structures [1]. Compatibility means that the action for any ‘time’ acts as a morphism of Lie groupoids. These actions encode a non-negative integer grading on the Lie groupoid compatible with the groupoid structure, and so we have a kind of ‘graded Lie groupoid’. Importantly, weighted Lie groupoids form a nice generalisation of VB-groupoids (VB = Vector Bundle), which can be defined as a Lie groupoids with regular homogeneity structures [2].

Based on our earlier work [3], in which we similarly define weighed Lie algebroids, we present the basics of weighted Lie theory. In particular we show that weighted Lie algebroids and weighted Lie groupoids are related by more-or-less standard Lie theory: we just need to use Lie II to integrate the action of the homogeneity structure on the weighted Lie algebroid.

The main point here is that we not only naturally generalise ‘VB-objects’, we simplify working with them. In particular, VB-objects require that the homogeneity structure be regular as this encodes a vector bundle structure [4]. The nice, but somewhat technical results of Bursztyn, Cabrera and del Hoyo [2] rely on showing that regularity of the homogeneity structure is preserved under ‘differentiation’ and ‘integration’. That is, when you pass from a groupoid to an algebroid and vice versa. Differentiation is no problem here, but integration is a much tougher question.

However, if we now consider VB-objects as sitting inside the larger category of weighted-objects then we can forget about the preservation of regularity during integration and simply check after that regularity is preserved. Bursztyn et al forced themselves to work in a smaller and not so nice category. We showed that working in this larger category of weighted-objects can simplify working with VB-objects.

Along side this, we show that there are plenty of nice and natural examples of weighted Lie groupoids. For example, the higher order tangent bundle of a Lie groupoid is a weighted Lie groupoid. This and other examples convince us that weighted Lie groupoids are important objects and that there is plenty of work to do.

References
[1] Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.

[2] Bursztyn H., Cabrera A., del Hoyo M., Vector bundles over Lie groupoids and algebroids, arXiv:1410.5135.

[3] Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439.

[4] Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.

Contribution to the conference proceedings "Geometry of Jets and Fields"

The contribution to the conference proceedings “Geometry of Jets and Fields” (Bedlewo, 10-16 May, 2015) as delivered by J. Grabowski is now on the arXiv.

The title is ‘New developments in geometric mechanics’. As well as myself the authors are K. Grabowska, J. Grabowski and P. Urbanski. We present a 16 page overview of our collective recent work in geometric mechanics. A little more specifically the main theme of the contribution is our application of graded bundles to geometric mechanics in the spirit of Tulczyjew.

For more details, consult the arXiv version and the original literature cited therein.

My work on sigma models with Lie algebroid targets gets cited!

I am always very happy when my work gets cited. I think I work in an area that is very specialised and slow to pick up citations. This is not great when starting out.

However, I am very pleased that a Japanese group, Tsuguhiko Asakawa, Hisayoshi Muraki and Satoshi Watamura [2] found my work interesting and cited my work on Lie algebroid sigma models [1].

I placed my preprint on the arXiv on June 25th and the first version of their preprint was placed on the arXiv on Aug 24th. This is a record for me (excluding self-citations that nobody counts).

I don’t always check my citation very regularly and the automatic notifications are not always very reliable. Anyway…

The Japanese group constructed a gravity theory on a Poisson manifold equipped with a Riemannian metric. They do this in the context of Poisson generalised geometry and use the Lie algebroid of a Poisson manifold. Fascinating stuff.

References
[1] Andrew James Bruce, Killing sections and sigma models with Lie algebroid targets, arXiv:1506.07738 [math.DG].

[2] Tsuguhiko Asakawa , Hisayoshi Muraki and Satoshi Watamura, Gravity theory on Poisson manifold with R-flux, arXiv:1508.05706 [hep-th].

A first look at Lie theory

A friend of mine made a request…

Any chance you could make an expository post on Lie Theory for those of us who only known some abstract algebra and calculus? The topic seems very inaccessible otherwise, but I hear Lie Groups and Lie Algebras mentioned regularly.

As your friendly neighbourhood mathematician I will try to oblige.

Disclaimer What I do is give an informal overview and not worry too much about details and proper proofs. Proofs you can find in textbooks. Rather I want to present the ideas and sketch some constructions.

I will build this account up over the period of a few weeks.

Rough Plan
The things I would like to cover are the following.

  1. Abstract Lie algebras
  2. Lie groups
  3. The Lie algebra of a Lie group
  4. Lie’s theorems
  5. Some odds and ends (Maybe a few words about Lie groupoids etc)

There maybe some changes here as the work develops.

I will also include some simple exercises for those that are interested. I will post solutions at the end.

Part 0: Introduction
Anybody who reads anything about modern physics will encounter the terms ‘Lie group’ and ‘Lie algebra’. Lie theory is all about the relation between these two structures.

A Lie group is a group that also has a smooth manifold structure, importantly the group operations are compatible with this smooth structure. Groups represent transformations and symmetries of mathematical objects. Lie groups are the mathematical framework for studying continuous symmetries of mathematical objects. Thus, Lie groups are fundamental in geometry and theoretical physics.

Now, every Lie group has associated with it a Lie algebra, whose vector space structure is the tangent space of the Lie group at the identity element. The Lie algebra describes the local structure of the group. Informally one can think of the Lie algebra as describing the elements of the Lie group that are ‘very close to the identity element’.

The theory of Lie groups and Lie algebras was initiated by Sophus Lie, and hence the nomenclature. Lie’s motivation was to extend Galois theory, which proved useful in the study of algebraic equations, to cope with continuous symmetries of differential equations. Lie laid down much of the basic theory of continuous symmetry groups.

The plan is with these notes is to sketch the relation between Lie groups and Lie algebras. I will stick to the finite dimensional case for this first look.

Part I: Abstract Lie algebras
Let us start with a completely algebraic set-up. Informally, a Lie algebra is a vector space with a non-associative product, known as a ‘bracket’ that satisfies some nice properties. We will only consider algebras over the reals or complex here, though everything will generalise to more arbitrary fields (with some minor modifications if necessary).

Definition
A Lie algebra is a vector space \(\mathfrak{g}\) together with a bilinear operation \([\bullet,\bullet]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}\), that satisfies the following conditions

  1. Skewsymmetry
    \([x,y] = -[y,x]\)
  2. Jacobi identity

\([x,[y,z]] + [z,[x,y]] +[y,[z,x]]=0\)

for all \(x,y, z \in \mathfrak{g}\).

Note that Lie algebras are non-associative. Thinking of the bracket as a form of multiplication we see that the Jacobi identity is related to the ‘associator’ which is non-zero in general

\([x,[y,z]] -[[x,y],z]= [x,[y,z]] + [z,[x,y]] = [[z,x],y] \neq 0\).

The Jacobi identity can also be written in ‘Loday form’

\([x,[y,z]] = [[x,y],z] + [y,[x,z]]\),

which means that the operator \(Ad_{x}:= [x, \bullet]\) satisfies the Leibniz rule, the so called adjoint operator is a derivation. Note that this form of the Jacobi identity has this interpretation even if the bracket is not skewsymmetric. In fact such bracket algebras are well studied and are usually called “Loday” or “Leibniz-Loday” algebras.

The dimension of a Lie algebra is defined to be the dimension of the underlying vector space. Elements of a Lie algebra are said to generate that Lie algebra if they form the smallest subalgebra that contains these elements is the Lie algebra itself.

Example Any vector space equipped with a vanishing bracket \([x,y]=0\), is a Lie algebra. We call any Lie algebra with a vanishing bracket an abelian Lie algebra.

Example The (real) vector space of all n×n skew-hermitian matrices together with the standard commutator is Lie algebra. This Lie algebra is denoted \(\mathfrak{u}(n)\).

Example The Heisenberg algebra is the Lie algebra generated by three elements x,y,z and the Lie brackets are defined as
\([x,y] =z\), \([x,z] =0\) and \([y,z] =0\).

Given a set of generators \(\{T_{a}\}\) we can define the Lie algebra in terms of its structure constants. As the Lie bracket of any pair of generators must be a linear combination of the generators we have

\([T_{a}, T_{b}] = C^{c}_{ab}\: T_{c}\),

and so the Lie algebra is determined by the structure constants \(C^{c}_{ab}\).

Exercise How many one dimensional Lie algebras are there up to isomorphisms?

Exercise There are exactly two Lie algebras of dimension two over the real numbers, up to isomorphism. Can you write these down in terms of generators?

Exercise What conditions do the structure constants need to satisfy in order to have a Lie algebra? (Hint: think about the two defining conditions of a Lie algebra)

People study Lie algebras in their own right, but historically they arose from the study of Lie groups. From my own perspective, it is the fact that Lie algebras are ‘infinitesimal Lie groups’ that makes them interesting and useful. In the next section I will move on to groups and in particular Lie groups.

Part II: Lie groups
Before we move on to Lie groups, let us recall the notion of a group. Generically, one thinks of groups as encoding transformations and symmetries of mathematical objects, so they arise all across mathematics.

Definition
A group is a set \(G\) together with a binary operation \(\circ: G \times G \rightarrow G\) that satisfies the following axioms

  1. Associativity
    For every \(a,b,c \in G\) we have \((a\circ b) \circ c = a\circ (b \circ c)\).
  2. Existence of the identity
    There exists an element \(e \in G\) such that \(e\circ a = a \circ e\) for all \(a \in G\).
  3. Existence of inverse elements
    For every \(a \in G\) there exists an element \(b := a^{-1}\) such that \(a\circ b = b \circ a =e\).

It can be shown that the identity element \(e\) is unique. There is only one identity element. Note we have said noting about commutativity. Generally \(a\circ b\) is not the same as \(b\circ a\). Groups for which these two expression are always equal are called abelian groups.

Example The set of integers \(\mathbb{Z}\) together with standard addition form an abelian group. The identity element is zero and the inverse of any element is \(a^{-1} = {-}a \).

Exercise Does the set of real numbers \(\mathbb{R}\) equipped with standard addition form a group? Does the set of real numbers with standard multiplication form a group?

Example A symmetric group a set consists of permutations on the given set; ie. bijective maps from the set to itself. The product is just composition of the permutations as functions. The identity element is just the identity function from the set to itself. The inverse of an element is just the inverse as a function.

Example Probably the simplest non-abelian group is the rotation group \(SO(3)\). This group consists of all rotations about the origin of three-dimensional Euclidean space and the composition is just standard composition of linear maps. Because all linear transformations can be represented by matrices (once a basis has been chosen) the group \(SO(3)\) can be represented by the set of orthogonal 3×3 matrices and standard matrix multiplication. This group is non-abelian as the order of which rotations are composed matters.

Now, Lie groups are both groups and smooth manifolds at the same time. Before we make this statement a bit more precise I should say a few words about manifolds…

For an informal overview of the idea of manifolds you can consult an earlier post I made here. I will assume everyone had read this, or is at least familiar with the basic idea. I will review the minimum needed to define a Lie group.

A manifold is a ‘space’ that is locally similar to \(\mathbb{R}^{n}\) for some n. A smooth manifold is a refinement of that notion to allow us to do calculus. Any manifold can be described by a collection of charts, also known as an atlas.

An atlas on a topological space \(X\) (say) is a collection of pairs \(\{(U_{\alpha},\phi_{\alpha})\} \) called charts, where the \(U_{\alpha}\) are open sets that cover the topological space, such that

\(\phi_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}^{n},\)

is a homomorphism of \(U_{\alpha}\) onto an open subset of \(\mathbb{R}^{n}\). Loosley this means that locally we can ways think about cutting our topological space up into small pieces of the real linear space.

The transition maps are defined as

\(\phi_{\alpha \beta}:= \phi_{\beta} \circ \phi^{-1}_{\alpha}|_{\phi_{\alpha}(U_{\alpha} \cap U_{\beta})}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \rightarrow \phi_{\beta}(U_{\alpha} \cap U_{\beta}).\)

Any topological space with an atlas is a topological manifold. Loosley, the transition maps allow you to sew together the local patches by telling you what happens on the overlap of such patches.

We will be interested smooth manifolds, that is we insist that the transition maps be infinitely differentiable in the standard sense. Because we can describe everything locally on a smooth manifold in terms of smooth transition functions and local patches of \(\mathbb{R}\) we can extend all our knowledge of standard multi-variable calculus to smooth manifolds.

In particular we know what a smooth map between two smooth manifolds is. As topological spaces a map between smooth manifolds is a continuous map. To define it as ‘smooth’ we compose the function with a chart on our source and target manifolds and as we know what smoothness means for map from \(\mathbb{R}^{n}\) to say \(\mathbb{R}^{m}\) we can accordingly define smoothness for maps between smooth manifolds.

Exercise Fill in details for the above paragraph.

We can now state what a Lie group is…

Definition A Lie group \(G\) is a smooth manifold that also carries a group structure whose product and inversion operations are smooth maps.

That is both

\(\mu : G \times G \rightarrow G\)
\((x,y) \mapsto \mu(x,y) = x\cdot y\)

and

\(inv : G \rightarrow G\)
\(x \mapsto x^{-1}\)

are smooth maps.

Examples to follow…

III Meeting on Lie systems

The III meeting on Lie systems is going to be held next week (21.09.2015 – 26.09.2015) here in Warsaw. It should be a great chance to catch up with some friends in the ‘Spanish Group’.

Of course you are all wondering what a Lie system is. Well, basically a Lie system is a systems of first-order ordinary differential equations whose general solution can be written in terms of a finite family of particular solutions and a superposition rule. There is a rich geometric theory here and many motivating examples that arise from physics.

From Poisson Geometry to Quantum Fields on Noncommutative Spaces

I will be attending the autumn school “From Poisson Geometry to Quantum Fields on Noncommutative Spaces” Oct 05–10, in Würzburg, Germany.

There will be a series of lectures:

  • Francesco D’Andrea (University of Naples)
    Topics in Noncommutative Differential Geometry
  • Martin Bordemann (Univ. Haute Alsace, Mulhouse)
    Algebraic Aspects of Deformation Quantization
  • Henrique Bursztyn (IMPA, Rio de Janeiro)
    Poisson Geometry and Beyond
  • Simone Gutt (ULB, Brussels)
    Symmetries in Deformation Quantization
  • Gandalf Lechner (University of Cardiff)
    Strict Deformation Quantization and Noncommutative Quantum Field Theories
  • Eva Miranda (University of Barcelona)
    Poisson Geometry and Normal Forms: A Guided Tour through Examples

It should be very interesting and I hope to learn a lot about subjects that are aligned with my general research area, but alas I have not yet looked into properly.

Also I will be presenting a poster on ‘Graded bundle in the category of Lie groupoids’ which is based on recent work with K. Grabowska and J. Grabowski (arXiv preprint)

The website for the school states that places may still be available.

On a variant of rhodonea curves

Rhodonea curves or rose curves are plots of a polar equation of the form
\(r = \cos(k \theta)\).

If we specialise to equations with

\(k= \frac{n}{d}\)

for n and d integers (>0), then we have plots of the form below. In the table n runs across and d down

Now, just for fun I considered a slight variant of this given by

\(r = \cos( k \theta) – k\)

The plots are as follows

For another variant I considered

\(r = \cos( k \theta) – k^{-1}\)

I am not sure there is anything mathematically deep here, I just like the images and classify this as some basic mathematical art.

On the physics of chocolate

Researchers at Technische Universität München, Germany, have reported that molecular dynamics can be used to gain new insights into the chocolate conching [1].

Chocolate conching is the stage of manufacturing where aromatic sensation, texture and mouthfeel are developed.

This work seems to be the first to attempt to properly understand the role of lecithins in chocolate production.

Physics, helping to build a tasty more palatable world.

Reference
[1] M Kindlein, M Greiner, E Elts and H Briesen, Interactions between phospholipid head groups and a sucrose crystal surface at the cocoa butter interface, 2015 J. Phys. D: Appl. Phys. 48 384002.

Link
Chocolate physics: how modelling could improve mouthfeel, IOP website.

Random thoughts on mathematics, physics and more…