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Inﬁnite-Dimensional Representations of 2-Groups
John C. Baez
1
, Aristide Baratin
2
, Laurent Freidel
3
,
4
, Derek K. Wise
5
1
Department of Mathematics, University of CaliforniaRiverside, CA 92521, USA
2
Max Planck Institute for Gravitational Physics, Albert Einstein Institute,Am M¨uhlenberg 1, 14467 Golm, Germany
3
Laboratoire de Physique, ´Ecole Normale Sup´erieure de Lyon46 All´ee d’Italie, 69364 Lyon Cedex 07, France
4
Perimeter Institute for Theoretical PhysicsWaterloo ON, N2L 2Y5, Canada
5
Institute for Theoretical Physics III, University of Erlangen–N¨urnbergStaudtstraße 7 / B2, 91058 Erlangen, Germany
Abstract
A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of agroup. Just as groups have representations on vector spaces, 2-groups have representationson ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the ﬁnite-dimensional 2-vector spaces introducedby Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certaininﬁnite-dimensional 2-vector spaces called ‘measurable categories’ (since they are closely relatedto measurable ﬁelds of Hilbert spaces), and used these to study inﬁnite-dimensional represen-tations of certain Lie 2-groups. Here we continue this work. We begin with a detailed studyof measurable categories. Then we give a geometrical description of the measurable represen-tations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensorproducts and direct sums for representations, and various concepts of subrepresentation. Wedescribe direct sums of intertwiners, and sub-intertwiners—features not seen in ordinary grouprepresentation theory. We study irreducible and indecomposable representations and intertwin-ers. We also study ‘irretractable’ representations—another feature not seen in ordinary grouprepresentation theory. Finally, we argue that measurable categories equipped with some extrastructure deserve to be considered ‘separable 2-Hilbert spaces’, and compare this idea to a ten-tative deﬁnition of 2-Hilbert spaces as representation categories of commutative von Neumannalgebras.
1
a r X i v : 0 8 1 2 . 4 9 6 9 v 2 [ m a t h . Q A ] 9 F e b 2 0 1 1
Contents
1 Introduction 3
1.1 2-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 2-Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Representations of 2-groups 16
2.1 From groups to 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.1 2-groups as 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 From group representations to 2-group representations . . . . . . . . . . . . . . . . . 202.2.1 Representing groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Representing 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 The 2-category of representations . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Measurable categories 28
3.1 From vector spaces to 2-vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Categorical perspective on 2-vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 303.3 From 2-vector spaces to measurable categories . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Measurable ﬁelds and direct integrals . . . . . . . . . . . . . . . . . . . . . . 343.3.2 The 2-category of measurable categories:
Meas
. . . . . . . . . . . . . . . . . 393.3.3 Construction of
Meas
as a 2-category . . . . . . . . . . . . . . . . . . . . . . 53
4 Representations on measurable categories 54
4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Invertible morphisms and 2-morphisms in
Meas
. . . . . . . . . . . . . . . . . . . . 584.3 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Structure of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Structure of intertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.3 Structure of 2-intertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Equivalence of representations and of intertwiners . . . . . . . . . . . . . . . . . . . . 774.5 Operations on representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5.1 Direct sums and tensor products in
Meas
. . . . . . . . . . . . . . . . . . . . 804.5.2 Direct sums and tensor products in
2Rep
(
G
) . . . . . . . . . . . . . . . . . . 854.6 Reduction, retraction, and decomposition . . . . . . . . . . . . . . . . . . . . . . . . 874.6.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.6.2 Intertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Conclusion 97A Tools from measure theory 99
A.1 Lebesgue decomposition and Radon-Nikodym derivatives . . . . . . . . . . . . . . . 100A.2 Geometric mean measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.3 Measurable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.4 Measurable
G
-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
1 Introduction
The goal of ‘categoriﬁcation’ is to develop a richer version of existing mathematics by replacing setswith categories. This lets us exploit the following analogy:
set theory category theoryelements objectsequations isomorphismsbetween elements between objectssets categoriesfunctions functorsequations natural isomorphismsbetween functions between functors
Just as sets have elements, categories have objects. Just as there are functions between sets, thereare functors between categories. The correct analogue of an equation between elements is not anequation between objects, but an isomorphism. More generally, the analog of an equation betweenfunctions is a natural isomorphism between functors.The word ‘categoriﬁcation’ was ﬁrst coined by Louis Crane [23] in the context of mathematical
physics. Applications to this subject have always been among the most exciting [9], since categori-
ﬁcation holds the promise of generalizing some of the special features of low-dimensional physics tohigher dimensions. The reason is that categoriﬁcation
boosts the dimension by one
.To see this in the simplest possible way, note that we can draw sets as 0-dimensional dots andfunctions between sets as 1-dimensional arrows:
S
ã
f
ã
S
If we could draw all the sets in the world this way, and all the functions between them, we wouldhave a picture of the category of all sets.But there are many categories beside the category of sets, and when we study categories
en masse
we see an additional layer of structure. We can draw categories as dots, and functors betweencategories as arrows. But what about natural isomorphisms between functors, or more generalnatural transformations between functors? We can draw these as 2-dimensional surfaces:
C
ã
f
f
ã
C
h
So, the dimension of our picture has been boosted by one! Instead of merely a category of allcategories, we say we have a ‘2-category’. If we could draw all the categories in the world this way,and all functors between them, and all natural transformations between those, we would have apicture of the 2-category of all categories.This story continues indeﬁnitely to higher and higher dimensions: categoriﬁcation is a processthan can be iterated. But our goal here lies in a diﬀerent direction: we wish to take a speciﬁcbranch of mathematics, the theory of inﬁnite-dimensional group representations, and categorify3
that just once. This might seem like a purely formal exercise, but we shall see otherwise. In fact,the resulting theory has fascinating relations both to well-known topics within mathematics (ﬁeldsof Hilbert spaces and Mackey’s theory of induced group representations) and to interesting ideas inphysics (spin foam models of quantum gravity, most notably the Crane–Sheppeard model).
1.1 2-Groups
To categorify group represenation theory, we must ﬁrst choose a way to categorify the basic notionsinvolved: the notions of ‘group’ and ‘vector space’. At present, categorifying mathematical deﬁni-tions is not a completely straightforward exercise: it requires a bit of creativity and good taste. So,there is work to be done here.By now, however, there is a fairly uncontroversial way to categorify the concept of ‘group’. Theresulting notion of ‘2-group’ can be deﬁned in various equivalent ways [8]. For example, we can think
of a 2-group as a category equipped with a multiplication satisfying the usual axioms for a group.Since categoriﬁcation involves replacing equations by natural isomorphisms, we should demand thatthe group axioms hold
up to natural isomorphism
. Then we should demand that these isomorphismsobey some laws of their own, called ‘coherence laws’. This is where the creativity comes into play.Luckily, everyone agrees on the correct coherence laws for 2-groups.However, to simplify our task in this paper, we shall only consider ‘strict’ 2-groups, where theaxioms for a group hold as
equations
—not just up to natural isomorphisms. This lets us ignore theissue of coherence laws. Another advantage of strict 2-groups is that they are essentially the sameas ‘crossed modules’ [34], which are structures already familiar in algebra. So, henceforth we shall
always use the term ‘2-group’ to mean a 2-group of this kind.Suppose
G
is a 2-group of this kind. Since
G
is a category, it has objects and morphisms. Theobjects form a group under multiplication, so we can use them to describe symmetries. The newfeature, where we go beyond traditional group theory, is the morphisms. For most of our moresubstantial results, we shall make a drastic simplifying assumption: we shall assume
G
is not onlystrict but also ‘skeletal’. This means that there only exists a morphism from one object of
G
toanother if these objects are actually equal. In other words, all the morphisms between object of
G
are actually automorphisms. Since the objects of
G
describe symmetries, their automorphismsdescribe
symmetries of symmetries
.The reader should not be fooled by the somewhat intimidating language. A skeletal 2-group isreally a very simple thing. Using the theory of crossed modules, explained in Section 2.1.2, we shall
see that a skeletal 2-group
G
consists of:
ã
a group
G
(the group of objects of
G
),
ã
an abelian group
H
(the group of automorphisms of any object),
ã
a left action
of
G
as automorphisms of
H
.A nice example is the ‘Poincar´e 2-group’, ﬁrst discovered by one of the authors [4]. But to
understand this, and to prepare ourselves for the discussion of physics applications later in thisintroduction, let us ﬁrst recall the ordinary Poincar´e group.In special relativity, we think of a point
x
= (
t,x,y,z
) in
R
4
as describing the time and locationof an event. We equip
R
4
with a bilinear form, the so-called ‘Minkowski metric’:
x
·
x
=
tt
−
xx
−
yy
−
zz
4

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