eBook ISBN:  9780821891162 
Product Code:  MEMO/219/1032.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9780821891162 
Product Code:  MEMO/219/1032.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 219; 2012; 120 ppMSC: Primary 20; Secondary 18; 22
A “\(2\)group” is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, \(2\)groups have representations on “\(2\)vector spaces”, which are categories analogous to vector spaces. Unfortunately, Lie \(2\)groups typically have few representations on the finitedimensional \(2\)vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional \(2\)vector spaces called “measurable categories” (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional representations of certain Lie \(2\)groups. Here they continue this work.
They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and \(2\)intertwiners for any skeletal measurable \(2\)group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and subintertwiners—features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study “irretractable” representations—another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered “separable \(2\)Hilbert spaces”, and compare this idea to a tentative definition of \(2\)Hilbert spaces as representation categories of commutative von Neumann algebras.

Table of Contents

Chapters

1. Introduction

2. Representations of 2Groups

3. Measurable Categories

4. Representations on Measurable Categories

5. Conclusion

A. Tools from Measure Theory


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A “\(2\)group” is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, \(2\)groups have representations on “\(2\)vector spaces”, which are categories analogous to vector spaces. Unfortunately, Lie \(2\)groups typically have few representations on the finitedimensional \(2\)vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional \(2\)vector spaces called “measurable categories” (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional representations of certain Lie \(2\)groups. Here they continue this work.
They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and \(2\)intertwiners for any skeletal measurable \(2\)group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and subintertwiners—features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study “irretractable” representations—another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered “separable \(2\)Hilbert spaces”, and compare this idea to a tentative definition of \(2\)Hilbert spaces as representation categories of commutative von Neumann algebras.

Chapters

1. Introduction

2. Representations of 2Groups

3. Measurable Categories

4. Representations on Measurable Categories

5. Conclusion

A. Tools from Measure Theory