Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Infinite-Dimensional Representations of 2-Groups
 
John C. Baez University of California, Riverside, Riverside, CA
Aristide Baratin Max Planck Institute for Gravitational Physics, Golm, Germany
Laurent Freidel Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Derek K. Wise University of Erlangen-Nurnberg, Erlagen, Germany
Infinite-Dimensional Representations of 2-Groups
eBook ISBN:  978-0-8218-9116-2
Product Code:  MEMO/219/1032.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Infinite-Dimensional Representations of 2-Groups
Click above image for expanded view
Infinite-Dimensional Representations of 2-Groups
John C. Baez University of California, Riverside, Riverside, CA
Aristide Baratin Max Planck Institute for Gravitational Physics, Golm, Germany
Laurent Freidel Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Derek K. Wise University of Erlangen-Nurnberg, Erlagen, Germany
eBook ISBN:  978-0-8218-9116-2
Product Code:  MEMO/219/1032.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2192012; 120 pp
    MSC: 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 finite-dimensional \(2\)-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional \(2\)-vector spaces called “measurable categories” (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional 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 sub-intertwiners—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 2-Groups
    • 3. Measurable Categories
    • 4. Representations on Measurable Categories
    • 5. Conclusion
    • A. Tools from Measure Theory
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2192012; 120 pp
MSC: 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 finite-dimensional \(2\)-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional \(2\)-vector spaces called “measurable categories” (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional 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 sub-intertwiners—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 2-Groups
  • 3. Measurable Categories
  • 4. Representations on Measurable Categories
  • 5. Conclusion
  • A. Tools from Measure Theory
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.