Download E-books Infinite-Dimensional Representations of 2-Groups (Memoirs of the American Mathematical Society) PDF
By John C. Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise
A "$2$-group" is a class outfitted with a multiplication enjoyable legislation like these of a gaggle. simply as teams have representations on vector areas, $2$-groups have representations on "$2$-vector spaces", that are different types analogous to vector areas. regrettably, Lie $2$-groups in most cases have few representations at the finite-dimensional $2$-vector areas brought by means of Kapranov and Voevodsky. accordingly, Crane, Sheppeard and Yetter brought definite infinite-dimensional $2$-vector areas referred to as "measurable different types" (since they're heavily regarding measurable fields of Hilbert spaces), and used those to check infinite-dimensional representations of convinced Lie $2$-groups. right here they proceed this paintings. they start with a close learn of measurable different types. Then they provide a geometric description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They examine tensor items and direct sums for representations, and numerous strategies of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features now not obvious in usual team illustration idea and learn irreducible and indecomposable representations and intertwiners. in addition they examine "irretractable" representations--another function now not noticeable in usual staff illustration conception. ultimately, they argue that measurable different types built with a few additional constitution should be thought of "separable $2$-Hilbert spaces", and examine this concept to a tentative definition of $2$-Hilbert areas as illustration different types of commutative von Neumann algebras.
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