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September 28, 2006

2-Groups and Algebras

Posted by Urs Schreiber

In another thread #, I am talking with Jim Stasheff and David Roberts about the question how to reconstruct a 2-bundle with connection from its local transition data #.

There are 1121\frac{1}{2} examples where I have some idea at least about certain aspects of the answer.

And there seems to be a pattern:

(1)for this structure 2-group the realization of its nerve is the automorphism group of this algebra which is the typical fiber of the (1-)bundle (U(1)1) PU(H) K(H) representing a (U(1)1)-2-bundle (Ω^GPG) String G A Ω^G representing a(Ω^GPG)-2-bundle \array{ \mathbf{\text{for this structure 2-group}} & \mathbf{\text{the realization of its nerve}} & \mathbf{\text{is the automorphism group of this algebra}} & \mathbf{\text{which is the typical fiber of the (1-)bundle}} \\ \href{}{(U(1)\to 1)} & \href{}{P U(H)} & \href{}{K(H)} & \href{} {\text{representing a }\;\; (U(1)\to 1)\text{-2-bundle}} \\ \href{}{(\hat \Omega G \to P G)} & \href{}{\mathrm{String}_G} & A_{\hat \Omega G} & \href{}{ \text{representing a}\;\; (\hat \Omega G \to PG) \text{-2-bundle} } \\ \\ }

Example 1 is this: start with transition data on some space XX with respect to the 2-group G 2G_2 coming from the crossed module U(1)1U(1)\to 1 (characterizing an abelian gerbe #). It is well known that this is equivalent to a (PU(H)K(,2))(P U(H) \simeq K(\mathbb{Z},2))-bundle on XX. PU(H)P U(H) happens to be the automorphism group of the algebra of compact operators on HH. Hence we can find the associated algebra bundle. Regarding each fiber not as a mere algebra, but as the category of modules of that algebra, we do obtain a 2-bundle of sorts. I think one can show that this is the 2-bundle whose local trivializations yields the 3-cocycle we started with #.

Example 2 is the string bundle with string connection by Stolz & Teichner #.

In both cases one can, I think, understand the algebra that the nerve acts on by automorphisms as the 2-vector space on which the 2-group is represented by its canonical 2-representation #.

So, clearly, there is some general mechanism at work which should generalize the above table from (U(1)1)(U(1)\to 1) and (Ω^GPG)(\hat \Omega G \to P G) to any strict 2-group. Which mechanism is that?

Posted at September 28, 2006 9:31 AM UTC

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Re: 2-Groups and Algebras

Urs generates ideas at a remarkable rate - I am still trying to get my head around some of the issues he raised in the “String coffee table” days. Interesting stuff.

Posted by: Bruce Bartlett on September 28, 2006 1:23 PM | Permalink | Reply to this

Re: 2-Groups and Algebras

I wrote:

Which mechanism is that?

Well, we know that

(1)(U(1)1) || PU(H) Inn(A Ω^G) (Ω^GPG) || String G Aut(A Ω^G) (1G) || G Out(A Ω^G). \array{ (U(1)\to 1) &\stackrel{|\cdot|}{\to}& P U(H) &\stackrel{\sim}{\to}& \mathrm{Inn}(A_{\hat \Omega G}) \\ \downarrow && \downarrow && \downarrow \\ (\hat \Omega G \to P G) &\stackrel{|\cdot|}{\to}& \mathrm{String}_G &\stackrel{\sim}{\to}& \mathrm{Aut}(A_{\hat \Omega G}) \\ \downarrow && \downarrow && \downarrow \\ (1 \to G) &\stackrel{|\cdot|}{\to}& G &\stackrel{\sim}{\to}& \mathrm{Out}(A_{\hat \Omega G}) } \,.

But we also know that

(2)(U(1)1) || PU(H) Inn(B(H))/U(1) (U(1)1) || PU(H) Aut(K(H)) (11) || 1 1, \array{ (U(1)\to 1) &\stackrel{|\cdot|}{\to}& P U(H) &\stackrel{\sim}{\to}& \mathrm{Inn}(B(H))/U(1) \\ \downarrow && \downarrow && \downarrow \\ (U(1) \to 1) &\stackrel{|\cdot|}{\to}& P U (H) &\stackrel{\sim}{\to}& \mathrm{Aut}(K(H)) \\ \downarrow && \downarrow && \downarrow \\ (1 \to 1) &\stackrel{|\cdot|}{\to}& 1 &\stackrel{\sim}{\to}& 1 } \,,

using the fact (e.g. p. 11 of Brodzki-Mathai-Rosenberg-Szabo #) that PU(H)P U(H) is isomorphic to the group of automorphisms of compact operators on HH.

Posted by: urs on September 29, 2006 2:34 PM | Permalink | Reply to this
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