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

And there seems to be a pattern:

(1)$\array{ \mathbf{\text{for this structure 2-group}} & \mathbf{\text{the realization of its nerve}} & \mathbf{\text{is the automorphism group of this algebra}} & \mathbf{\text{which is the typical fiber of the (1-)bundle}} \\ \href{http://golem.ph.utexas.edu/category/2006/08/on_ntransport_part_i.html#c004898}{(U(1)\to 1)} & \href{http://en.wikipedia.org/wiki/Projective_unitary_group}{P U(H)} & \href{http://en.wikipedia.org/wiki/Compact_operator}{K(H)} & \href{http://golem.ph.utexas.edu/category/2006/09/on_ntransport_2vector_transpor.html} {\text{representing a }\;\; (U(1)\to 1)\text{-2-bundle}} \\ \href{http://golem.ph.utexas.edu/string/archives/000547.html}{(\hat \Omega G \to P G)} & \href{http://golem.ph.utexas.edu/string/archives/000571.html}{\mathrm{String}_G} & A_{\hat \Omega G} & \href{http://golem.ph.utexas.edu/string/archives/000712.html}{ \text{representing a}\;\; (\hat \Omega G \to PG) \text{-2-bundle} } \\ \\ }$

Example 1 is this: start with transition data on some space $X$ with respect to the 2-group $G_2$ coming from the crossed module $U(1)\to 1$ (characterizing an abelian gerbe #). It is well known that this is equivalent to a $(P U(H) \simeq K(\mathbb{Z},2))$-bundle on $X$. $P U(H)$ happens to be the automorphism group of the algebra of compact operators on $H$. 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 $(U(1)\to 1)$ and $(\hat \Omega G \to P G)$ 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)$\array{ (U(1)\to 1) &\stackrel{|\cdot|}{\to}& P U(H) &\stackrel{\sim}{\to}& \mathrm{Inn}(A_{\hat \Omega G}) \\ \downarrow && \downarrow && \downarrow \\ (\hat \Omega G \to P G) &\stackrel{|\cdot|}{\to}& \mathrm{String}_G &\stackrel{\sim}{\to}& \mathrm{Aut}(A_{\hat \Omega G}) \\ \downarrow && \downarrow && \downarrow \\ (1 \to G) &\stackrel{|\cdot|}{\to}& G &\stackrel{\sim}{\to}& \mathrm{Out}(A_{\hat \Omega G}) } \,.$

But we also know that

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

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

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