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March 16, 2006

Remarks on 2-Reps

Posted by Urs Schreiber

Here is a first refinement of some ideas related to the representation theory of the String(n)\mathrm{String}(n)-2-group which I mentioned recently (\to).

Actually, I won’t talk about String(n)\mathrm{String}(n) here at all, but instead try to make some general facts about 2-reps explicit. I don’t claim that the following is particularly deep. In fact, the considerations are quite elementary. The motivation for spelling this out is that by slightly enriching the following discussion (in particular by replacing vector spaces by Hilbert spaces) this should describe 2-reps of the String(n)\mathrm{String}(n)-2-group in terms of bimodules of vonNeumann algebras.

Claim: Every faithful representation

(1)ρ:HVect \rho : H \to \mathrm{Vect}

of any group HH on a vector space VV (over some field) gives rise in a canonical way to a 2-representation

(2)Aut(H) VectMod \mathrm{Aut}(H) \to {}_{\mathrm{Vect}}\mathrm{Mod}

of the automorphism 2-group of HH on a 2-vector space (a module category over Vect\mathrm{Vect}).

This representation factors through the 2-category of bimodules.

The simple idea is this. Last time (\to) I tried to indicate how every faithful representation ρ\rho of HH on the vector space VV gives rise to a 2-representation of Aut(H)\mathrm{Aut}(H) on Im(ρ)\mathrm{Im}(\rho) by

- sending an object gg of Aut(H)\mathrm{Aut}(H) (an automorphism of HH) to the autofunctor

(3)Im(ρ) Im(ρ) V V ρ(f) ρ(g(f)) V V \array{ \mathrm{Im}(\rho) &\to& \mathrm{Im}(\rho) \\ \\ V && V \\ \rho(f) \downarrow\;\;\; &\mapsto& \;\;\;\;\downarrow\rho(g(f)) \\ V && V }

- sending a morphism ghgg \overset{h}{\to} g' of Aut(H)\mathrm{Aut}(H) to the natural isomorphism given by these naturality squares:

(4)V ρ(h) V ρ(g(f)) ρ(g(f)) V ρ(h) V. \array{ V &\overset{\rho(h)}{\to}& V \\ \rho(g(f)) \downarrow\;\; && \;\;\; \downarrow \rho(g'(f)) \\ V &\overset{\rho(h)}{\to}& V } \,.

But we may want a representation on a proper 2-vector space. By this I shall mean, somewhat loosely, any module category over some monoidal category CC.

Here we can choose C=VectC = \mathrm{Vect}. There is a close relation between the 2-category of module categories and the 2-category of bimodules of algebras in Vect\mathrm{Vect} (\to). So let’s first construct a representation in terms of bimodules from the above one.

This is obvious. We let A ρEnd(V)A_\rho \subset \mathrm{End}(V) be the algebra generated from the operators in Im(ρ)\mathrm{Im}(\rho) – nothing but the category algebra of Im(ρ)\mathrm{Im}(\rho)

(5)A ρ:=ρ(h)|hH A_\rho := \langle \rho(h) | h \in H \rangle

(In the more general case where we have representations on infinite dimensional Hilbert spaces we’d take the double commutant of Im(ρ)\mathrm{Im}(\rho) and obtain a vonNeumann algebra A ρA_\rho.)

The above autofunctors sending ρ(h)\rho(h) to ρ(g(h))\rho(g(h)) extend to automorphisms

(6)A ρoversetϕ gA ρ A_\rho overset{\phi_g}{\to} A_\rho

of this algebra. By a standard construction, from every morphism AϕAA \overset{\phi}{\to} A' of algebras we obtain an AA-AA' bimodule

(7) AN A=(A,ϕ) {}_A N_{A'} = (A',\phi)

which, as a vector space, is simply AA' itself, where the right action by AA' is simply the product in AA' and where the left action by AA is obtained by first sending AA to AA' using ϕ\phi and then acting from the left on AA' by multiplication.

Hence for any two objects gg and gg' of Aut(H)\mathrm{Aut}(H) we obtain two A ρA_\rho-A ρA_\rho bimodules N g=(A ρ,ϕ g)N_g = (A_\rho,\phi_g) and N g=(A ρ,ϕ g)N_{g'} = (A_\rho,\phi_{g'}).

Now, one can easily check that for every morphism ghgg \overset{h}{\to} g' in Aut(H)\mathrm{Aut}(H) we get a homomorphism of A ρA_\rho-A ρA_\rho-bimodules

(8)ϕ h:N g N g A ρa ρ(h)a \array{ \phi_h \;:\; N_g &\to & N_{g'} \\ A_\rho \ni a &\mapsto& \rho(h)\cdot a }

by multiplying from the left with ρ(h)\rho(h). The condition for this to preserve the left A ρA_\rho action is precisely the commutativity of the above naturality squares. (The right action is preserved trivially.)

One also checks that horizontal and vertical composition of bimodule homomorhisms N gϕ hN gN_g \overset{\phi_h}{\to} N_{g'} reproduces the horizontal and vertical composition in Aut(H)\mathrm{Aut}(H) and Aut(Im(ρ)))\mathrm{Aut}(\mathrm{Im}(\rho))).

Finally, we can regard homomorphisms of bimodules as 2-morphisms in VectMod{\mathrm{Vect}}\mathrm{Mod} by a standard construction (\to).

Spelled out, the 2-representation

(9)Aut(H) VectMod \mathrm{Aut}(H) \to {}_\mathrm{Vect}\mathrm{Mod}

obtained this way from ρ:HVect\rho : H \to \mathrm{Vect} works as follows.

The 2-group is represented on the 2-vector C=Mod A ρC = \mathrm{Mod}_{A_\rho}, which is the category of right modules over the algebra A ρ=ρ(h)|hHA_\rho = \langle \rho(h) \;|\; h \in H\rangle.

Every object gObj(Aut(H))g \in \mathrm{Obj}(\mathrm{Aut}(H)) is sent to the Vect\mathrm{Vect}-linear functor

(10)Mod A ρ Mod A ρ M M AN g. \array{ \mathrm{Mod}_{A_\rho} &\to& \mathrm{Mod}_{A_\rho} \\ M &\mapsto& M \otimes_A N_g } \,.

Every morphism ghgg \overset{h}{\to} g' \in \mathrm{Mor}(\mathrm{Aut}(H)) is sent to the obvious natural isomorphism of these functors. (See the diagram at the end of these notes.)

Posted at March 16, 2006 2:49 PM UTC

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