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February 5, 2007

Amplimorphisms

Posted by Urs Schreiber

In some parts of the quantum field theory literature, that concerned with algebraic formulations, one finds the concept of an amplimorphism of algebras.

For AA a kk-algebra and VV a kk-vector space (kk some field), people call an algebra homomorphism of the form μ:AA kEnd(V) \mu : A \to A \otimes_k \mathrm{End}(V) an amplimorphism.

Without actually admitting it, people are talking about an entire 2-category of such amplimorphisms. (Update: the 2-category of amplimorphisms is made explicit in section 2.1 of P. Zito’s thesis.)

All this is done – “old school” – in components (as far as I am aware), which makes it a little hard to see what amplimorphisms really are.

After thinking about it for a while, it seems I found that the 2-category of amplimorphisms is a certain natural 2-category of rectangles internal to the 3-category Σ(Bim(Vect k))\Sigma(\mathrm{Bim}(\mathrm{Vect}_k)) – the 1-object 3-category obtained from the monoidal 2-category of algebras and bimodules over kk.

This is described here:

arrow-theory of amplimorphisms

Does anyone here know anything about amplimorphisms? And how best to think about them? What they really are?

The term was apparently first used in

K. Szlachanyi, K. Vecsernyes,
Quantum symmetry and braid group statistics in GG-spin models
Commun. Math. Phys. 156, 127-168 (1993)
(pdf)

To quickly see the definition, go to the beginning of

G. Mack, V. Schomerus
Models of Quantum Space Time: Quantum Field Planes
hep-th/9403170

For lots of further details check out section 3 of

Florian Nill, Kornel Szlachanyi
Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry
hep-th/9509100.

Some hints concerning the use of amplimorphisms in the study of symmetries of (rational conformal) field theories can be found here:

J. Fuchs, A. Ganchev, P. Vecsernyes
On the quantum symmetry of rational field theories
hep-th/9407013


Update: Feb 6, 2007

The point of introducing amplimorphisms originally, was that people were looking for a braided category of representations of some Hopf algebra, which would be equivalent to the category of representations of local observable algebras in 2-dimensional QFT (compare the next entry).

It seems that K. Szlachanyi and K. Vecsernyes somehow just guessed this concept. And it apparently took a while to demonstrate that it does indeed yield the desired equivalence.

In

Peter Vecsernyés
On the Quantum Symmetry of the Chiral Ising Model
hep-th/9306118

we can see the author intoduce the concept of an amplimorphism. As a motivation, he writes (on p. 3)

[in order to get nontrivial braiding] we are to study algebra embeddings of ν:HM n(H)\nu : H \to M_n(H) type, that is amplifying monomorphisms ν\nu of HH, where M n(H)M_n(H) is an n×nn \times n matrix with entries in HH, in order to mimic the endomorphisms ρ:AA\rho : A \to A of the observables.

I cannot yet quite follow this motivation. In which sense does this “mimic endomorphisms”?

Later, on p. 8, we are told that

[…] we will use amplifying monomorphisms, or amplimorphisms, for short, of HH instead of representations. The benefit of this choice stems from the existence of a left inverse of an amplimorphism, which can lead to the notion of conditional expectations, statistics parameter and index.

Finally, the thesis mentioned above is

Pasquale A. Zito
2-C *C^*-Categories with non-simple units

On p. 13, P. Zito makes one important aspect of amplifunctors explicit (p. 13):

for infinite-dimensional algebras HH for which HEnd(V)H H \otimes End(V) \simeq H the notion of amplimorphisms reduces to that of ordinary endomorphisms. Hence we can understand amplimorphisms as a way to obtain certain features of infinite-dimensional algebras already for finite-dimensional algebras.

Posted at February 5, 2007 1:40 PM UTC

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15 Comments & 2 Trackbacks

Re: Amplimorphisms

I’ll admit this is the first time I’ve heard of the concept, but the first thing that leaps out at me is the top line of your diagram.

μ:AAEnd(V)\mu:A\rightarrow A\otimes \End(V)

I think some insight can come from writing this a little differently.

μ:End(k)AAEnd(V)\mu:End(k)\otimes A\rightarrow A\otimes \End(V)

Now AA looks like it’s “intertwining” the two endomorphism algebras, and μ\mu looks like the intertwinor morphism. Whether this is obvious, useless, or interesting I don’t know offhand.

Posted by: John Armstrong on February 5, 2007 2:55 PM | Permalink | Reply to this

Re: Amplimorphisms

I think some insight can come from writing this a little differently. μ:End(k)AAEnd(V) \mu : \mathrm{End}(k) \otimes A \to A \otimes \mathrm{End}(V)

Yes, that’s good. In fact, that’s equivalent to writing Id A μ A End(V) \array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ A\downarrow\;\; &\;\Downarrow \mu& \; \downarrow A \\ \bullet &\stackrel{\mathrm{End}(V)}{\to}& \bullet } in Σ(Bim(Vect k))\Sigma(\mathrm{Bim}(\mathrm{Vect}_k)):

the identity 1-morphism in Σ(Bim(Vect k))\Sigma(\mathrm{Bim}(\mathrm{Vect}_k)) is the kk-algebra such that tensoring any other algebra with it does nothing (up to …). This implies precisely Id kEnd(k). \mathrm{Id}_\bullet \simeq k \simeq \mathrm{End}(k) \,.

So the above diagram really reads End(k) A μ A End(V) . \array{ \bullet &\stackrel{\mathrm{End}(k)}{\to}& \bullet \\ A\downarrow\;\; &\;\Downarrow \mu& \; \downarrow A \\ \bullet &\stackrel{\mathrm{End}(V)}{\to}& \bullet } \,.

That’s a lesson that I have learned a while ago: whenever you see something involving a twist of roughly the sort we are seeing here in the defintion of amplimorphisms, chances are good that you are really dealing with a pseudonatural transformation of some sort.

That’s what these square diagrams suggest: that they are really components of some sort of transformation of some kind of (2 or 3?)-functor taking values in some world of algebras.

This is vaguely interesting, because I already do know # that such transformations do control correlators of CFTs – and these amplimorphisms are supposed to know about symmetries of such CFTs.

But even if there is a relation, I don’t see it yet in detail.

Posted by: urs on February 5, 2007 3:20 PM | Permalink | Reply to this

Re: Amplimorphisms

So to force us to pay attention to the intertwining interpretation, should we work in a double category whose horizontal morphisms are required to be of the form End(V)?

(Actually in that case, it’s cleaner just to say that the horizontal morphisms are vector spaces, while the vertical morphisms are algebras; only when you get to specifying what the squares are do you have to mention the operator End.)

Posted by: Toby Bartels on February 5, 2007 9:48 PM | Permalink | Reply to this

Re: Amplimorphisms

So to force us to pay attention to the intertwining interpretation, should we work in a double category whose horizontal morphisms are required to be of the form End(V)\mathrm{End}(V)?

I am still trying to better understand the role played by these amplimorphisms in some of the QFT literature. A couple of authors claim that they are “a very powerful tool” - so apparently they do capture some important structure that appears in these applications.

So while I don’t know yet, I would not be surprised if it turns out that some aspects of the original definition of amplimorphisms are more restrictive than really necessary.

I mean, the above formulation in terms of squares generalizes the original definition without any further work in several respects:

- algebra morphisms are generalized to bimodules

- there is no real need to restrict attention to horizontal morphisms of the form End(V)\mathrm{End}(V).

- there is no need to require all vertical morphisms to be labeled by the same algebra

- there are actually even higher morphisms naturally in the game, namely bimodule homomorphisms going perpendicular to the drawing plane.

As I said, I don’t know yet if any of these generalizations is actually useful for those applications. But from the point of view of the structure at hand, there seems to be no reason to exclude these.

Posted by: urs on February 5, 2007 10:02 PM | Permalink | Reply to this

Re: Amplimorphisms

So, the moral of the story is that diamonds are beautiful?

Man, watching you guys think through this stuff in the open is better than YouTube anyday :)

Posted by: Eric on February 5, 2007 11:35 PM | Permalink | Reply to this

a good advice

I read on your homepage about what you call “higher dimensional algebra”

I’m afraid the concept of dimension makes no sense at all when considering morphisms of morphisms. Arrows between arrows do in general not correspond to surfaces! Only (maybe) in the special type of category you are considering.

Maybe it is a good idea to go more into this, but depending on the category you are considering, it may happen that for all objects in it the class of morphisms of morphisms is already empty. Or that there are objects with an infinitely non-trivival sequence of morphisms of morphisms of morphisms…

Also: There is a canonical way to assign to an arbitrary category the associated category of the morphisms of its objects, whose morhisms are what you may call 2-morphisms (and so on for n>2). To give it a new name (n-category) does not produce any new insight, so don’t make it look like a new mathematical concept, even if the idea to apply the special case you are thinking of to physics may be new.

Talk to an actual algebraist about your ideas, but to be taken seriously you really should rename your term “higher dimensional algebra”.

Even if in the special category you are considering there is a direct connection between n-dimensional Topology and “n-morphisms”, the category-theoretic construction still has nothing to do with dimension.

The discussions on here sound a lot like esoteric/pseudoscience and I don’t think that is anybodys intent here. Talk to mathematicians!

Posted by: a friend on February 5, 2007 7:42 PM | Permalink | Reply to this

“higher dimensional algebra”

Arrows between arrows do in general not correspond to surfaces!

That’s right!

Just as maps between sets do not in general correspond to curves – and are still usually displayed in a “1-dimensional” notation: f:ST. f : S \to T \,.

Since you post this as a comment to the above entry, you might notice that, incidentally, above we do discuss an example of 2-morphisms that do not correspond to surfaces – but to bimodules.

But what this also shows is: the notation for these 2-morphisms is very naturally one that makes full use of the 2-dimensionality of paper and screen.

You might want to read an account of the history of the term “higher dimensional algebra”. Maybe

Higher Dimensional Algebra and its Applications

by Tim Porter.

Posted by: urs on February 5, 2007 8:02 PM | Permalink | Reply to this
Read the post Amplimorphisms and Quantum Symmetry, I
Weblog: The n-Category Café
Excerpt: Localized endomorphisms of quantum observables in arrow-theoretic terms.
Tracked: February 6, 2007 12:08 PM

Re: Amplimorphisms

Am I supposed to think of an amplimorphism

AAEnd(V)A \to A \otimes End(V)

as a funny sort of morphism from AA to itself, or a funny sort of morphism from VV to itself?

Your definition of how they compose suggests it’s more like the former.

However, if so, it might clarify the situation if we consider more general amplimorphisms that go from an algebra AA to some other algebra BB:

ABEnd(V)A \to B \otimes End(V)

As far as I can tell, these compose just as well. I now see you already agree:

there is no need to require all vertical morphisms to be labeled by the same algebra

Anyway, I don’t feel I understand this stuff at all, mainly because I don’t understand the application to physics. But, I suspect that some nice algebraic idea is being developed in a somewhat ‘squashed’ way, and mathematicians will need to unfold it a bit to see its inner beauty. The above is my feeble attempt to help you do this…

Posted by: John Baez on February 8, 2007 4:08 AM | Permalink | Reply to this

Re: Amplimorphisms

But, I suspect that some nice algebraic idea is being developed in a somewhat ‘squashed’ way, and mathematicians will need to unfold it a bit to see its inner beauty.

Yes, exactly. My diagrams above were supposed to be a first step in that direction. Compare with the formulation of the definition in the literature to see the difference…

Am I supposed to think of an amplimorphism AAEnd(V) A \to A \otimes \mathrm{End}(V) as a funny sort of morphism from AA to itself

Yes. That seems to be what it is used for in applications.

I have added an update to the end of the above entry two days ago which adresses this vaguely:

The point is apparently that for many examples of infinite-dimensional algebras tensoring with a matrix algebra does not change anything AEnd(V)A. A \otimes \mathrm{End}(V) \simeq A \,. Then in this case the concept of an amplimorphism reduces to that of an ordinary endomorphism AAA \to A.

In some parts of the literature one finds hence comments along the lines that amplimorphisms are a way to do with finite dimensional algebras what is otherwise only possible with infinite dimensional algebras.

So people working with these seem to have an intuition for why it must be precisely this way. But it remains a little obscure to the outsider.

The only solid fact that I could extract is this:

it is possible to put a braiding on the category of amplimorphisms by using some braiding on the underlying algebras. This way amplimorphisms for an algebra AA form a braided monoidal category.

Fact: if we choose AA to be a certain Hopf algebra, then this braided monoidal category (which we should think of as a category of reps of that Hopf algebra) is equivalent to the braided monoidal category of local observables in 2-dimensional AQFT.

That was the point of introducing amplimorphisms: people were looking for an analog of the Doplicher-Robberts statement for the case that the category in question has nontrivial braiding.

Still, when I look at the original literature, the definition of amplimorphisms still seems to come out of the blue sky, justifying itself only by this end result.

I am hoping that there is also a nicer explanation of the raison d’etre of amplimorphisms.

Posted by: urs on February 8, 2007 10:24 AM | Permalink | Reply to this
Read the post Some Notes on Local QFT
Weblog: The n-Category Café
Excerpt: Some aspects of the AQFT description of 2d CFT.
Tracked: April 1, 2007 5:33 AM

Re: Amplimorphisms

The amplimorphisms that we discussed above are supposed to be the substitute for ordinary endomorphisms of observable algebras in the case that we have a finite physical model.

More in detail:

given a local net of observable algebras A:O(X)Algebras \mathbf{A} : O(X) \to Algebras such that each algebra is infinite-dimensional, the (localized, but never mind for the moment) endomorphisms on the inductive limit algebra A tot=colim O(X)A A_{tot} = colim_{O(X)} \mathbf{A} play an important role.

If, however, the local net is locally finite in that it factors through FiniteAlgebrasAlgebras FiniteAlgebras \hookrightarrow Algebras one wants to use not endomorphisms A totA totA_{tot} \to A_{tot} of A totA_{tot}, but amplimorphims A totA totEnd(V) A_{tot} \to A_{tot} \otimes End(V) for VV some finite dimensional vector space.

The point to notice is that for infinite dimensional algebras amplimorphisms coincide with ordinary endomorphisms, so amplimorphisms are a hallmark of finite models.

Now, I am trying to learn more about the existing work on constructions of AQFTs from continuum limits of lattice models.

In this context one comes across

Florian Nill, Kornel Szlachanyi, Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry.

On p.4 of the introduction they review the following idea, discussed in detail later in the article:

given a locally finite algebra A totA_{tot}, i.e. one that arises as the inductive limit over a net of finite algebras, one can ask for an embedding

A totF A_{tot} \hookrightarrow F

into a larger algebra, such that the larger algebra regards amplimorphisms as ordinary endomorphisms in that for each amplimorphism μ:A totA totEnd(V) \mu : A_{tot } \to A_{tot} \otimes End(V) there is a unitary finFEnd(V)f in F \otimes End(V) such that

μ(a)=Ad f(aId V). \mu(a) = Ad_f (a \otimes Id_V) \,.

I suppose, therefore, that passing from A totA_{tot} to FF is like taking the continuum limit of the original finite model.

Is that right?

I’d be grateful for whatever comment on this you might have.

Posted by: Urs Schreiber on June 18, 2008 3:19 PM | Permalink | Reply to this

Re: Amplimorphisms

Life has returned to a state of being merely “extremely hectic” as opposed to “ludicrously unbearably insanely hectic” so I’m hoping to take advantage of the relative calmness to learn something about “arrow theory”. In particular, I really hope to understand

An Exercise in Groupoidification: The Path Integral

If it is not too much trouble, is there any chance of getting a copy of arrow-theory of amplimorphisms in .pdf format? I’m currently in an usual situation where I do not have admin privileges on either my home or work computers (don’t ask!) and cannot install anything that will read .ps or convert .ps to .pdf. It’s not crucial, but thought I would ask in case it was easy.

I’ll use the search feature (well, I will use Google since the search feature does not work for me), but any pointers on how to get started with “arrow theory” would be much appreciated.

Cheers!

Posted by: Eric on June 18, 2008 5:05 PM | Permalink | Reply to this

Re: Amplimorphisms

Sounds like you need ps2pdf.com!

Posted by: Jamie Vicary on June 18, 2008 5:43 PM | Permalink | Reply to this

Re: Amplimorphisms

Hah! Thank you Jamie :)

Posted by: Eric on June 18, 2008 6:55 PM | Permalink | Reply to this

Re: Amplimorphisms

That ps-file does not contain much beyond the above figure and a little introduction for how to read it.

What is worse, for me, is that this insight that amplimorphisms have an interpretation in terms of components of lax-natural transformations hasn’t really, so far, helped me to see what they really are. But I didn’t look into this question for a while and am only now coming back to it.

Concerning the geometric representation theory picture (aka groupoidification) of the path integral obtained from “taking sections”: that’s something I want to get back to as soon things on my side here have become less hectic. I am thinking this might happen in about two weeks or so.

What generally might help more than converting old ps files to new pdf files is that you post concrete questions, if any, on what I wrote. Then I can try to reply and we can try to walk through what I said in small steps. Or else, walk through your ideas in small steps. Or through anyone else’s ideas we find worth looking at, for that matter. :-)

Posted by: Urs Schreiber on June 18, 2008 5:57 PM | Permalink | Reply to this

Re: Amplimorphisms

I wasn’t brave enough to ask, but if you are willing, I think it is a great idea :) It might even help you to explain things to someone not in the middle of it. I understand that you are busy so no need to explain delays in responses. That is what RSS and asynchronous comments are for :)

Cheers!

Posted by: Eric on June 18, 2008 6:59 PM | Permalink | Reply to this

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