### 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 $A$ a $k$-algebra and $V$ a $k-$vector space ($k$ some field), people call an algebra homomorphism of the form
$\mu : A \to A \otimes_k \mathrm{End}(V)$
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 $\Sigma(\mathrm{Bim}(\mathrm{Vect}_k))$ – the 1-object 3-category obtained from the monoidal 2-category of algebras and bimodules over $k$.

This is described here:

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 $G$-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 $\nu : H \to M_n(H)$ type, that is amplifying monomorphisms $\nu$ of $H$, where $M_n(H)$ is an $n \times n$ matrix with entries in $H$, in order to mimic the endomorphisms $\rho : A \to A$ 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 $H$ 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-$C^*$-Categories with non-simple units

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

for infinite-dimensional algebras $H$ for which $H \otimes End(V) \simeq H$ 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.

## 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.

$\mu:A\rightarrow A\otimes \End(V)$

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

$\mu:End(k)\otimes A\rightarrow A\otimes \End(V)$

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