From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: This week in the mathematics archive (10 Jan - 14 Jan)
Date: 20 Jan 2000 15:21:02 -0800
Newsgroups: sci.math.research
Summary: [missing]

[deletia --djr]

After a hiatus, I have more follow-ups to report than usual in this first
non-empty discussion in the new millenium.  First, Varshavsky's claimed proof
of the Jacobian conjecture [math.AG/9912196] was retracted almost immediately
after it was posted.  Nonetheless, as with all retractions for scholarly
reasons, the first version remains available at the parent arXiv site:

    http://arxiv.org/abs/math.AG/9912196v1

I have to admit that I was not very surprised.  This was not because I
have any doubts about Varshavsky as a mathematician or because I know
anything about the Jacobian conjecture or his argument; on the contrary I
don't know anything about the question and I have heard on good authority
that he is very good.  However hard problems have a way of snatching
defeat from the jaws of victory.  Varshavsky's introduction suggests
that Milman and Spivakovsky listened to his arguments before he archived
them, but apparently it was not enough. I can only applaud Varshavsky for
permanently recording his claimed proof and for retracting it promptly.
At the very least, just because there is a mistake, that doesn't mean
that there is nothing to learn from the article.

Moreover there is always the chance that he will be able to un-retract it.
There is the famous example of an article by Dmitry Tamarkin, "Another
proof of M. Kontsevich formality theorem" [math.QA/9803025], which
was first submitted on March 8, 1998, then revied on March 10, then
retracted on March 13, and finally replaced with the current version
(version 4) on September 24, a version which is now widely accepted as
correct. Indeed Kontsevich himself [math.QA/9904055] is apparently very
impressed by Tamarkin's work.

After Thanksgiving I mentioned that a basic understanding of the heat
equation is relevant to cooking.  In particular its invariance under
rescaling tells you that heat penetration time grows quadratically with
thickness. There was an interesting article in the November issue of
Physics Today (http://www.aip.org/pt/nov99/current.htm) that took this
topic much further.  The authors modelled heat penetration in cooking
meat with a PDE simulation; it mentioned that the quadratic rule is
called Fourier's Law. It had some specific recommendations for how to
slice meat for frying (thinly) and how often to flip it (frequently). I
am also reminded that in French an oven is called a "four".

"Biperfect Hopf algebras", by Etingof, Gelaki, Guralnick, and Saxl
[math.QA/9912068] has an interesting counterexample in the theory of
finite-dimensional Hopf algebras. Etingof and Gelaki have had a number of
archive articles in this area over the past two years [math.QA/9801129,
math.QA/9805106, math.QA/9812118, math.QA/9812151, math.QA/9905168].  A
finite-dimensional Hopf algebra is a finite-dimensional vector space V such
that both V and V^* have the structure of an associative algebra, with natural
compatibility relations.  If one of the two structures is commutative, it must
then be the group algebra of a finite group G, with the commutative side
corresponding to pointwise multiplictation of functions on G.  So you can
think of a general finite-dimensional Hopf algebra as a "doubly
non-commutative" generalized finite group.  The merit of the articles by
Etingof and Gelaki (and also some that are solely authored by one of the two)
is that they have tried valiantly to extend the theory of finite groups to
finite-dimensional Hopf algebras.  For example they generalized the theorem
that the dimension of an irreducible complex representation of a group divides
its order.  

Observe that a group is perfect if the trivial representation is the only
1-dimensional one; stated this way the definition makes sense for any
semisimple algebra.  Here the authors a find a Hopf algebra which is doubly
perfect, or biperfect, meaning that both algebra structures are semisimple and
perfect.  The example is a semi-dualized form of the Mathieu group M_24.  In
one sense, then, this Hopf algebra is not far from a finite group.  But in
another sense, it is as far as away as can be.  A one-dimensional
corepresentation of a group algebra k[G] is given by a group element; such a
corepresentation of an arbitrary Hopf algebra is called a group-like element. 
Thus, neither the semi-dualized M_24 Hopf algebra nor its dual has any
group-like elements other than the identity. This entirely defeats one
possible approach to a structure theory  for finite-dimensional Hopf algebras;
it is why the construction is an interesting counterexample.

"This week in the mathematics archive" may be freely redistributed
with attribution and without modification.

[deletia -- djr]

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