Nicolás Andruskiewitsch on the classification of Finite Dimensional Hopf Algebras
New Hot Paper Commentary, May 2011
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Article: Article Title: On the classification of finite-dimensional pointed Hopf algebras
Authors: Andruskiewitsch, N;Schneider,
HJ |
Nicolás Andruskiewitsch talks with ScienceWatch.com and answers a few questions about this month's New Hot Paper in the field of Mathematics.
Why do you think your paper is highly
cited?
The paper contains the classification of a family of certain mathematical objects, namely "finite-dimensional pointed Hopf algebras with abelian group with some restrictions on its order." The area of Hopf algebras is relatively young and received a strong impulse with the discovery of quantum groups by Drinfeld and Jimbo. Our result is the first general classification theorem in this area, and I think this is why it was cited profusely.
Does it describe a new discovery, methodology, or
synthesis of knowledge?
I would say it is a synthesis of knowledge. Schneider and I introduced a method to classify pointed Hopf algebras in 1997 and this paper is a culmination of our work. It also confirms the deep connection of Hopf algebras with Lie theory.
Would you summarize the significance of your paper
in layman's terms?
Mathematicians like to classify mathematical objects in a class of interest, that is to give a full list of them. In our work, we give a substantial step towards the classification of finite dimensional Hopf algebras, mathematical objects that earned popularity because of their connections with some questions in physics.
How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?
"Mathematicians like to classify mathematical objects in a class of interest, that is to give a full list of them."
This is a long story. In 1988, Pierre Cartier visited the University of Córdoba, Argentina, and advocated the new subject of quantum groups. I followed his suggestion and spent some time studying them, first in Argentina and then as a postdoc in Paris (1990-1991) and Bonn (1992-1994). By 1992 I realized that an interesting question would be to elucidate the role of quantum groups in the full picture of classification of Hopf algebras—but this last part was just beginning!
At the end of my stay in Germany as a postdoc, in November 1993, I visited Hans-Jürgen Schneider in Munich, where we found out that we had common interests. This was the starting point of a long-time collaboration—and a close friendship. We wrote several papers together; in 1997, we designed a method to classify pointed Hopf algebras and used it to prove several results, including a negative answer to a conjecture by Kaplansky (similar negative answers were obtained independently and simultaneously by other people). The paper was published in 1998 in the Journal of Algebra.
Next we were able to understand how Lie theory enters into the picture, through some invariants attached to pointed Hopf algebras with abelian group; these invariants are matrices with certain properties. Via these invariants, we were able to connect pointed Hopf algebras with the finite quantum groups of Lusztig (derived from those introduced by Drinfeld and Jimbo). The paper with these ideas appeared in the Advances in Mathematics in 2000.
We were able to give several partial classification results in the years 2000-2004 but we lacked the classification of those invariants. In 2004, István Heckeberger from Leipzig (now in Marburg) was able to classify those matrices in the rank 2 case. This was crucial for finishing one of the steps in our method; the techniques to solve the other steps have been more or less developed by us before, so that with extra effort we were able to finish the classification of finite dimensional pointed Hopf algebras with abelian group, provided that the order of this group is not divisible by 2, 3, 5, 7.
Where do you see your research leading in the
future?
I am still interested in the classification of Hopf algebras. Let me comment first that István Heckeberger was able to classify the matrices that are invariants of pointed Hopf algebras without restriction on the rank in 2006. As we see it now, pointed Hopf algebras with abelian group are related either to Lie groups, or else to Lie supergroups (as suggested by Hiroyuki Yamane), and there is a third set of "mysterious," not yet explained examples.
Other important results solving other steps of the method were recently obtained by my student Iván Angiono. As of today, there remains only one last problem to complete the classification of finite dimensional pointed Hopf algebras with abelian group, without restrictions on the order of the group. I hope to contribute to this question.
There has also been intense activity in the classification of pointed Hopf algebras with non-abelian group. Although I see the final result as very far off at this point, there have recently been many partial results that led to very interesting questions on finite groups and representation theory. It is my intention to keep working in this line of research.
Do you foresee any social or political
implications for your research?
This is hard for me to justify, but I feel that doing mathematical research
with international visibility in a country like Argentina would help to
motivate people to do science, then would help to improve the teaching of
mathematics and at the end, would help to improve the quality of living of
our people. This would certainly be a long way, and you may say I am a
dreamer... but I am not the only one.
Nicolás Andruskiewitsch, Doctor in Mathematics
Full Professor, National University of Córdoba
Principal Researcher, CONICET
Córdoba, Argentina
KEYWORDS: FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS, CLASSIFICATION, QUANTUM GROUPS, NICHOLS ALGEBRAS, ROOTS.