## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 1 - Algebra**

Montag, 17. September 2001, 15.00, Hörsaal 21

**Second quantized quantum groups and their applications**
**Karl-Georg Schlesinger**, **Erwin-Schrödinger-Institut, Wien**
(Koautor: Harald Grosse)

The idea of so called trialgebras - algebraic
structures with a coassociative coproduct and two
associative products, all pairwise compatible -
was introduced in 1995 by Crane and Frenkel. We
construct explicit examples of trialgebras by a
procedure which amounts to an anewed quantization
of some of the classical quantum group examples. We
show that one of our examples is realized as a
symmetry of a two dimensional spin system much in
the same way as quantum group symmetries arise for
spin chains. Finally, we show that with second
quantization of quantum groups one has reached the
end of the story: Trialgebras are stable in the
sense that one can not deform them nontrivially once
again to structures involving four levels of
algebraic structure (e.g. two products and two
coproducts) in a compatible way.

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