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.