Quantum Deformations of Quantum Groups

So called trialgebras, i.e. algebraic structures with two associative products and a coassociative coproduct, all joined in a pairwise compatible way, have been suggested in the context of four dimensional topological field theory by Crane and Frenkel. We show how to construct explicit examples of trialgebras by applying once more deformation theory to the algebra of functions on the Manin plane and to some of the classical examples of quantum groups and quantum algebras. For a certain class of trialgebras we show how to formulate the relevant data in terms of a system of coupled matrix equations. For one of the trialgebras which we construct, we will see that it is realized as a symmetry of a two dimensional spin system, generalizing the quantum group symmetry of the XXZ model (with suitable boundary conditions) known from the one dimensional case. The same trialgebra is also realized as a symmetry of a system of infinitely many q-oscillators. Finally, we give a result which shows that with trialgebras the final level of quantum deformations is reached. There are no nontrivial deformations of trialgebras to algebraic structures involving two associative products and two coassociative coproducts in a compatible way.