Gauge transformations and quasitriangularity

Natural conditions on a Poisson/quantum group G to implement Poisson/quantum gauge transformations on the lattice are investigated. In addition to our previous result that transformations on one lattice link require G to be coboundary, it is shown that for a sequence of links one needs a quasitriangular G. 0 Introduction: one link results Is it possible to formulate lattice (group valued) gauge fields with a Poisson (or quantum) group as the structure group, in such a way that the gauge transformations are Poisson or quantum actions? Which conditions should such a group satisfy? In [1] we have answered the last question in case of gauge transformations performed on one link only. We recall these results. 1. Let (G, π) be a Poisson Lie group. There exists a Poisson structure σ on G such that the map G×G×G ∋ (x, y, z) 7→ xyz ∈ G (1) is Poisson as a map from (G, π)×(G, σ)×(G, π) to (G, σ) if and only if (G, π) is coboundary. In this case σ is just the ‘plus’ structure σ(g) = π+(g) := rg + gr, (2)