ALGEBRAIC CLASSIFICATION OF ACTIONS INVARIANT UNDER GENERALIZED FLIP MOVES OF 2-DIMENSIONAL GRAPHS

Statistical models defined on 2-dimensional graphs are classified whic h are invariant under flip moves, i.e., certain local changes of the a djacency structure of the graphs. The special case of regular graphs o f degree 3-which are duals of 2-dimensional triangulations-corresponds to topological models and the classification leads to metrized, assoc iative algebras. As a novel feature flip invariant models on regular g raphs of degree 4 are classified by Z(2)-graded metrized associative a lgebras. They give rise to invariants for checkered graphs. Moreover, the general case of graphs with vertices of arbitrary degree (where de gree 3 does occur) is discussed. Using structure theorems for (graded, ) metrized, associative algebras we prove that only the simple ideals contribute to the partition function of such models. The partition fun ctions art computed explicitly and reveal the invariant structures of the graph under the flip moves.