A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right-and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of 'extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a 'dual connection' which exists on the dual bimodule (as defined in this work).

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a 'reduction' of the 'universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a 'Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an 'internal' discrete space (à la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a 'symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a 'noncommutative differential calculus' on manifolds.

Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order". We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N -simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.

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A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsionfree connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.LPTHE Orsay 95/79 * Laboratoire associé au CNRS, URA D0063