Z2 x Z2 lattice as a Connes-Lott-quantum group model

We apply quantum group methods for noncommutative geometry to the Z(2) X Z(2) lattice to obtain a natural Dirac operator on this discrete space. This then leads to an interpretation of the Higgs fields as the discrete part of space-time in the Connes-Lott formalism for elementary particle Lagrangians. The model provides a setting where both the quantum groups and the Connes approach to noncommutative geometry can be usefully combined, with some of Connes' axioms, notably the first-order condition, replaced by algebraic methods based on the group structure. The noncommutative geometry has nontrivial cohomology and moduli of flat connections, both of which we compute. (C) 2002 Elsevier Science B.V. All rights reserved.