By A. Jaffe (Chief Editor)

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Math. Phys. a) 89, 571–597 (1983); b) 119, 243–285 (1988); c) 122, 175–202 (1989); d) 122, 355–392 (1989) 7. : A Short course in Cluster Expansions. In Critical Phenomena, Random Systems, Gauge Theories. Les Houches (1984), London–New York: Elsevier Science Publishers, 1986 8. : Weak Perturbations of Gaussian Measures. In: Mathematical Quantum Theory I. Field Theory and Many Body Theory. CRM Proceedings Lecture Notes 9. : Commun. Math. Phys. 85, 517–528 (1982) 10. , S´en´eor. : Commun. Math. Phys.

This is evident after establishing the following claim: For each f ∈ B there is a finite dimensional H-module W so that f ∈ {h ◦ ρ | h ∈ (Hom(W, W )) } ⊂ B. Fix a nonzero f ∈ B. Choose I to be maximal among ideals of H contained in ker f . Let ρ : H → Hom(W, W ) be the representation induced by left multiplication of H on W = H/I. Define T to be the linear span of the functionals {y → f (xyz) | x, z ∈ H}. Since I is an ideal and it lies in the kernel of f , T may be thought of as a subspace of W ∗ .

Kania-Bartoszy´nska Part 1. Lattice Gauge Field Theory Herein we develop, from a self contained and axiomatic approach, the machinery of gauge field theory on an abstract, oriented, ciliated graph. For basic background on Hopf algebras we rely on Sweedler [18] and Kassel [13]. The discussion here is restricted to the definitions and basic results confirming that the theory is consistent and computationally viable. For origins of the ideas we refer the reader to [1, 3, 4, 5, 8, 10]. 1. Objects The elementary objects of a lattice gauge field theory are: a ribbon Hopf algebra; an abstract graph which is oriented and ciliated; discretized connections, gauge transformations, and gauge fields.

### Communications in Mathematical Physics - Volume 198 by A. Jaffe (Chief Editor)

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