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# Unitary representations of gauge groups

Abstract (Summary)
I generalize to the case of gauge groups over non-trivial principal bundles representations that I. M. Gelfand, M. I. Graev and A. M. Versik constructed for current groups. The gauge group of the principal G-bundle P over M, (G a Lie group with an euclidean structure, M a compact, connected and oriented manifold), as the smooth sections of the associated group bundle is presented and studied in chapter I. Chapter II describes the symmetric algebra associated to a Hilbert space, its Hilbert structure, a convenient exponential and a total set that later play a key role in the construction of the representation. Chapter III is concerned with the calculus needed to make the space of Lie algebra valued 1-forms a Gaussian $L\sp2$-space. This is accomplished by studying general projective systems of finitely measurable spaces and the corresponding systems of $\sigma$-additive measures, all of these leading to the description of a promeasure, a concept modeled after Bourbaki and classical measure theory. In the case of a locally convex vector space E, the corresponding Fourier transform, family of characters and the existence of a promeasure for every quadratic form on $E\sp\prime$ are established, so the Gaussian $L\sp2$-space associated to a real Hilbert space is constructed. Chapter III finishes by exhibiting the explicit Hilbert space isomorphism between the Gaussian $L\sp2$-space associated to a real Hilbert space and the complexification of its symmetric algebra. In chapter IV taking as a Hilbert space H the $L\sp2$-space of the Lie algebra valued 1-forms on P, the gauge group acts on the motion group of H defining in an straight forward fashion the representation desired.
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