Let M be a connected differentiable manifold. Denote by Omega(m)(M) the space of H-1-loops based at a fixed point m is an element of M. Associated to Omega(m)(M) one has <(Omega)over tilde>(m)(M), the group of unparameterized loops. Given a bundle-connection pair (E, del) over M with fiber the finite-dimensional vector space V and structure group G subset of GL(V) we get (up to equivalence) a smooth representation of <(Omega)over tilde>(m)(M) in G given by the parallel transport operator P-del. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of <(Omega)over tilde>(m)(M) arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids.

Gibilisco P. (1997). Bundle-connection pairs and loop group representations. MATHEMATICAL NOTES OF THE ACADEMY OF SCIENCES OF THE USSR, 61, 417-429.

Bundle-connection pairs and loop group representations

GIBILISCO, PAOLO
1997

Abstract

Let M be a connected differentiable manifold. Denote by Omega(m)(M) the space of H-1-loops based at a fixed point m is an element of M. Associated to Omega(m)(M) one has <(Omega)over tilde>(m)(M), the group of unparameterized loops. Given a bundle-connection pair (E, del) over M with fiber the finite-dimensional vector space V and structure group G subset of GL(V) we get (up to equivalence) a smooth representation of <(Omega)over tilde>(m)(M) in G given by the parallel transport operator P-del. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of <(Omega)over tilde>(m)(M) arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/06 - Probabilita' e Statistica Matematica
English
Con Impact Factor ISI
Bundle; Connection; Differentiable manifold; Induced representation; Loop group
Gibilisco P. (1997). Bundle-connection pairs and loop group representations. MATHEMATICAL NOTES OF THE ACADEMY OF SCIENCES OF THE USSR, 61, 417-429.
Gibilisco, P
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/49738
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