- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:14
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
10
2016
1
Projective families of Dirac operators on a Banach Lie groupoid
Pedram
Hekmati
University of Adelaide, ADELAIDE, SA, AUSTRALIA
Jouko
Mickelsson
University of Helsinki, HELSINKI, FINLAND
Twisted K-theory, Dirac operators, Banach Lie groups
We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $G \ltimes \frak k$, where $\frak k$ denotes the Hilbert space of self-adjoint Hilbert–Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator $\mathbb D$ in a suitable completion of the quantum Weil algebra $\mathcal{U}(\frak{g}) \otimes Cl(\frak{k})$, which is subsequently extended to a projective family of self-adjoint operators $\mathbb D_A$ on $G \ltimes \frak k$. While the kernel of $\mathbb D_A$ is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.
Quantum theory
$K$-theory
Topological groups, Lie groups
1
28
10.4171/JNCG/227
http://www.ems-ph.org/doi/10.4171/JNCG/227