AG Lie-Gruppen: Milan Niestijl, TU Delft

May 16
16-05-2022 2:15 PM Uhr bis 3:45 PM Uhr
Übungsraum Ü2, Cauerstr. 11, Erlangen

Positive Energy Representations of Gauge Groups With Support at a Fixed

Vortragender: Milan Niestijl, TU Delft
Veranstalter: Karl-Hermann Neeb

Let $mathcal{K} to M$ be a locally trivial smooth bundle of Lie groups
equipped with an action of some Lie group $P$ by bundle automorphisms.
Complementing recent progress B. Janssens and K.H. Neeb on the case where the
$P$-action on $M$ has no fixed-points, projective unitary representations
$overline{rho}$ of the locally-convex Lie group $mathcal{G} :=
Gamma_c(mathcal{K})$ are studied which are of “positive energy” and factor
entirely through the germs at some fixed point $a in M$ of the $P$-action.
Under suitable assumptions, it is shown that the kernel of a particular
quadratic form on $R[[x_1,cdots, x_d]]$ generates an ideal in $mathfrak{G} :=
Lie(mathcal{G})$ on which the derived representation $drho$ must vanish. This
leads in particular to sufficient conditions for $drho$ to factor through a
finite jet space $J^k_x(mathcal{K})$ or through $J^infty_a(N,mathcal{K})$ for
some usually lower-dimensional submanifold $N$. Some examples are considered. If
time permits, we will have a closer look at the special case where $P=S^1$.

Friedrich-Alexander-Universität Erlangen-Nürnberg