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- W4297923339 abstract "The irreducible unitary highest weight representations $(pi_lambda,mathcal{H}_lambda)$ of the group $U(infty)$, which is the countable direct limit of the compact unitary groups $U(n)$, are classified by the orbits of the weights $lambda in mathbb{Z}^{mathbb{N}}$ under the Weyl group $S_{(mathbb{N})}$ of finite permutations. Here, we determine those weights $lambda$ for which the first cohomology space $H^1(U(infty),pi_lambda,mathcal{H}_lambda)$ vanishes. For finitely supported $lambda neq 0$, we find that the first cohomology space $H^1(U(infty),pi_lambda,mathcal{H}_lambda)$ never vanishes. For these $lambda$, the highest weight representations extend to norm-continuous irreducible representations of the full unitary group $U(mathcal{H})$ (for $mathcal{H}:= ell^2(mathbb{N},mathbb{C})$) endowed with the strong operator topology and to norm-continuous representations of the unitary groups $U_p(mathcal{H})$ ($pin [1,infty]$) consisting of those unitary operators $gin U(mathcal{H})$ for which $g-mathbb{1}$ is of $p$th Schatten class. However, not every 1-cocycle on $U(infty)$ automatically extends to one on these unitary groups, so we may not conclude that the first cohomology spaces of the extended representations are non-vanishing. On the contrary, for the groups $U(mathcal{H})$ and $U_infty(mathcal{H})$, all first cohomology spaces vanish. This is different for the groups $U_p(mathcal{H})$ with $1leq p <infty$, where only the natural representation on $mathcal{H}$ and on its dual have vanishing first cohomology spaces." @default.
- W4297923339 created "2022-10-01" @default.
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- W4297923339 creator A5079684489 @default.
- W4297923339 date "2016-07-01" @default.
- W4297923339 modified "2023-09-27" @default.
- W4297923339 title "On the First Order Cohomology of Infinite--Dimensional Unitary Groups" @default.
- W4297923339 doi "https://doi.org/10.48550/arxiv.1607.00181" @default.
- W4297923339 hasPublicationYear "2016" @default.
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