Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the First Order Cohomology of Infinite--Dimensional Unitary Groups   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
$U_p(\mathcal{H})$ ($p\in [1,\infty]$) consisting of those unitary operators
$g\in U(\mathcal{H})$ 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 $1\leq
p <\infty$, where only the natural representation on $\mathcal{H}$ and on its
dual have vanishing first cohomology spaces.

28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg