Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive energy representations for locally finite split Lie algebras   Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of
type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a
splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with
$\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary
highest weight representation $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$
extends to a representation $\tilde\rho_\lambda$ of the semidirect product
$\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a
positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is
bounded from below. In the present note we characterise all pairs $(\lambda,D)$
with $\lambda$ bounded for which this is the case.
  If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an
infinite dimensional real, complex or quaternionic Hilbert space and $\lambda$
is bounded, then we accordingly obtain a characterisation of those highest
weight representations $\pi_\lambda$ satisfying the positive energy condition
with respect to the continuous $\mathbb R$-action induced by $D$. In this
context the representation $\pi_\lambda$ is norm continuous and our results
imply the remarkable result that, for positive energy representations, adding a
suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of
$\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding
pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to
describe explicitly.

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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th