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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Finite dimensional semigroups of unitary endomorphisms of standard subspaces
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by Karl-H. Neeb
Represent. Theory 25 (2021), 300-343
Published electronically: April 27, 2021


Let $\mathtt {V}$ be a standard subspace in the complex Hilbert space $\mathcal {H}$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $\mathcal {H}$ containing the modular group $(\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}}$ of $\mathtt {V}$ and the corresponding modular conjugation $J_{\mathtt {V}}$. We study the semigroup \[ S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge $\operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $\mathfrak {g}$ of $G$. The semigroup $S_{\mathtt {V}}$ is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form $G \exp (iC)$, where $C \subseteq \mathfrak {g}$ is an $\operatorname {Ad}(G)$-invariant closed convex cone.

Our main results assert that the Lie wedge $\operatorname {\textbf {L}}(S_{\mathtt {V}})$ spans a $3$-graded Lie subalgebra in which it can be described explicitly in terms of the involution $\tau$ of $\mathfrak {g}$ induced by $J_{\mathtt {V}}$, the generator $h \in \mathfrak {g}^\tau$ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup $S_{\mathtt {V}}$ itself.

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Bibliographic Information
  • Karl-H. Neeb
  • Affiliation: Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
  • MR Author ID: 288679
  • Email:
  • Received by editor(s): May 21, 2019
  • Received by editor(s) in revised form: June 29, 2020, and February 9, 2021
  • Published electronically: April 27, 2021
  • Additional Notes: The author was supported by DFG-grant NE 413/10-1.
  • © Copyright 2021 American Mathematical Society
  • Journal: Represent. Theory 25 (2021), 300-343
  • MSC (2020): Primary 22E45; Secondary 81R05, 81T05
  • DOI:
  • MathSciNet review: 4248936