RECORD
Oscillator algebras with semi-equicontinuous coadjoint orbits
Representation Theory · Submitted 7 Oct 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group \gamma. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator D of \gamma. A central point of our investigations is that we make no assumption on the structure of the spectrum of D. In particular, D can be any skew-adjoint operator on a Hilbert space.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. | background 1.00 | 0.06 | 0.03 | asserted 1.00 |
| s1 | Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. | background 0.82 | 0.05 | 0.03 | asserted 1.00 |
| s2 | In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group \gamma. | contribution 0.94 | 0.94 | 0.64 | asserted 1.00 |
| s3 | For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. | result 0.98 | 0.92 | 0.76 | asserted 1.00 |
| s4 | This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator D of \gamma. | result 0.76 | 0.77 | 0.15 | asserted 0.99 |
| s5 | A central point of our investigations is that we make no assumption on the structure of the spectrum of D. | contribution 0.76 | 0.86 | 0.05 | asserted 0.99 |
| s6 | In particular, D can be any skew-adjoint operator on a Hilbert space. | interpretation 0.46 | 0.56 | 0.04 | asserted 0.87 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group \gamma. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator D of \gamma. A central point of our investigations is that we make no assumption on the structure of the spectrum of D. In particular, D can be any skew-adjoint operator on a Hilbert space.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.