Details
| Original language | English |
|---|---|
| Pages (from-to) | 501-545 |
| Number of pages | 45 |
| Journal | Journal of Noncommutative Geometry |
| Volume | 18 |
| Issue number | 2 |
| Early online date | 7 Oct 2023 |
| Publication status | Published - 20 Mar 2024 |
Abstract
In this article, we develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the “correct” notion of positivity on a *-algebra A is not necessarily the algebraic one, for which positive elements are sums of Hermitian squares a *a with a 2 A, but it can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra A μ-red should be such that states on A μ-red are obtained as reductions of certain states on A. We discuss three examples in detail: reduction of the *-algebra of smooth functions on a Poisson manifold M, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a U(1)-action.
Keywords
- -algebras, non-commutativity, positivity, states, Symmetry reduction
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Mathematical Physics
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In: Journal of Noncommutative Geometry, Vol. 18, No. 2, 20.03.2024, p. 501-545.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Symmetry reduction of states I
AU - Schmitt, Philipp Lothar
AU - Schötz, Matthias
N1 - Funding Information: Funding. This work was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO) under EOS Project no. 30950721.
PY - 2024/3/20
Y1 - 2024/3/20
N2 - In this article, we develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the “correct” notion of positivity on a *-algebra A is not necessarily the algebraic one, for which positive elements are sums of Hermitian squares a *a with a 2 A, but it can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra A μ-red should be such that states on A μ-red are obtained as reductions of certain states on A. We discuss three examples in detail: reduction of the *-algebra of smooth functions on a Poisson manifold M, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a U(1)-action.
AB - In this article, we develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the “correct” notion of positivity on a *-algebra A is not necessarily the algebraic one, for which positive elements are sums of Hermitian squares a *a with a 2 A, but it can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra A μ-red should be such that states on A μ-red are obtained as reductions of certain states on A. We discuss three examples in detail: reduction of the *-algebra of smooth functions on a Poisson manifold M, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a U(1)-action.
KW - -algebras
KW - non-commutativity
KW - positivity
KW - states
KW - Symmetry reduction
UR - http://www.scopus.com/inward/record.url?scp=85188422285&partnerID=8YFLogxK
U2 - 10.4171/JNCG/534
DO - 10.4171/JNCG/534
M3 - Article
VL - 18
SP - 501
EP - 545
JO - Journal of Noncommutative Geometry
JF - Journal of Noncommutative Geometry
SN - 1661-6952
IS - 2
ER -