Symmetry Reduction of States I

  • Philipp Schmitt

    Leibniz Universität Hannover, Germany
  • Matthias Schötz

    Instytut Matematyczny PAN, Warsaw, Poland
Symmetry Reduction of States I cover
Download PDF

This article is published open access under our Subscribe to Open model.


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 . The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares with , but 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 thus depends on this choice of positivity on , and the notion of positivity on the reduced algebra should be such that states on are obtained as reductions of certain states on . We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold , which reproduces the coisotropic reduction of ; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.

Cite this article

Philipp Schmitt, Matthias Schötz, Symmetry Reduction of States I. J. Noncommut. Geom. 18 (2024), no. 2, pp. 501–545

DOI 10.4171/JNCG/534