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Quantization of the Momentum Map via $\frak{g}$-adapted Formalities

Chiara Esposito, Ryszard Nest, Jonas Schnitzer, Boris Tsygan

Abstract

In this note, we provide a proof of the existence and complete classification of $G$-invariant star products with quantum momentum maps on Poisson manifolds by means of an equivariant version of the formality theorem.

Quantization of the Momentum Map via $\frak{g}$-adapted Formalities

Abstract

In this note, we provide a proof of the existence and complete classification of -invariant star products with quantum momentum maps on Poisson manifolds by means of an equivariant version of the formality theorem.

Paper Structure

This paper contains 15 sections, 27 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homological Perturbation
  4. $L_\infty$-algebras
  5. A zig-zag of $G_\infty$-quasi-isomorphisms
  6. A $\mathfrak{g}$-adapted quasi-inverse
  7. Equivariant Complexes as curved Lie algebras
  8. Compatibility with Curvatures
  9. $\mathfrak{g}$-adapted $L_\infty$-algebras
  10. Equivariant formality
  11. A $\mathrm{Der}(A)$-adapted zig zag
  12. A $\mathfrak{g}$-adapted formality map
  13. Cofree Lie coalgebras
  14. The dual of the Poincaré-Birkhoff-Witt Theorem for cofree conilpotent Lie coalgebras
  15. Chevalley-Eilenberg cohomology of cofree Lie coalgebras

Key Result

Proposition 1

Let $(i, p, h)$ be a homotopy retract given by the diagram eq:homretract and let $b$ be a small perturbation. Then the perturbed diagram \begin{tikzcd} (C^\bullet, \mathrm{D} _C) \arrow[r," I ", shift left = 3pt] &(D^\bullet,\D_D + b ) \arrow[l," P ", shift left = 3pt] \arrow[loop

Theorems & Definitions (70)

  • Definition 1: Homotopy retract
  • Definition 2: Perturbation
  • Proposition 1: Homological perturbation
  • Corollary 1: Morphism of perturbed retracts
  • Definition 3: Special deformation retract
  • Corollary 2: Perturbation of special deformation retract
  • Definition 4: (Curved) $L_\infty$-algebra
  • Example 1: Multivector fields
  • Example 2: Multidifferential operators
  • Example 3: Curved Lie algebra
  • ...and 60 more