[Paper Review] A categorical approach to quantum moment maps

Pavel Safronov|arXiv (Cornell University)|Jan 25, 2019

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a categorical framework for quantum moment maps using quantum Manin pairs and triples, unifying quantum group and Lie algebra actions. It defines quantum moment maps that generalize Lu's and Varagnolo–Vasserot's constructions, showing they degenerate to classical moment maps under the classical limit, and establishes their compatibility with fusion and quasi-Poisson geometry via shifted symplectic structures.

ABSTRACT

We introduce quantum versions of Manin pairs and Manin triples and define quantum moment maps in this context. This provides a framework that incorporates quantum moment maps for actions of Lie algebras and quantum groups for any quantum parameter. We also show how our quantum moment maps degenerate to known classical versions of moment maps and describe their fusion.

Motivation & Objective

• To develop a unified categorical framework for quantum moment maps that incorporates actions of quantum groups and Lie algebras.
• To define quantum moment maps in the context of quantum Manin pairs and triples, generalizing known constructions such as those by Lu and Varagnolo–Vasserot.
• To show that quantum moment maps degenerate to classical moment maps (e.g., Alekseev–Kosmann-Schwarzbach) in the classical limit.
• To establish compatibility of quantum moment maps with fusion operations and shifted Poisson geometry.
• To provide a systematic quantum deformation of moment map theory using formal deformation quantization and coideal subalgebras.

Proposed method

• Introduces quantum Manin pairs and triples as a categorical generalization of classical Poisson-Lie group structures.
• Defines quantum moment maps as algebra maps μ: H′ → A satisfying a twisted covariance condition involving the Hopf algebra action and antipode.
• Uses formal deformation quantization with parameter ℏ to model quantum structures, with Aℏ a flat ℋ-algebra over a formal power series ring.
• Applies the theory of shifted symplectic and Poisson structures to interpret moment maps categorically via Lagrangian correspondences in ∞-categories.
• Employs the coquasitriangular structure r = ε⊗ε + ℏr to encode the quantum R-matrix and derive moment map equations at order ℏ.
• Uses the classical limit (ℏ → 0) to show that quantum moment maps recover known classical moment map equations, including those of Alekseev–Kosmann-Schwarzbach.

Experimental results

Research questions

• RQ1 How can quantum moment maps be systematically defined for actions of quantum groups and Lie algebras in a unified categorical framework?
• RQ2 What is the relationship between quantum moment maps and classical moment maps in the ℏ → 0 limit?
• RQ3 How do quantum moment maps relate to the theory of quasi-Poisson groups and their doubles?
• RQ4 Can the fusion of quantum moment maps be described categorically using Lagrangian correspondences in shifted symplectic geometry?
• RQ5 What role do coideal subalgebras and coquasitriangular structures play in defining consistent quantum moment maps?

Key findings

• Quantum moment maps defined via quantum Manin pairs and triples generalize both Lu’s and Varagnolo–Vasserot’s constructions in a single framework.
• The classical limit of the quantum moment map equation recovers the Alekseev–Kosmann-Schwarzbach moment map equation at order ℏ.
• The map r₂: 𝔤* → 𝔡 is shown to be an injective Lagrangian embedding, ensuring compatibility with the canonical r-matrix structure.
• The existence of a quantum moment map μℏ: ℱ → Aℏ that lifts μ* modulo ℏ implies that μ is a classical moment map, establishing consistency with classical geometry.
• The coaction on 𝒪(D/G) is compatible with the Poisson bracket, and the bracket is uniquely determined by the bracket on 𝒪(D).
• The construction shows that the quantum moment map structure is preserved under the classical limit, confirming consistency with established classical theories.