Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral-Geometric Deformations of Function Algebras on Manifolds

Amandip Sangha

Abstract

We propose a novel method for deforming the algebra of smooth functions on a compact Riemannian manifold based on spectral data rather than group actions or Poisson structures. The deformation is defined through spectral coefficients of the Laplacian and a weight function satisfying a fusion-type cocycle condition, producing a noncommutative product that depends intrinsically on Laplacian spectrum of the manifold. We develop the analytic and algebraic foundations of this construction, establishing associativity, continuity, and the existence of a group structure on admissible weights. We show that the construction is functorial with respect to spectrum-preserving maps of manifolds. This spectral-geometric deformation quantization approach provides a fully intrinsic analytic model of noncommutative geometry based solely on spectral data of the manifold.

Spectral-Geometric Deformations of Function Algebras on Manifolds

Abstract

We propose a novel method for deforming the algebra of smooth functions on a compact Riemannian manifold based on spectral data rather than group actions or Poisson structures. The deformation is defined through spectral coefficients of the Laplacian and a weight function satisfying a fusion-type cocycle condition, producing a noncommutative product that depends intrinsically on Laplacian spectrum of the manifold. We develop the analytic and algebraic foundations of this construction, establishing associativity, continuity, and the existence of a group structure on admissible weights. We show that the construction is functorial with respect to spectrum-preserving maps of manifolds. This spectral-geometric deformation quantization approach provides a fully intrinsic analytic model of noncommutative geometry based solely on spectral data of the manifold.

Paper Structure

This paper contains 49 sections, 10 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. From symmetry to spectrum
  4. Outline of the paper
  5. Spectral–Geometric Setup
  6. Analytic and Algebraic Foundations
  7. Analytic extension to smooth functions
  8. Associativity and continuity theorems
  9. Group structure on admissible 3--cocycles
  10. Equivariance under Group Actions
  11. Equivariance of the fusion coefficients.
  12. Equivariance of the deformation.
  13. Compatibility with the internal automorphisms.
  14. Internal Actions on the Deformed Algebra
  15. Fourier–Side Description as Twisted Convolution
  16. ...and 34 more sections

Key Result

Lemma 3.3

Let $(M,g)$ be a compact $n$–manifold and $s>\tfrac{n}{2}$. Assume $\omega:I^3\to\mathbb T$ is unimodular, satisfies the algebraic cocycle/adjoint axioms of the paper, and obeys the log–Lipschitz control eq:LL with constant $L$. Then the twisted product $\star_\omega$ extends uniquely to a bounded b with $C_s$ depending only on $s$ and $(M,g)$.

Theorems & Definitions (33)

  • Definition 2.1: Spectral–Geometric Deformation
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Log–Lipschitz spectral control on $\omega$
  • Remark 3.2
  • Lemma 3.3: Sobolev boundedness of the twisted product
  • proof
  • Corollary 3.4: Extension to smooth functions
  • proof
  • ...and 23 more