[Paper Review] Projectively equivariant symbol calculus

Pierre Lecomte, Valentin Ovsienko|ArXiv.org|Sep 11, 1998

Advanced Topics in Algebra16 references20 citations

TL;DR

This paper introduces a projectively equivariant symbol calculus that constructs a unique (up to normalization) isomorphism between the space of differential operators on λ-densities and polynomial functions on the cotangent bundle, equivariant under the action of sl(n+1,R). The key result is that this isomorphism provides a canonical quantization map on manifolds with a flat projective structure, enabling classification of quotient modules of differential operators and leading to sl(n+1,R)-equivariant star-products.

ABSTRACT

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $λ$ and the space of functions on $T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on ${\mathbb{R}}^n$. However, these modules are isomorphic as $sl(n+1,{\mathbb{R}})$-modules where $sl(n+1,{\mathbb{R}})\subset \Vect({\mathbb{R}}^n)$ is the Lie algebra of infinitesimal projective transformations. In addition, such an $sl_{n+1}$-equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the $sl_{n+1}$-equivariant symbol map to study the $\Vect(M)$-modules of linear differential operators acting on tensor densities, for an arbitrary manifold $M$.

Motivation & Objective

• To construct a canonical, sl(n+1,R)-equivariant symbol map between differential operators on λ-densities and polynomial functions on the cotangent bundle.
• To resolve the lack of natural quantization maps by restricting symmetry from Diff(M) to the smaller group SL(n+1,R) of projective symmetries.
• To classify the quotient modules D^k_λ(M)/D^ℓ_λ(M) for k−ℓ≥2 using the sl(n+1,R)-equivariant symbol map.
• To establish that such equivariant maps are necessarily local and differential, implying they are given by differential operators.
• To extend the construction to define a 1-parameter family of sl(n+1,R)-equivariant star-products on T*M via deformation quantization.

Proposed method

• Define the space of λ-densities and the space of polynomial functions on T*R^n, and show they are isomorphic as sl(n+1,R)-modules.
• Construct an sl(n+1,R)-equivariant symbol map σ_λ: D^k_λ(R^n) → Pol^k(T*R^n) using representation theory and the action of infinitesimal projective transformations.
• Prove uniqueness of the symbol map up to normalization by analyzing irreducible components of the modules under sl(n+1,R) action.
• Establish locality of the symbol map by showing that any sl(n+1,R)-equivariant linear map on Pol(T*R^n) must be differential, due to invariance under the affine group R*⋉R^n.
• Use the symbol map to define a star-product ⋆_ħ on Pol(T*M) by rescaling the quantization map via ℏ^k, leading to a formal deformation quantization.
• Compute cohomology classes in H^1(Vect(M), Hom(S^k, S^{k−ℓ})) associated with the quotient modules, showing nontrivial deformations for k−ℓ≥2.

Experimental results

Research questions

• RQ1 Can a natural quantization map be defined between differential operators on λ-densities and polynomial functions on the cotangent bundle, equivariant under a large symmetry group?
• RQ2 Is there a maximal group of symmetries under which the modules of differential operators and polynomial symbols become isomorphic?
• RQ3 What is the structure of the quotient modules D^k_λ(M)/D^ℓ_λ(M) for k−ℓ≥2, and how do they deform the space of symbols?
• RQ4 Does sl(n+1,R)-equivariance force the symbol map to be local and differential?
• RQ5 Can the sl(n+1,R)-equivariant symbol map be used to construct a star-product on T*M that is equivariant under projective transformations?

Key findings

• The spaces D_λ(R^n) and Pol(T*R^n) are isomorphic as sl(n+1,R)-modules, and this isomorphism is unique up to normalization.
• The sl(n+1,R)-equivariant symbol map is necessarily local and given by a differential operator, implying it is a differential map.
• For k−ℓ≥2, the quotient modules D^k_λ(M)/D^ℓ_λ(M) are nontrivial deformations of the module of symbols, with cohomology classes in H^1(Vect(M), Hom(S^k, S^{k−ℓ})) computed explicitly.
• The construction yields a 1-parameter family of sl(n+1,R)-equivariant star-products on T*M via deformation quantization, with the first-order term being the Poisson bracket.
• The symbol map is globally well-defined on any manifold M endowed with a (flat) projective structure, via local formulae that are invariant under projective coordinate changes.
• In the case λ=1/2 and k−ℓ=2, the quotient module is an exception and does not yield a nontrivial deformation, distinguishing it from other cases.