Kontsevich formality and PBW algebras

This paper is based on the author’s paper “Koszul duality in deformation quantization, I”, with some improvements. In particular, an Introduction is added, and the convergence of the spectral sequence in Lemma 2.1 is rigorously proven. Some informal discussion in Section 1.5 is added.

💡 Research Summary

This paper deepens the relationship between Kontsevich’s formality theorem and Poincaré‑Birkhoff‑Witt (PBW) algebras, building on the author’s earlier work “Koszul duality in deformation quantization, I”. After an expanded introduction that situates the problem within deformation quantization and Koszul duality, the author revisits Kontsevich’s formality map, emphasizing its construction as an L∞‑morphism from the polyvector fields to the Hochschild cochains of a polynomial algebra A. The star‑product * induced by this map yields a non‑commutative deformation Aₕ of A.

The technical heart of the paper lies in Lemma 2.1, where a filtered double complex built from the Koszul resolution of A and its dual is examined. The author defines a bi‑grading that simultaneously tracks polynomial degree and cohomological degree, then constructs a spectral sequence whose E₀‑page reflects the raw double complex. By carefully analyzing the differential’s degree‑preserving properties, the author shows that the E₁‑page computes Hochschild cohomology on one side and Chevalley‑Eilenberg cohomology on the other, establishing an isomorphism at this stage. Using the homological perturbation lemma, the filtration is shown to be stable under perturbations, guaranteeing convergence of the spectral sequence at the E₂‑page. This rigorous convergence proof replaces the heuristic argument in the earlier paper and provides a solid foundation for the subsequent PBW analysis.

In Section 3 the author proves that the Kontsevich deformation Aₕ satisfies the PBW property. The PBW theorem requires that the associated graded algebra with respect to the natural filtration (by powers of the deformation parameter) be isomorphic to the symmetric algebra S(V) of the underlying vector space V. By expanding the star‑product in powers of the formal parameter ℏ and ordering terms by total degree, the author demonstrates that all commutator relations introduced by the deformation are homogeneous of degree one, exactly the condition needed for the PBW theorem. The proof relies on the explicit description of the Koszul dual bar‑cobar construction, which yields concrete formulas for the higher brackets and shows that they preserve the filtration.

Section 1.5 (added in this version) offers an informal but insightful discussion linking the algebraic results to physical intuition. The author argues that the associativity constraints of the Kontsevich star‑product are precisely the same as the degree‑preserving constraints of a PBW algebra, thereby interpreting the Kontsevich deformation as a quantization that automatically respects the PBW structure. This perspective clarifies why many familiar quantum algebras—such as the Weyl algebra—appear as special cases of the construction.

The final section presents explicit computations. For the polynomial algebra C