BV generators for polynomial Weyl subalgebras

Abstract

Let k be any field of characteristic zero and let 0≠h∈k[x]. For A_h=k⟨x,Y⟩/(Yx−xY−h(x)), the ordinary Hochschild cohomology, with its canonical cup product and Gerstenhaber bracket, admits a Batalin–Vilkovisky generator if and only if h is squarefree. In the squarefree case its bracket is zero. A repeated irreducible factor produces a cup-zero, noncommuting pair of degree-one classes, and a central annihilator forces every putative generator to vanish on degree one. Thus in that case no degree-minus-one bracket generator exists, even without requiring its square to be zero. The argument uses the published resolution and derivation structure of this family, and answers the concrete-family question recorded by Lopes in Oberwolfach Report 46/2023, p. 2702, Question (2).

The idea in plain English

Hochschild cohomology packages information about an algebra and its possible deformations. It comes with both a product and a bracket. A Batalin–Vilkovisky, or BV, generator is a single operator that recovers the bracket from the product and whose square is zero.

This paper considers a family of algebras governed by one polynomial h. The classification is simple to state: a BV generator exists exactly when h has no repeated irreducible factor. Nonzero constant polynomials are included in the positive case.

The negative case is stronger than failure to find a suitable formula. A repeated factor produces cohomology classes whose product is zero but whose bracket is nonzero. Together with a central annihilating class, these relations rule out every degree-minus-one bracket generator, even if its square is not required to vanish.

The precise result

For any characteristic-zero field k and nonzero h∈k[x], let A_h=k⟨x,Y⟩/(Yx−xY−h(x)). Theorem 1 states that h being squarefree, the canonical Gerstenhaber bracket on ordinary HH•(A_h,A_h) being zero, existence of a bracket generator, and existence of a BV generator are equivalent.

For squarefree h, the zero operator is a BV generator. If h has a repeated irreducible factor over k, no bracket generator exists. The argument works over the original field and does not require the polynomial to split into linear factors.

Why the result holds

Published results describe the algebra’s resolution and derivations. The paper compares that resolution with normalized bars to identify the canonical cup product needed for the obstruction.

When d=gcd(h,h′) is nonconstant, a central degree-one class annihilates every other degree-one class under the cup product. This forces any possible bracket generator to vanish on degree one. An explicit cup-zero pair with nonzero bracket then gives a contradiction.

When d=1, the second and higher cohomology groups vanish and the remaining degree-one derivations commute. The entire bracket is therefore zero.

Scope and limitations

The classification concerns this one-variable polynomial family over fields of characteristic zero, with its ordinary Hochschild cohomology and canonical product and bracket. It does not classify arbitrary Weyl subalgebras, positive-characteristic cases, twisted-coefficient variants, or Poisson analogues.

For h≠0, A_h is the subalgebra of the first Weyl algebra generated by x and yh(x). The literal embedded h=0 case instead gives k[x], which the paper treats separately using Δ(f∂x)=−f′. Setting h=0 in the abstract two-generator presentation gives a different algebra, k[x,Y].

The result uses established resolution and derivation theorems. The accompanying finite exact computations support selected identities; they are not proof-assistant certification or a replacement for the universal argument.

Earlier work already ruled out bracket generators for other ordinary Hochschild cohomology algebras. This paper gives a classification for the stated family, with bounded historical coverage. It makes no first-discovery claim.

Status and AI use

AI tools were used extensively in mathematical exploration, proof development, computation, literature research, manuscript preparation and adversarial auditing. Internal AI reviews do not constitute formal independent human peer review, journal acceptance or proof-assistant certification.

The manuscript and declared Zenodo edition are dated 22 October 2026. The Zenodo record was created on the same date. These dates identify this edition; they do not establish first priority.

Read the paper

Open PDF in a full window · Download the original from Zenodo

Collaborate on this result

I welcome substantive collaboration from researchers in Hochschild cohomology, noncommutative algebra and BV structures on independent verification, comparison with earlier results and preparation for journal review. Any coauthorship would reflect a substantive contribution to the research.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). BV generators for polynomial Weyl subalgebras (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23572690

Download BibTeX

Background and prior work

• Lopes, with Solotar — Gerstenhaber algebra structure on Hochschild cohomology: Question (2), printed page 2702, asks about BV structures for this family when its Nakayama automorphism is not semisimple. The workshop took place in 2023; the report was published in July 2024.
• Lopes and Solotar — Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra: Provides the free resolution, characteristic-zero cohomology quotient, derivation structure and Nakayama automorphism used by the note.
• Benkart, Lopes and Ondrus — Derivations of a parametric family of subalgebras of the Weyl algebra: The established derivation and normalizer classification supplies the degree-one spanning description and center.
• Herscovich and Li — Gerstenhaber structure on Hochschild cohomology of the Fomin–Kirillov algebra on 3 generators: Proposition 4.15 is an earlier ordinary-Hochschild no-generator result and a predecessor of the cup/bracket obstruction strategy.
• Kowalzig and Krähmer — Batalin–Vilkovisky structures on Ext and Tor: Theorem 1.7 assumes a semisimple modular automorphism; this hypothesis cannot be silently dropped for the nonconstant-polynomial branch.
• Bian, Itagaki, Kou, Lyu and Zhou — BV structures for self-injective Nakayama algebras: Theorem 7.11 treats truncated basic cycle algebras, a different family. Its Criterion 7.7 includes a commuting-odd-generators hypothesis.