BV Structures with Nonsemisimple Nakayama Automorphisms

Definitions and mathematical statement

Let ${\mathbb{F}}$ be a field of characteristic $0$, and let $h(x)\in{\mathbb{F}}[x]$. Define the first Weyl algebra and the subalgebra appearing in the supplied extract by \[ A_1={\mathbb{F}}\langle x,y\rangle/(yx-xy-1),\qquad A_h={\mathbb{F}}\langle x,h(x)y\rangle\subseteq A_1. \] Here the second expression denotes the unital subalgebra generated by the indicated elements. The extract allows an arbitrary polynomial $h$; no restriction excluding $h=0$ is supplied.

For an associative ${\mathbb{F}}$-algebra $A$, put $A^e=A\otimes_{{\mathbb{F}}}A^{\mathrm{op}}$ and \[ HH^n(A)=\operatorname{Ext}_{A^e}^n(A,A),\qquad HH^\bullet(A)=\bigoplus_{n\geq0}HH^n(A). \] Use the canonical cup product $\smile$ and Gerstenhaber bracket $[-,-]_G$, of degrees $0$ and $-1$, respectively. A Batalin--Vilkovisky structure compatible with these operations means an ${\mathbb{F}}$-linear operator \[ \Delta:HH^n(A)\longrightarrow HH^{n-1}(A),\qquad \Delta^2=0, \] where $HH^{-1}(A)=0$, such that, for all homogeneous $a\in HH^p(A)$ and $b\in HH^q(A)$, \[ [a,b]_G=(-1)^p\left(\Delta(a\smile b)-\Delta(a)\smile b-(-1)^p a\smile\Delta(b)\right). \] This fixes a sign convention for the bracket-generating operator.

The supplied reconstructed statement additionally assumes that $A_h$ is twisted Calabi--Yau. In the usual terminology, this means that $A_h$ has a bounded resolution by finitely generated projective $A_h^e$-modules and, for some integer $d\geq0$ and automorphism $\nu$, \[ \operatorname{Ext}_{A_h^e}^{i}(A_h,A_h^e)\cong \begin{cases} 0,&i\ne d,\\ {}_1(A_h)_{\nu},&i=d, \end{cases} \] as bimodules, using the convention that the right action on ${}_1(A_h)_{\nu}$ is $u\cdot a=u\nu(a)$. Such an automorphism is called a Nakayama automorphism.

Question, subject to the qualifications below. For $A_h$ as above, under the reconstructed twisted Calabi--Yau hypothesis and the source condition that its Nakayama automorphism is not semisimple, does $HH^\bullet(A_h)$ admit an operator $\Delta$ satisfying the displayed conditions?

Unresolved qualifications. The extract specifies the Weyl subalgebra family, whereas the reconstructed statement describes $A_h$ only as a twisted Calabi--Yau algebra. It is therefore unclear whether the intended scope is precisely this family or all twisted Calabi--Yau algebras. The extract does not supply the additional twisted Calabi--Yau hypothesis or its applicable range of $h$. It also does not specify the meaning of semisimplicity for the automorphism on this potentially infinite-dimensional algebra, the field over which it is to be tested, or whether the question seeks existence for every qualifying $h$, for some qualifying $h$, or a classification. The displayed question records the conditional existence problem for a qualifying member without resolving those scope issues. Compatibility with the canonical Gerstenhaber structure is the interpretation used here; the extract does not explicitly define its intended BV structure.

Short English statement

For the algebra generated by $x$ and $h(x)y$ inside the first Weyl algebra over a characteristic-zero field, ask whether its Hochschild cohomology has a square-zero operator of degree $-1$ generating the usual Gerstenhaber bracket when the Nakayama automorphism is nonsemisimple. The supplied reconstruction adds a twisted Calabi--Yau assumption. Its scope, the precise semisimplicity condition, and the quantification over $h$ require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30005740$, source problem code catalogue entry, title BV Structures with Nonsemisimple Nakayama Automorphisms. The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is named.
• Supplied report citation: Mini-Workshop: Poisson and Poisson-type algebras (2024), Oberwolfach Reports, pp. 2681--2715, DOI 10.4171/owr/2023/46, https://doi.org/10.4171/owr/2023/46. The supplied literature-source label is Oberwolfach Report 46/2023. The extract includes the page marker 2703. The record gives proposed year 2023; the difference between that year and the citation's 2024 label has not been resolved here.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that BV structures are established for twisted Calabi--Yau or Frobenius algebras with semisimple Nakayama automorphism, while the nonsemisimple $A_h$ case in the extract remains unresolved. This is supplied dated curation, not an independently verified literature assessment. Its listed source is https://doi.org/10.4171/owr/2023/46.
• The record labels its statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims; no independent source visit or verification is asserted here.
• The extract attributes the cup product and Gerstenhaber bracket to Gerstenhaber via its reference [2], but supplies no bibliographic details for [2]. No additional citation is inferred.