BV algebra with a degree-one generator

Let $k$ be a field of characteristic $0$. A unital $\mathbb{Z}$-graded associative $k$-algebra $A={⨁}_{r\in \mathbb{Z}}{A}^r$ is graded commutative when its unit $1$ has degree $0$ and homogeneous $a,b$ satisfy $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$. A linear map $\mathrm{\Delta }:A^r\to A^{r+1}$ is a BV generator of degree $+1$ when $\mathrm{\Delta }(1)=0$, ${\mathrm{\Delta }}^2=0$, and it has order at most two in the following graded-commutator convention. For homogeneous operators $P,Q$, set $[P,Q]=PQ-(-1{)}^{\mathrm{\mid }P\mathrm{\mid }\mathrm{\mid }Q\mathrm{\mid }}QP,$ and for homogeneous $a\in A$ let $m_a(x)=ax$. The order condition is $[[[\mathrm{\Delta },m_a],m_b],m_c]=0$ for all homogeneous $a,b,c\in A$. The pair $(A,\mathrm{\Delta })$ with these data is a BV algebra in the degree-$+1$ convention. Its derived bracket is the bilinear operation of degree $+1$ $\{a,b\}=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}(\mathrm{\Delta }(ab)-(\mathrm{\Delta }a)b-(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a\mathrm{\Delta }b)$ on homogeneous elements, extended by linearity.

This item fixes the grading, commutator order condition, and bracket formula. It does not include a separate differential $Q$ or assert compatibility with one. The associated odd-skew, biderivation, and Jacobi identities are proved in The derived BV bracket is an odd Poisson bracket.

Getzler writes the second-order condition as a seven-term identity and also includes a separate differential in his larger structure. Seidel uses a degree-$-1$ operator, and Kosmann-Schwarzbach–Monterde formulate the general odd bracket first in a $\mathbb{Z}\mathrm{/}2$-graded setting. This item fixes its own degree-$+1$ commutator and bracket conventions; no result in a different grading is imported without a conversion.