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Homotopy Batalin-Vilkovisky algebras. (English) Zbl 1258.18005
Recall that a Batalin-Vilkovisky algebra is a graded commutative algebra $$A$$ equipped with a unary operator $$\Delta: A\rightarrow A$$, homogeneous of degree $$1$$, satisfying $$\Delta^2 = 0$$, and so that the operation $\lambda(x_1,x_2) = \Delta(x_1 x_2) - (\Delta(x_1) x_2 + x_1\Delta(x_2)){(*)}$ satisfies the relations of an odd Poisson bracket (a Gerstenhaber bracket) on $$A$$. The operad of Batalin-Vilkovisky algebras $$\mathcal{BV}$$ is defined by generators and relations, with a two-ary generating operation $$\mu\in\mathcal{BV}(2)$$ representing the product underlying a Batalin-Vilkovisky structure, a second two-ary generating operation $$\lambda\in\mathcal{BV}(2)$$ representing the Gerstenhaber bracket, and a unary generating operation $$\Delta\in\mathcal{BV}(1)$$ which represents the Batalin-Vilkovisky operator.
The Koszul resolution of an operad, in the sense of V. Ginzburg and M. Kapranov [Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)], is a quasi-free operad $$\mathcal P_{\infty} = (\mathcal F(s^{-1}\mathcal P^{\text{<}}),d_2)$$ whose differential $$d_2$$ is quadratic on generators $$\mathcal P^{\text{<}}$$. The definition of the operad $$\mathcal{BV}$$ involves a relation (*) which is not homogeneous quadratic with respect to the composition of generating operations, since this equation involves the Gerstenhaber bracket as a linear term. The existence of this inhomogeneous relation gives an obstruction for the construction of a Koszul resolution in the above sense. The authors therefore consider a graded operad $$q\mathcal{BV}$$, with the same generating operations as in the operad $$\mathcal{BV}$$, but where this linear term in (*) is canceled. The crux of their construction relies on the observation that this operad $$q\mathcal{BV}$$ is Koszul, and that a linear term can be added to the differential of the Koszul resolution of this graded operad to get a quasi-free operad $$\mathcal{BV}_{\infty} = (\mathcal F(s^{-1}q\mathcal{BV}^{\text{<}}),d_1+d_2)$$ defining a resolution of the operad $$\mathcal{BV}$$.
The authors considers the category of algebras associated to the operad $$\mathcal{BV}_{\infty}$$ to get an explicit definition of the notion of a homotopy $$\mathcal{BV}$$-algebra. The authors notably use this effective approach to prove that any non-negatively graded topological vertex operator algebra in the sense of B. H. Lian and G. J. Zuckerman [Commun. Math. Phys 154, 613–646 (1993; Zbl 0780.17029)] inherits a homotopy $$\mathcal{BV}$$-algebra structure.