## Curved Koszul duality of algebras over unital versions of binary operads.(English)Zbl 1508.18017

The theory of Koszul operads gives a reduced construction of cofibrant resolutions in algebra categories. These resolutions can be used to compute derived functors on algebras. The constructions of such resolutions involves two steps. Let $$P$$ be the operad that governs our category of algebras. In the first step, we have to compute the Koszul dual cooperad of the operad $$P$$ from a presentation of this operad by generators and relations. Then the Koszul duality functors return a natural cofibrant resolution functor on the category of $$P$$-algebras. But we can also compute, in a second step, a reduced cofibrant resolution of $$P$$-algebras when we have good algebras with a presentation of this algebra.
The definition of the Koszul dual of an operad only works for good (Koszul) operads. The Koszul duality theory was originally defined for operads equipped with a presentation with binary generating operations and homogeneous quadratic relations. The author of the paper under review deals with operads $$uP$$ obtained by adding a unit operation (a constant) to such a binary quadratic Koszul operad $$P$$. The main purpose of paper is precisely to give a construction of reduced resolutions of algebras equipped with quadratic-linear-constant presentation over $$uP$$.
The author also explains the application of his construction to the case of symplectic Poisson $$n$$-algebras $$A_{n;D} = \mathbb{R}[x_1,\dots,x_D,\xi_1,\dots,\xi_D]$$, which is generated by variables $$(x_i,\xi_i)$$ such that $$\deg(x_i) = 0$$, $$\deg(\xi_i) = n-1$$, and which are equipped with a Poisson bracket of degree $$n-1$$ satisfying $$\{x_i,\xi_j\} = \delta_{i j}$$. This algebra admits a presentation with quadratic-linear-constant relations over the operad $$uPois_n$$ that governs unital Poisson $$n$$-algebras. He proves that this algebra is Koszul, and describes its reduced cofibrant resolution. He uses this resolution to compute the factorization homology $$\int_M A_{n;D}$$ of a simply connected parallelized closed $$n$$-manifold $$M$$ with coefficients in $$A_{n;D}$$. (He proves that this factorization homology is one-dimensional.)

### MSC:

 18M70 Algebraic operads, cooperads, and Koszul duality 16S37 Quadratic and Koszul algebras 18G10 Resolutions; derived functors (category-theoretic aspects) 17B63 Poisson algebras
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