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.)


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


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