Related papers: Poincar\'e/Koszul Duality for General Operads

We establish that $E_n$-operads satisfy a rational intrinsic formality theorem for $n\geq 3$. We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in…

Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category…

In this paper we prove that the linear Koszul duality equivalence constructed in a previous paper provides a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras.

We prove that truncations of nonstandard graded polynomial rings are (nonstandard) Koszul modules in the sense of Herzog and Iyengar. This provides an analogue of the fact that such truncations have linear resolutions in the standard graded…

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology…

We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…

We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial…

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the…

We show a variation of the usual homological freeness criterion for operadic modules over a Koszul operad. We then apply this result to decorated partition posets for some operads, showing that their augmentation is Cohen-Macaulay and…

Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in equal characteristic zero with a differential graded algebra structure, we introduce a…

We construct a monoidal model structure on the category of all curved coalgebras and show that it is Quillen equivalent, via the extended bar-cobar adjunction, to another model structure we construct on the category of curved algebras. When…

Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul…

A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a…

We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We…

The main result of this paper is to establish precisely which blocks in the Category $\mathcal{O}$ of the periplectic Lie superalgebra $\mathfrak{pe}(2)$ are Koszul. It is known that $\mathcal{O}(\mathfrak{pe}(2))$ has three blocks up to…

A dual pre-Poisson algebra is an algebraic structure that integrates a permutative algebra and a Leibniz algebra under certain compatibility conditions. As the Koszul dual notion of the pre-Poisson algebra, this structure serves as a…

Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for $A_\infty$-algebras, we give an explicit proof that the universal enveloping algebra $UL$ of a differential graded Lie algebra $L$ is Koszul, via an…

We explore the constraints imposed by Poincar\'e duality on the resonance varieties of a graded algebra. For a 3-dimensional Poincar\'e duality algebra $A$, we obtain a fairly precise geometric description of the resonance varieties…

We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many…

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we…