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Register a new userGraph homology: Koszul and Verdier duality
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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 of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the space of metric ribbon graphs, known to be homotopy equivalent to the moduli space of Riemann surfaces, and the cohomology of a certain sheaf on the space of usual metric graphs.
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Let $alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $fstar g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_iotimes x_j-x_jotimes x_i=R_{ij}(hbar)$ where $R_{ij}(hbar)in T(V^*)otimeshbar mathbb{C}[[hbar]]$, $R_{ij}=hbar Sym(alpha_{ij})+mathcal{O}(hbar^2)$, with one relation for each pair of $i,j=1...dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. At the same time we get a free resolution of the deformed algebra (for an arbitrary $alpha$). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra.
We establish a categorical version of Vogan duality for quasi-split real groups. This proves a conjecture of Soergel in the quasi-split case.
Let $alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkins theory [T1].
Given a Serre class $mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $mathcal{S}$ up to a given bound $s geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about the literature.
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Wittens conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.
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