arXiv subjects

Alexander Vitanov

Publications and source records attributed to Alexander Vitanov.

4 recordsLinked to original sources

arXiv2022

Construction of Sheaves of Cherednik Algebras via Formal Geometry

In this note we realize the sheaf of Cherednik algebras $H_{1, c, X, G}$ on a general good complex orbifold $X/G$, originally introduced by Etingof for smooth complex varieties with an action by a finite group, by gluing sheaves of flat sections of flat holomorphic vector bundles on orbit type strata in $X$ which result from a localization procedure. In the case, when $c$ is formal, this construction can be interpreted as a formal deformation of $D_X\rtimes\mathbb CG$ via Gel'fand-Kazhdan formal geometry. Contrary to the original definition of $H_{1, c, X, G}$ the presented construction permits the computation of trace densities, Hochschild homologies and an algebraic index theorem for formal deformations of $D_X\rtimes\mathbb CG$. We also hope that the methods developed here will contribute towards a full proof of Dolgushev-Etingof's conjecture.

math.AG↗

arXiv2021

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb C_Y[[\hbar]]$-modules which computes the hypercohomology of the Hochschild chain complex $\mathcal{C}_{\bullet}$ of the sheaf of formal Cherednik algebras. We show that this hypercohomology is isomorphic to the Chen-Ruan cohomology of the orbifold $X/G$ with values in the ring of formal power series $\mathbb C[[\hbar]]$. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathcal{D}_X\rtimes G$ has a well-defined Euler characteristic which is proportional to the topological Euler characteristic of $X/G$. Finally, we prove an algebraic index theorem.

math.QA↗

arXiv2021

Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations

According to a statement by Pavel Etingof, in the special case of an affine variety $X$ with a faithful action by a finite group $G$, the sheaf of (twisted) Cherednik algebras $\mathcal{H}_{1, c, ψ, X, G}$ with formal parameters $c, ψ$ is a universal formal deformation of $\mathcal{D}_X\rtimes G$ where $\mathcal{D}_X$ is the sheaf of differential operators on $X$. In the current note, we generalize Etingof's result to the non-affine case. We prove that for a generic smooth analytic or algebraic variety $X$, the sheaf $\mathcal{H}_{1, c, ψ, X, G}$ with formal $c$ and $ψ$ is a universal filtered formal deformation of $\mathcal{D}_X\rtimes G$. To that aim, we first construct quasi-isomorphisms between the Hochschild (co)chain complex of $\mathcal{D}_X\rtimes G$ and the $G$-invariant part of the direct sum over all elements $g$ in $G$ of sheaves of holomorphic differential forms on the cotangent bundles of the $g$-fixed point submanifolds in $X$. Finally, we combine these quasi-isomorphisms with results from the theory of algebraic extensions for sheaves of filtered associative algebras to establish a bijective correspondence between the space of isomorphism classes of filtered infinitesimal deformations of $\mathcal{D}_X\rtimes G$ and the parameter space of $\mathcal{H}_{1, c, ψ, X, G}$.

math.KT↗

arXiv2013

Chain Rules for Smooth Min- and Max-Entropies

The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here we consider the chain rule for the more general smooth min- and max-entropy, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality, but manifests itself as a set of inequalities that reduce to the chain rule for the von Neumann entropy in the i.i.d. case.

quant-ph↗