Tannakian formalism over fields with operators

Model theory and the Tannakian formalism

We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other. More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We also formulate the notion of a differential tensor category, and a version of the Tannakian formalism for differential linear groups, and show how the same techniques can be used to deduce the analogous results in that context.

Tannakian QFT：从spark代数到量子群

Tannakian QFT: from spark algebras to quantum groups

本文提出一种基于spark代数的非微扰构造，实现拓扑量子场论中线算子范畴的Hopf代数表示，并应用于3d规范理论，获得新结果。

Zariski Closures of Reductive Linear Differential Algebraic Groups

Linear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all i…

Hecke operators on quasimaps into horospherical varieties

Let $G$ be a connected reductive complex algebraic group. This paper is part of a project devoted to the space $Z$ of meromorphic quasimaps from a curve into an affine spherical $G$-variety $X$. The space $Z$ may be thought of as an algebraic model for the loop space of $X$. The theory we develop associates to $X$ a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category $Q(Z)$ of perverse sheaves on $Z$ with the category of finite-dimensional representations of $\ch…

Baxter operator formalism for Macdonald polynomials

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by degenerat…