Deformation cohomology of Schur-Weyl categories. Free symmetric categories

TL;DR

This paper investigates the deformation cohomology of Schur-Weyl categories, providing explicit computations for free symmetric tensor categories and relating them to Lie algebra invariants.

Contribution

It offers a detailed description of the deformation cohomology for tensor categories generated by one object, especially free symmetric categories, and compares results with Lie algebra invariants.

Findings

Computed deformation cohomology for free symmetric tensor categories.

Established connections between deformation cohomology and exterior invariants of Lie algebras.

Provided explicit formulas for categories with endomorphism algebras free of zero-divisors.

Abstract

The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra.