Middle convolution for Lie algebra representations

TL;DR

This paper develops a Lie algebra version of the middle convolution functor, connecting it to existing concepts like the Long-Moody functor and hyperplane arrangement complements.

Contribution

It introduces a Lie algebra analogue of middle convolution, linking it to various algebraic and geometric structures and establishing a Riemann-Hilbert correspondence.

Findings

Generalizes the Long-Moody functor for Lie algebras.

Recovers Dettweiler-Reiter middle convolution for Fuchsian systems.

Shows compatibility with Haraoka's middle convolution for hyperplane arrangements.

Abstract

This paper introduces a Lie algebra analogue of the middle convolution functor, which is defined on the category of modules over certain Lie algebras, including, as typical motivating examples, free Lie algebras, Drinfeld-Kohno Lie algebras, and the holonomy Lie algebras of complements of hyperplane arrangements. First, we demonstrate that the middle convolution for Lie algebra representations can be regarded as a natural generalization of the infinitesimal analogue of the Long-Moody functor for Drinfeld-Kohno Lie algebras. Second, we show that our middle convolution recovers the Dettweiler-Reiter additive middle convolution for Fuchsian systems on the punctured Riemann sphere as a special case. Furthermore, we show that when applied to the holonomy Lie algebra of the complement of a hyperplane arrangement, our functor is compatible with Haraoka's middle convolution for logarithmic connections on such complements. Finally, we establish a Riemann-Hilbert correspondence between the middle convolution for the holonomy Lie algebra and the middle convolution for local systems on complements of hyperplane arrangements.