The Hopf Algebraic Structure of Finitely Supported Currents on a Lie Group

TL;DR

This paper explores the structure of finitely supported de Rham currents on a Lie group, revealing a filtered differential graded Hopf algebra with explicit formulas for algebraic operations.

Contribution

It introduces a Hopf algebraic framework for finitely supported currents on Lie groups, including explicit formulas and the convolution product as a smash product.

Findings

The space of finitely supported currents forms a filtered differential graded Hopf algebra.

Convolution of currents corresponds to a Hopf-algebraic smash product.

Explicit formulas for bundle maps of algebraic operations are derived.

Abstract

The space of de Rham currents supported in finitely many points in a Lie group $G$ has the structure of a filtered differential graded Hopf algebra. The product is given by convolution of compactly supported currents, and the co-product dualizes to wedge product on differential forms. This space arises as the finitely supported sections functor $ \Gamma^{finite} $ applied to the bundle $ \mathcal{U}(G) $ of currents on $ G $ supported at a single (variable) point, and the differential Hopf algebra operations pull back via $ \Gamma^{finite} $ to bundle maps. Explicit formulas for these bundle maps are obtained, and we show in particular that the convolution product takes the form of a Hopf-algebraic smash product.