A global theory of algebras of generalized functions

TL;DR

This paper develops a geometric, intrinsic construction of the Colombeau algebra of generalized functions on manifolds, preserving differential structure and embedding distributions while maintaining compatibility with Lie derivatives.

Contribution

It introduces a global, differential algebraic framework for Colombeau algebras on manifolds using convenient vector spaces, unifying local and global theories.

Findings

Constructed an intrinsic algebra $ ilde{ ext{G}}(M)$ on manifolds.

Embedded distributions into the algebra preserving Lie derivatives.

Ensured the algebra contains smooth functions as a faithful subalgebra.

Abstract

We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of $\hat{\mathcal G}(M)$. $\hat{\mathcal G}(M)$ is a{\em differential} algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ${\mathcal D}'(M)$ into $\hat{\mathcal G}(M)$ that renders ${\mathcal C}^\infty (M)$ a faithful subalgebra of $\hat{\mathcal G}(M)$. Finally, it is shown that this embedding commutes with Lie derivatives. Thus $\hat{\mathcal G}(M)$ retains all the distinguishing properties of the local theory in a global context.