A note on actions of some monoids

TL;DR

This paper explores how smooth actions of various monoids related to the real and complex numbers can define and unify structures like vector bundles, graded bundles, and supermanifolds, simplifying their understanding.

Contribution

It introduces a unified approach to defining geometric structures via monoid actions, extending previous work to complex, jet, matrix, and supermanifold contexts.

Findings

Reveals connections between monoid actions and classical geometric structures.

Shows how to recover holomorphic and complex vector bundles from monoid actions.

Extends the framework to supermanifolds and matrix actions.

Abstract

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid $(\mathbb{R},\cdot)$ . Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps $(\mathbb{R},0)\rightarrow (\mathbb{R},0)$, smooth action of the monoid of real 2 by 2 matrices and smooth actions of multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.