# Lie groupoids and the Frolicher-Nijenhuis bracket

## TL;DR

This paper explores the structure of multiplicative vector-valued forms on Lie groupoids, demonstrating they form a graded Lie subalgebra and discussing various examples and characterizations.

## Contribution

It introduces the concept that multiplicative vector-valued forms on Lie groupoids form a natural graded Lie subalgebra, extending the understanding of the Frolicher-Nijenhuis bracket.

## Key findings

- Multiplicative vector-valued forms form a graded Lie subalgebra on Lie groupoids.
- Various examples of multiplicative vector-valued forms are discussed.
- Different characterizations of these forms are provided.

## Abstract

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different characterizations of multiplicative vector-valued forms.