# Rank $n$ swapping algebra for Grassmannian

## TL;DR

This paper establishes a connection between the Poisson algebra of the Grassmannian and the rank n swapping algebra, providing a new algebraic framework for understanding Grassmannian structures.

## Contribution

It introduces an injective Poisson homomorphism from the Grassmannian's Poisson algebra to the rank n swapping algebra, linking geometric and algebraic structures.

## Key findings

- Constructed an injective Poisson homomorphism.
- Linked Grassmannian Poisson algebra with swapping algebra.
- Enhanced understanding of algebraic structures on Grassmannians.

## Abstract

The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$ is its geometric mode. In this paper, we find an injective Poisson homomorphism from the Poisson algebra on Grassmannian $G_{n,r}$ arising from boundary measurement map to the rank $n$ swapping fraction algebra.