On Samelson products in $SU(n)$-and $Sp(n)$

TL;DR

This paper investigates the order of specific Samelson products in special unitary and symplectic groups, providing new results and applications in algebraic topology related to these products.

Contribution

It determines the order of certain Samelson products in $SU(n)$ and $Sp(n)$, offering new insights and applications in the topology of Lie groups.

Findings

Computed the order of Samelson products in $SU(n)$ for $a+b=n+k$

Analyzed the order of Samelson products in $Sp(n)$ involving symplectic quasi-projective spaces

Provided applications to the structure of these products in algebraic topology.

Abstract

Let $a$ and $b$ be two positive integers such that $a, b < n$. We denote the inclusion $\Sigma \mathbb{C}P^a\rightarrow SU(n)$ by $\varepsilon_{a,n}$. Also, let $m$ and $n$ be two positive integers such that $m < n$. This article has two parts. In the first part, we will study the order of the Samelson product $\langle \varepsilon_{a,n}, \varepsilon_{b,n}\rangle$ where $a+b=n+k$, for $k \geq 0$. Also, we will give two applications. In the second part, we will study the order of the Samelson product $S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$, where $Q_{n-m+1}$ is the symplectic quasi-projective space of rank $n-m+1$.