Samelson products in quasi-$p$-regular exceptional Lie groups

TL;DR

This paper investigates the fundamental Samelson products in the $p$-localization of quasi-$p$-regular exceptional Lie groups, providing a detailed analysis of their triviality or non-triviality for low-rank factors.

Contribution

It determines the (non-)triviality of Samelson products in $p$-localized exceptional Lie groups with low-rank factors, advancing understanding of their multiplicative structures.

Findings

Identifies when Samelson products are trivial or non-trivial in these groups.

Provides explicit results for groups with rank ≤ 2.

Enhances knowledge of the multiplicative structure of $p$-local exceptional Lie groups.

Abstract

There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samelson products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular.