An algebra generated by two sets of mutually orthogonal idempotents

TL;DR

This paper studies a universal algebra generated by two sets of orthogonal idempotents, providing explicit bases, ideal structures, and a linear map with a free subalgebra kernel, advancing understanding of its algebraic properties.

Contribution

It introduces four explicit bases for the algebra, explores their relations, and describes its ideal structure and a key linear map, revealing new structural insights.

Findings

Four explicit bases for the algebra are constructed.

An infinite nested sequence of two-sided ideals is described.

The kernel of a specific linear map is a free subalgebra of rank d.

Abstract

For a field $F$ and an integer $d\geq 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in $A$ looks with respect to our bases. Using our bases we obtain an infinite nested sequence of 2-sided ideals for $A$. Using our bases we obtain an infinite exact sequence involving a certain $F$-linear map $\partial: A \to A$. We obtain several results concerning the kernel of $\partial$; for instance this kernel is a subalgebra of $A$ that is free of rank $d$.