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arXiv · 0906.3839

An algebra generated by two sets of mutually orthogonal idempotents

Tatsuro Ito·Paul Terwilliger
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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 $d$.

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Tatsuro Ito, Paul Terwilliger. 2009-06-21. An algebra generated by two sets of mutually orthogonal idempotents. https://arxiv.org/abs/0906.3839

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