An algebra generated by two sets of mutually orthogonal idempotents

Details An algebra generated by two sets of mutually orthogonal idempotents An algebra generated by two sets of mutually orthogonal idempotents

2009-06-21

arxiv.org/abs/0906.3839v1

Mathematics

Rings and Algebras

20 pages

Scientific paper

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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