# Classifying algebras with two idempotent generators and involution

Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.

For example, the group algebra of a dihedral group is dihedral in this sense. In my paper Nonstandard braid relations and Chebyshev polynomials, I study a family of dihedral algebras in which $P_1$ and $P_2$ satisfy a fancy version of the braid relation involving Chebyshev polynomials evaluated at $\frac{1}{u + u^{-1}}$, where $K=\mathbb{Q}(u)$.

I am working with a student on generalizing this paper and we have first decided to classify all dihedral algebras. We are willing to assume split semi-simplicity but it would be nice to not require this. We think we can do the classification in the split semi-simple case: all irreducibles are one or two dimensional and there are eight cases depending on which kinds of one-dimensional irreducibles appear.

Since this class of dihedral algebras seems quite natural and the classification has turned out to be more interesting than expected, we are wondering

Have dihedral algebras been studied and/or classified before?

Of course, many examples of dihedral algebras have been studied, but we are wondering if this entire class of algebras has been studied.

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