I've been led to consider certain finitely-generated algebras that arise from some Coxeter groups (finite and affine Weyl groups at least). As a very concrete example, consider the infinite dihedral group $G = \langle s,t : s^2 = t^2 = 1 \rangle$. In this case, I am led to consider the associative unital algebra (over whatever base field $k$): $$A = k \langle u,v : u^3 = u, \ v^3 = v, \ u^2 v^2 = v^2 u^2 \rangle.$$
Has anyone seen or studied such algebras before? I'm interested in its finite-dimensional modules. I can keep chugging... but maybe it has a name?
