“Degenerate” 3-dimensional Sklyanin algebras are monomial algebras

Abstract

The 3-dimensional Sklyanin algebras, Sa,b,c, form a flat family parametrized by points (a,b,c)∈P2−D where D is a set of 12 points. When (a,b,c)∈D, the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are called “degenerate Sklyanin algebras”. C. Walton showed they do not have the same properties as the non-degenerate ones. Here we prove that a degenerate Sklyanin algebra is isomorphic to the free algebra on u, v, and w, modulo either the relations u2=v2=w2=0 or the relations uv=vw=wu=0. These monomial algebras are Zhang twists of each other. Therefore all degenerate Sklyanin algebras have the same category of graded modules. A number of properties of the degenerate Sklyanin algebras follow from this observation. We exhibit a quiver Q and an ultramatricial algebra R such that if S is a degenerate Sklyanin algebra then the categories QGrS, QGrkQ, and ModR, are equivalent. Here QGr(−) denotes the category of graded right modules modulo the full subcategory of graded modules that are the sum of their finite-dimensional submodules. The group of cube roots of unity, μ3, acts as automorphisms of the free algebra on two variables, F, in such a way that QGrS is equivalent to QGr(F⋊μ3).