Degenerate 3-dimensional Sklyanin algebras are monomial algebras

TL;DR

This paper proves that degenerate 3-dimensional Sklyanin algebras are monomial algebras, specifically isomorphic to certain zero-relations algebras, and explores their module categories and related quiver and ultramatricial algebra structures.

Contribution

It establishes that degenerate Sklyanin algebras are monomial algebras with specific relations and characterizes their module categories via quivers and ultramatricial algebras.

Findings

Degenerate Sklyanin algebras are isomorphic to monomial algebras with specific relations.

All degenerate Sklyanin algebras have equivalent categories of graded modules.

The categories QGr(S), QGr(kQ), and Mod(R) are equivalent for a fixed quiver Q and ultramatricial algebra R.

Abstract

The 3-dimensional Sklyanin algebras, S(a,b,c), over a field k, form a flat family parametrized by points (a,b,c) lying in P^2-D, the complement of a set D of 12 points in the projective plane, P^2. When (a,b,c) is in D the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are said to be "degenerate". Chelsea Walton showed the degenerate 3-dimensional Sklyanin algebras 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,w, modulo either the relations u^2=v^2=w^2=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 QGr(S), QGr(kQ), and Mod(R), are equivalent; neither Q nor R depends on S. 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.