鈥淒egenerate鈥?3-dimensional Sklyanin algebras are monomial algebras
- 作者：S. Paul Smith smith@math.washington.edu
- 关键词：16S38
- 刊名：Journal of Algebra
- 出版年：2012
- 期刊代码：69_00218693
- 类别：mt
- 出版时间：15 May, 2012
- 卷：358
- 期：Complete
- 页码：74-86
- 文件大小：182 K
摘要
The 3-dimensional Sklyanin algebras, , form a flat family parametrized by points where is a set of 12 points. When , the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are called 鈥渄egenerate Sklyanin algebras鈥? 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 or the relations . 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 , , and , are equivalent. Here 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, , acts as automorphisms of the free algebra on two variables, F, in such a way that is equivalent to .