Sklyanin代数与1的立方根
Sklyanin algebras and a cubic root of 1
摘要：
本文考虑具有3个生成元的Sklyanin代数$S$，它是域$\K$上的具有$3$生成元$x,y,z$的二次代数，由$3$关系$pxy+qyx+rzz=0$，$pyz+qzy+rxx=0$和$pzx+qxz+ryy=0$给出，其中$p,q,r\in\K$ .这类代数受到了广泛的关注。特别是，使用工具从代数几何阿廷，泰特和范登伯格\cite{ATV2}表明，如果至少有两个参数$p$，$q$和$r$是非零和至少有两个三个数字$p^3$，$q^3$和$r^3$是不同的，那么$S$是阿廷-Schelter规则。更具体地说，$S$是Koszul，并且具有与3不定式中的交换多项式代数相同的希尔伯特级数。它已成为普遍接受的是，这是不可能实现相同的目标，纯粹的代数和组合手段，如Gröbner基础技术。作者以前已经消除了这种信念。然而，我们以前的证明并不比基于代数几何的证明复杂。它在$S$上的一个合适的单侧模中使用了Gröbner基的构造，并且有相当多的情况要考虑。在本文中，我们展示了一个线性替代后，它成为可能，以确定领导单项式的减少Gröbner基础的理想的关系$S$本身（而不通过一个模块）。我们还发现明确（在参数方面）Sklyanin代数是同构的。新技术的唯一缺点是，如果地面场的特性等于3，它就失败了。
英文摘要：
We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. This class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry Artin, Tate and Van Den Berg \cite{ATV2} showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates. It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Gröbner basis technique. The authors have previously dispelled this belief. However our previous proof was no less complicated than the one based on algebraic geometry. It used a construcion of a Gröbner basis in a suitable one-sided module over $S$ and had quite a number of cases to consider. In this paper we exhibit a linear substitution after which it becomes possible to determine the leading monomials of a reduced Gröbner basis for the ideal of relations of $S$ itself (without passing to a module). We also find out explicitly (in terms of parameters) which Sklyanin algebras are isomorphic. The only drawback of the new technique is that it fails if the characteristic of the ground field equals 3.
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