A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation
A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation
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DOI：
10.1016/j.aim.2004.03.019
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发表时间：
2005-05
影响因子：
1.7
通讯作者：
W. Rump
中科院分区：
文献类型：
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作者：
W. Rump
已知每个具有Gateva-Ivanova（trans.Amer.Math.Soc.343（1994）203）意义下的生成集X和二项式关系的斜多项式环是整体维数的Artin-Schelter正则整环|X|.此外，每一个这样的环都会产生量子杨-巴克斯特方程的非退化酉集合论解R：X2→ X2，它固定了X2的对角线。Gateva-Ivanova的猜想（Talk at the International Algebra Conference，Miskolc，Hungary，1996）指出，相反地，每个这样的解R来自一个具有二项式关系的斜多项式环。一个等价的猜想（杜克数学杂志100（1999）169）说，基础集合X是R-可分解的。我们证明了这些定理，并构造了一个不可分解的解R，|X| =∞，这表明无限X的扩展为假。
It is known that every skew-polynomial ring with generating set X and binomial relations in the sense of Gateva-Ivanova (Trans. Amer. Math. Soc. 343 (1994) 203) is an Artin-Schelter regular domain of global dimension |X|. Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution R : X2→X2of the quantum Yang-Baxter equation which fixes the diagonal of X2. Gateva-Ivanova's conjecture (Talk at the International Algebra Conference, Miskolc, Hungary, 1996) states that conversely, every such solution R comes from a skew-polynomial ring with binomial relations. An equivalent conjecture (Duke Math. J. 100 (1999) 169) says that the underlying set X is R-decomposable. We prove these conjectures and construct an indecomposable solution R with |X|=∞ which shows that an extension to infinite X is false.