Centralizer Coalgebras, FRT-Construction, and Symplectic Monoids
Centralizer Coalgebras, FRT-Construction, and Symplectic Monoids
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DOI：
10.1006/jabr.2001.8909
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发表时间：
2001-10
影响因子：
0.9
通讯作者：
Sebastian Oehms
中科院分区：
文献类型：
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作者：
Sebastian Oehms
L. Faddeev， N. Reshetikhin和L. Takhtadjian [RTF]引入了一种构造方法来获得经典群坐标环的量子变形。关于这种所谓的frt构造的一般考虑可以在[Ma， Ta, Ha， Su]和许多关于量子群的教科书中找到。我们的方法与以往的方法有以下三个方面的不同：首先，我们关注的是在构造的第一步出现的梯度矩阵双代数。这意味着我们更愿意研究适当闭单群的量子化，而不是经典群。我们特别关注这些分级双代数的齐次和。这些是可以用对偶方式定义为自同态环上子集的中心化代数的余代数。因此，我们称它们为扶正代数，并研究它们与相应的扶正代数的关系。进一步，我们在任意的诺瑟积分域上作为基环进行研究。这是有意义的，因为所有已知的例子都已经很好地定义在积分劳伦多项式环上，即XX−1（在不定式X中）。我们将看到在场上的理论有很大的不同，特别是关于扶正代数和余代数的比较。例如，扶正器协代数可能具有r -扭转。我们给出了r -投射性的以下判据：当且仅当扶正代数在基变化下是稳定的，这个性质成立。此外，我们将看到后一个性质对于中心化代数总是有效的。
L. Faddeev, N. Reshetikhin, and L. Takhtadjian [RTF] introduced a construction to obtain quantum deformations of coordinate rings of classical groups. General considerations about this so called FRT-construction can be found for instance in [Ma, Ta, Ha, Su] and also in many textbooks on quantum groups. Our approach differs from former ones in the following three aspects:First, we focus attention to the graded matric bialgebra which arises in the first step of the construction. This means that we rather look at quantizations of appropriate closed monoids instead of classical groups. Especially we look at the homogenous summands of these graded bialgebras. These are coalgebras which can be defined in a dual way to centralizer algebras of subsets in an endomorphism ring. We therefore call them centralizer coalgebras and investigate their relationship to the corresponding centralizer algebras. Further, we work over arbitrary noetherian integral domains as base rings. This makes sense since all known examples are already well defined over rings of integral Laurent polynomials, ie, XX− 1 (in the indeterminate X). We will see that there are tremendous differences to the theory over fields, especially concerning the comparison of centralizer algebras and coalgebras. For instance the centralizer coalgebra may have R-torsion. We present the following criterion for R-projectivity: This property holds if and only if the centralizer algebra is stable under base changes. Furthermore, we will see that the latter property is always valid for centralizer coalgebras.