Hecke algebras of typeAn and subfactors
Hecke algebras of typeAn and subfactors
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An 型赫克代数和子因子
DOI：
10.1007/bf01404457
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发表时间：
1988
影响因子：
3.1
通讯作者：
H. Wenzl
中科院分区：
文献类型：
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作者：
H. Wenzl
V. Jones在他的论文[j - 1]中引入了一个指数，用来“衡量”II1因子中子因子的大小。那篇论文的主要结论是子因子的指标必须大于或等于4，或者它必须等于4cosZ（x//）对于某个l~N， I>3并且所有这些指标值都存在子因子。与子群类似，单指标不能表征子因子直至自同构共轭。只有可数的几个可能的索引值< 4，这一事实似乎与另一个不变量有关。索引小于4的子因子总是有平凡的中心子（或相对交换子），即与子因子的每个元素交换的因子的唯一元素是单位元的倍数。另一方面，在i - j - 1]中给出的索引大于4的子因子的例子都有非平凡的集中器。此外，所有已知的具有平凡相对交换子的子因子的例子都有一个代数整数作为索引。在目前的知识状态下，对于具有平凡中心化器的子因子的指标是否只有可数的可能值仍然是未知的。但是，注意，任意II~因子中具有平凡中心化子因子的所有可能的指标值的集合必须是R的闭子集（见[HW]）。我们最初的动机是研究如何通过AF代数构造超有限因子II1的子因子。给出了一种计算索引的方法，并给出了所构造子因子的中心化器大小的上界。然后将我们的一般结果应用于A，_I型的复Hecke代数H，(q)， n~ n系列。他们的标准发生器gx， g2, -…， gn1满足与对称群S的一组简单反射相同的关系，只不过反射性质g~ = 1被g~ = (q1) g i + q所取代。众所周知，如果q不是单位的根，则H，(q)与C S同构。如果参数是单位的根，则Hn(q)可能不再是简单的，其结构一般是未知的。然而，就子因子而言，这是最有趣的情况。我们定义Ha(q)的表示p使得p（H,(q)）对所有n~ n都是半简单的。一起
In his paper [J-l] V. Jones introduced an index, which 'measures' the size of a subfactor in a II1 factor. The main result of that paper is that the index of a subfactor has to be either greater or equal than 4 or it has to be equal to 4cosZ(x//) for some l~N, I>3 and that there exist subfactors for all these index values. Similarly as for subgroups, the index alone does not characterize the subfactor up to conjugacy by automorphisms. The fact that there are only countably many possible index values < 4 seems to be related to another invariant. Subfactors with index less than 4 always have trivial centralizers, (or relative commutants), i.e. the only elements of the factor which commute with every element of the subfactor are multiples of the identity. On the other hand, the examples given in I-J-l] for subfactors with index greater than 4 all have nontrivial centralizers. Furthermore, all known examples of subfactors with trivial relative commutants have as index an algebraic integer. At the current state of knowledge, it is still unknown whether there are only countably many values possible for the index of subfactors with trivial centralizers. Note however, that the set of all possible index values of a subfactor with trivial centralizer in an arbitrary II~ factor has to be a closed subset of R (see [HW]). Our original motivation for this paper was to study how subfactors of the hyperfinite II1 factor can be constructed via AF algebras. We provide a method of computing the index and we give an upper bound for the size of the centralizer of the constructed subfactor. Our general results will then be applied to the series of complex Hecke algebras H,(q), n~N of type A,_I. Their standard generators gx, g2, -.., gn1 satisfy the same relations as a set of simple reflections of the symmetric group S, except that the reflection property g~ = 1 is replaced by g ~ = ( q 1 ) g i + q . It is well-known that H,(q) is isomorphic to C S , if q is not a root of unity. If the parameter is a root of unity, Hn(q) may no longer bc sernisimple and its structure is not known in general. This is, however, the most interesting case as far as subfactors are concerned. We define representations p of Ha(q) such that p(H,(q)) is semisimple for all n~N. Together with