Constructing representations of braid groups
Constructing representations of braid groups
复制标题
构建辫子组的表示
DOI：
10.4310/cag.1994.v2.n2.a3
复制
发表时间：
1994
影响因子：
0.7
通讯作者：
D. Long
中科院分区：
文献类型：
--
作者：
D. Long
大量的工作一直致力于辫子群的线性表示的研究，这似乎是一个既丰富又有趣的主题，其应用于纽结理论和统计力学。中心的兴趣是琼斯表示[6]和工作[8]给出了一个同调结构的被加数在这个表示有两个行。在本文中，我们将给出一个结构，它与劳伦斯的结构有关，但比劳伦斯的结构更一般。我们的方法有一些优势。例如，一个基本问题是，这样的表示是否可以被选择为忠实的，并且该表示是否以足够几何的方式构造，以便它能够按照[10]的精神对忠实性问题提供一些见解。另一个优点是计算变得相当常规。该方法的另一个结果是，它将产生具有许多参数的表示。中心思想有两个主要成分，第一个是经典马格努斯结构的概括，第二个也是[8]的核心，是迭代的。这些联合收割机结合起来，给出了一种方法，可以用一种相当不平凡的方式从旧的表示构造新的表示。第一个结果如下。回想一下，我们有一个自然的描述Bn作为一个子群Aut（Fn ∈“，所以我们有一个典型的方式来形成一个分裂扩张Fn xi Bn。该定理使用这个分裂扩展来产生一个表示：
Much work has been devoted to the study of the linear representations of the braid groups, a subject which appears to be both rich and of interest for its applications to knot theory and statistical mechanics. Of central interest is the Jones representation [6] and the work of [8] gives a homological construction for the summands in this representation which have two rows. In this paper we shall give a construction which is related to, but more general than that offered by Lawrence. Our approach has some advantages. For example, one fundamental question is whether such a representation can be chosen to be faithful and the representation is constructed in a way that is sufficiently geometric that it affords some insight into the faithfulness questions in the spirit of [10]. Another advantage is that calculations become rather routine. One further consequence of the method is that it yrill produce representations with many parameters. The central idea has two main ingredients, the first being a generalisation of the classical Magnus construction and the second, which also lies at the heart of [8] is iterative. These combine to give a method for constructing new representations from old in a rather nontrivial way. The first result is the following. Recall that we have a natural description of Bn as a subgroup of Aut(Fn)', so that we have a canonical way of forming a split extension Fn xi Bn. The theorem uses this split extension to produce a representation:
DOI：
--
发表时间：
1990
期刊：
影响因子：
--
作者：
R. Lawrence
通讯作者：
R. Lawrence