Integrality and symmetry of quantum link invariants
Integrality and symmetry of quantum link invariants
复制标题
量子链接不变量的完整性和对称性
DOI：
10.1215/s0012-7094-00-10224-4
复制
发表时间：
2000
影响因子：
2.5
通讯作者：
Thang T. Q. Lê
中科院分区：
文献类型：
--
作者：
Thang T. Q. Lê
0.导论.其分量被简单李代数g的模着色的框架链接的量子不变量是v1/D中的劳伦多项式（具有整数系数），其中v是量子参数，D是依赖于g的整数。我们表明，量子不变量，适当的归一化，是洛朗多项式在v2。我们还建立了量子链不变量在单位根处的两个对称性质。第一个断言，量子链路不变量，在第r个单位根，是不变的仿射Weyl群Wr的作用下，它的作用在重量格。Wr的基本域是基本凹室Cr，一个单形。设G是对应的单连通复李群的中心。G对Cr有自然作用。第二个对称性质，在其最简单的形式，断言量子链接不变量是不变的G的作用下，如果链接有零链接矩阵。第二个对称性性质将Kirby和Melvin（sl 2情形）和Kohno和Takata（sln情形）的对称性原理推广到任意单李代数。
0. Introduction. Quantum invariants of framed links whose components are colored by modules of a simple Lie algebra g are Laurent polynomials in v1/D (with integer coefficients), where v is the quantum parameter and D an integer depending on g. We show that quantum invariants, with a suitable normalization, are Laurent polynomials in v2. We also establish two symmetry properties of quantum link invariants at roots of unity. The first asserts that quantum link invariants, at rth roots of unity, are invariant under the action of the affine Weyl group Wr , which acts on the weight lattice. A fundamental domain of Wr is the fundamental alcove Cr , a simplex. Let G be the center of the corresponding simply connected complex Lie group. There is a natural action of G on Cr . The second symmetry property, in its simplest form, asserts that quantum link invariants are invariant under the action ofG if the link has zero linking matrix. The second symmetry property generalizes symmetry principles of Kirby and Melvin (the sl2 case) and Kohno and Takata (the sln case) to arbitrary simple Lie algebra.