Modified quantum dimensions and re-normalized link invariants
Modified quantum dimensions and re-normalized link invariants
复制标题
修改的量子维度和重新归一化的链接不变量
作者：
Nathan Geer;Bertrand Patureau;V. Turaev
摘要本文利用修正的量子维数，给出了链环的Reshetikhin-Turaev量子不变量的一个重规范化。在单李代数的情况下，这些修改的量子维数与通常的量子维数成比例。更有趣的是，我们给出了两个例子，通常的量子尺寸消失，但修改后的量子尺寸是非零的，并导致非平凡的链接不变量。第一个例子是一类不变量所产生的李超代数以前定义的前两个作者。这些链接不变量是多变量的，并且推广了多变量亚历山大多项式。第二个例子是一个层次的链接不变量所产生的幂零表示的量化$mathfrak {sl}（2）$在一个根的单位。这些不变量包含Kashaev的量子双对数不变量的纽结。
Abstract In this paper we give a re-normalization of the Reshetikhin–Turaev quantum invariants of links, using modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly, we give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example is a hierarchy of link invariants arising from nilpotent representations of quantized $mathfrak {sl}(2)$ at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knots.