Modified quantum dimensions and re-normalized link invariants

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 sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

“…The resolution of this problem is to choose a basepoint on the link, and cut the link at that point to yield a .1; 1/-tangle. The idea of cutting basepoints to obtain link invariants is well-established, and appears for example in [19] and explored in a more general context in [7]. Then we can apply the MOY moves until we have a polynomial times a single strand, and define .L/ to be this polynomial.…”

“…, where .L/ 2 C.q/ (see [7] for more discussion on the role of such traces). In fact, one finds that .L/ is a Laurent polynomial and is independent of the choice of tensor factors involved in the trace, and the braid presentation of L (this is essentially [19,Theorem 4.6]).…”

The trace on projective representations of quantum groups

“…The usual construction of quantum invariants do not directly apply to these categories because of the following obstructions: the categories are not semi-simple and have vanishing quantum dimensions. Partial results overcoming these obstructions have been obtained in [9,17,18,19,20,22]. In this paper we generalize some of these results using a new concept called generically semi-simple (loosely meaning the category is graded and semi-simple on a dense portion of the graded pieces).…”

“…The motivation of this paper is to provide the underpinnings for the construction of topological invariants. With this in mind, in this subsection, we recall the notion of re-normalized colored ribbon graph invariants introduced and studied in [19,17,20,21,22]. This subsection is independent of the rest of the paper.…”

“…It follows that C sl (2) , C H sl (2) and C sl (2) are all pivotal C-categories. Moreover, U H is braided, see [19] for the odd case and [20] for the even.…”