Modified quantum dimensions and re-normalized link invariants
By Nathan Geer, Bertrand Patureau-Mirand, Vladimir Turaev. Compositio Mathematica, 2007
136 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/0a5def465656f0c1c97852ff548a9b0d4fc58ee5
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Traces on ideals in pivotal categories (2011, background): “, (θX) ∗ = θX∗ for all X ∈ C), then s(V ) = 1 for all V ∈ A (see [6]), and so t∨ = t on A.”
- Generalized trace and modified dimension functions on ribbon categories (2010, methodology): “Also, similar arguments show that the representations of quantum sl2 at a root of unity considered in [29] also fit within our framework.”