Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are ...

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6
votes
2answers
311 views

How do quantum knot invariants change when I pick a funny ribbon element?

So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...
3
votes
4answers
740 views

An inner product that makes the R-matrix unitary

So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...
2
votes
3answers
356 views

What's the best reference for actual formulas for RT invariants?

If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference ...
3
votes
3answers
430 views

What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
4
votes
2answers
420 views

Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
2
votes
2answers
383 views

How does one think about the “off-diagonal” part of the R-matrix?

The universal R-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and ...
3
votes
1answer
296 views

What is the “right” hermitian structure on tensor products of quantum group representations?

This is pretty specific, but there are some experts around. So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...