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18 votes
2 answers
1k views

Why does Drinfeld Unitarization work?

In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
Olivia Borghi's user avatar
5 votes
1 answer
902 views

What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?

For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if $$(R\otimes id)(id\otimes R)(R\otimes id) = (id\otimes R)(R\otimes ...
Jake Wetlock's user avatar
  • 1,144
2 votes
1 answer
276 views

Infinitesimal categories and left duality

I have been reading Kassel's Quantum groups and there is something I can not understand. In Section 4 of chapter $XX$, he introduces the notion of a Infinitesimal symmetric category, that is a ...
Vik S.'s user avatar
  • 437
4 votes
1 answer
445 views

A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

Does anyone have a proof for the following Lemma? Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
cl4y70n____'s user avatar
4 votes
0 answers
103 views

Scaling Yetter--Drinfeld Modules

A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
Nadia SUSY's user avatar
9 votes
2 answers
362 views

Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
Bas Winkelman's user avatar
4 votes
0 answers
310 views

Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
Abo Kutis-Felan's user avatar
7 votes
0 answers
172 views

When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
Noah Snyder's user avatar
  • 28.1k
4 votes
0 answers
134 views

Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...
Janos Erdmann's user avatar
8 votes
2 answers
819 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 ...
S. Carnahan's user avatar
  • 45.7k