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Questions tagged [qa.quantum-algebra]

Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

831 questions
24
votes
1 answer

Question about the Yangian

I've a slightly technical question about the Yangian which I'm hoping an expert out there can answer. Recall that the Yangian $Y(\mathfrak{g})$ is a Hopf algebra quantizing $U(\mathfrak{g}[z])$. Drinfeld, in his quantum groups paper, explains…
Kevin Costello
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11
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3 answers

Are the “identity object axioms” in the definition of a braided monoidal category needed? (Answered: No)

I am asking because the literature seems to contain some inconsistencies as to the definition of a braided monoidal category, and I'd like to get it straight. According Chari and Pressley's book ``A guide to quantum groups," a braided monoidal…
Peter Tingley
  • 603
11
votes
1 answer

Automorphic forms and quantum groups

The paper Eisenstein series and quantum affine algebras by Kapranov makes contact between automorphic forms and quantum groups. I haven't found even one other paper devoted to this theme. Have other authors come at this, perhaps from other…
David Feldman
  • 17,466
10
votes
2 answers

The other classical limit of a quantum enveloping algebra?

Let $\mathbb K$ be a field (of characteristic 0, say), $\mathfrak g$ a Lie bialgebra over $\mathbb K$, and $\mathcal U \mathfrak g$ its usual universal enveloping algebra. Then the coalgebra structure on $\mathfrak g$ is equivalent to a co-Poisson…
Theo Johnson-Freyd
  • 52,873
5
votes
3 answers

q-deformation of the permutation group?

The only definition of a quantum group I know of involves q-deforming the relation $EF-FE=H$ or for SL(2): \[ \left[ \left( \begin{array}{cc} 0 & 1 \\\\ 0 & 0 \end{array} \right), \left( \begin{array}{cc} 0 & 0 \\\\ 1 & 0 \end{array} \right)…
john mangual
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5
votes
1 answer

Classical limit and Drinfelds realization of quantum groups

Let $$\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\oplus\mathbb{C}d$$ be untwisted affine Lie algebra (as defined in V.G.Kac, Infinite-Dimensional Lie Algebras, 3d ed. Cambridge University Press, 1990). For…
Neph
  • 51
4
votes
0 answers

Intuition for pointed Hopf algebras

I have familiarized myself with various definitions (one-dimensionality of simple left comodules, generated as an algebra by group-like and skew-like elements...) and examples of pointed Hopf algebras (small quantum groups, universal enveloping…
crackplot
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3
votes
2 answers

A property of quantum group R matrices?

Assume Q is a quantum Lie group which allows a R matrix (with the usual quantum Yang-Baxter equation). Take the Jordan normal form J of R. Does a Q exist with offdiagonal J elements (i.e. R has defective eigenvalues)?
Hauke Reddmann
  • 4,701
3
votes
1 answer

Are Quantum Clebsch-Gordan coeffients quantum group dependent?

Background: I'm a quantum chemist and even once wrote a proggie for 3j and 6j symbols. Imagine my gaping mouth when I read the paper (Reshetikhin/Turaev, I think) who let them pop up in my fave, knot theory. My ultimate goal would be to rewrite my…
Hauke Reddmann
  • 4,701
3
votes
0 answers

Embedding Quantum SL(2) into the Quantum Matrices

Let $M_q[2]$ be the algebra of quantum matrices over the complex numbers with the usual generators $a,b,c,d$ and the relations $ab = qba$, ... etc. Moreover, let $SL_q(2)$ be the quotient of $M_q(2)$ by the ideal generated by det$_q-1$, where det$_q…
John McCarthy
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3
votes
2 answers

The relations between quantum affine algebras and Yangians

Drinfeld had said that quantum loop algebras can degenerate into Yangians.But i can not find any proof about that .Can anyone give me some reference about this question.
Ming L
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3
votes
0 answers

Quotient of the free Poisson algebra

Assume is the free Poisson algebra on the set of generators $=\{_{1},_{2},…,_{}\}$. It is well-known that is the polynomial algebra with infinitely many generators $y_{1}$, $y_{2}$, ... where $_{1}=_{1}$,…,$_{}=_{}$,$_{+1}=[_{1},_{2}]$,... is a…
user100
  • 31
3
votes
1 answer

Pictorial explanation of Dynkin index and quadratic Casimir?

Draw a colored loop. You explained quantum dimension. :-) OK, now you may use trivalent nodes and knot crossings too (with edges colored by irreps). You can now pictorialize 6j symbols, writhe normalizers, structure constants etc. My question is…
Hauke Reddmann
  • 4,701
2
votes
1 answer

associative Yang-Baxter on U(g)

Consider $\mathfrak{g}$ a finite-dimensional Lie algebra over the field $\textbf{k}$. If A is an associative algebra, we are searching for functions from $\textbf{C}\times \textbf{C}$ to $A\otimes A$ such…
bob
  • 21
2
votes
0 answers

Changing the sign in the definition of the cocommutator of a coboundary Lie bialgebra

A Lie bialgebra is a Lie algebra additionally equipped with a 1-cocycle $\delta: {\mathfrak g}\to \Lambda^2 {\mathfrak g}$ that satisfies the co-Jacobi identity. Non-trivial Lie bialgebras can be obtained from trivial ones (with zero cocommutator…
Kirill Krasnov
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