Questions tagged [quantum-groups]
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 called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
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Compatibility conditions for Yetter-Drinfeld modules
In the paper, page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is
$$
h_{(1)} v_{(-1)} \otimes h_{(2)}.v_{(0)} = (h_{(1)}.v)_{(-1)}h_{(2)} \otimes (h_{(1)}....
7
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Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...
7
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1
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Is there a good differential calculus for quantum SU(3)?
For quantum $\operatorname{SU}(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $\operatorname{SU}(2)$ by $a$, $b$, $c$, $d$, then the ideal of $\ker(\epsilon)$...
7
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Fusion Classification of $U_q(sl_N)$ Categories
Frohlich and Kerler classify categories with $SU(2)_k$ fusion rules and Kazdhan-Wenzl expand this to $SU(N)_k$ categories. In both cases, unless I am missing something, the classification are ...
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2
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The Irreducible Representations of the Sekine Quantum Groups
Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
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Nontrivial examples of locally compact quantum groups
What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
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How to define $U_q \mathfrak{g}$ without generators and relations?
I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
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Are there attempts to numerically finding algebraic structures over finite-dimensional vector spaces?
By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of ...
7
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0
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What is the kernel of the action of the Iwahori-Hecke algebra?
The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy ...
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Relationship between R-matrix and Casimir element?
Given a simple Lie algebra $\mathfrak{g}$, is there any relation between its Casimir element and the $R$-matrix of the related Yangian $Y(\mathfrak{g})$?
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Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
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Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?
This is a follow up of this question.
Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...
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An alternative Cauchy theorem on Hopf algebras
Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].
We are interesting in an alternative ...
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Is there an E8 symmetry in the zero-field Ising model?
In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules $\sigma^...
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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 ...
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Does the braid group act faithfully on the quantized enveloping algebra?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where $...
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Embedding quantum affine algebras at different levels
For $l$ a positive integer, an affine Lie algebra
$\widehat{\mathfrak{g}}$ has a level $l$ embedding $\phi_l: \widehat{\mathfrak{g}} \longrightarrow \widehat{\mathfrak{g}}$ which takes $x\otimes t^k$ ...
7
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519
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Where can I find tables of dual canonical basis vectors?
Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra.
Now presumably this algorithm has been implemented ...
7
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Explicit Braid Group Reps from quantum SO(N) at roots of unity
This question is related to this one (and indeed the goals are similar).
Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ ...
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Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?
This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the ...
6
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Kontsevich Integral without associators?
Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
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2
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What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied
It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group ...
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Understanding "Decategorified" symplectic Khovanov homology
In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
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Alternative Definition of the Quantum Determinant?
Let $M_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i_j$ for $i,j = 1, \ldots ,N$, and reations
$$
u^i_ju^k_j = qu^k_ju^i_j, \text{ for } i < k, ~~~~~~~ u^...
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Quantum exterior algebra
In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed:
$$
K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i),
$$
with nonzero field elements $q_{i,j}...
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Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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Confusion around the reflection equation algebra
I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
6
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Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?
For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
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Norm of contragredient of unitary representations of compact quantum groups
Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a ...
6
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1
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Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
6
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1
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Corepresentations of Tensor Products of Hopf Algebras
Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
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Bialgebraic structure of Sklyanin algebra
Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
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What vector space does the Kauffman bracket skein algebra of FxI act on?
The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). ...
6
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Nuclearity of $C(\partial_F \mathbb{G})$
Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$....
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Invertible elements of the Hopf algebra quantum $SU(2)$
Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see
https://en.wikipedia.org/wiki/Compact_quantum_group
(Note that on the ...
6
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2
answers
275
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When are the braid relations in a quasitriangular Hopf algebra equivalent?
Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
6
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1
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Weyl Character Formula for Quantum Groups
How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...
6
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2
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522
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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 ...
6
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1
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Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?
Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
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Commutative and Cocommutative Quantum Groups
I am using this definition:
An algebra of functions on a finite quantum group $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
I have the following (very well known --...
6
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1
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Question about unusual highest weight modules for $U_q(sl(2))$
Background
Let $U_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations
$$KK^{-1}=K^{-1}K=1$$
$$Kx^+K^{...
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1
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Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?
I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it.
Question: Given
a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and
...
6
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1
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302
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Compact Quantum Groups and FRT-Algebras
As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...
6
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1
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Relation between TQFT representations and factorizable sheaves
I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks.
More ...
6
votes
2
answers
250
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
6
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1
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172
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How to translate multi-segments to Drinfeld polynomials?
Let $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the ...
6
votes
1
answer
157
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The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$
For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...
6
votes
1
answer
218
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Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
6
votes
1
answer
172
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Matrix model or cocycle twist construction for q-deformations of compact simple Lie groups in $q=-1$?
S. Zakrzewski constructed compact or locally compact quantum groups with deformation parapmeter $q=-1$ for subgroups of $GL(2;C)$, e.g. $SU_{-1}(2)$, $SU_{-1}(1,1)$, $SL_{-1}(2,R)$, as algebras of ...
6
votes
0
answers
118
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How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?
Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...