Questions tagged [qa.quantum-algebra]
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
831
questions
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Fermionic Wick Theorem
Assume we are given are fermionic many-particle system with creation operators $c_1^\dagger,c_2^\dagger,...$ acting on some Hilbert space $\mathcal{H}$. That is, the $c_i^\dagger$ and their ...
8
votes
1
answer
526
views
q-analog of a combinatorial identity involving binomial coefficients
Using, e.g., properties of iterated finite differences it is easy to show that for any pair of integers $n$ and $m$ with $n>\!>m$ one has the identity
$$
\sum_{k=0}^m(-1)^{k-m} {n-k\choose m}{m\...
6
votes
1
answer
385
views
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 ...
4
votes
0
answers
159
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Is the Nichols-Richmond theorem true for integral fusion rings?
The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper.
It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here):
Theorem: ...
3
votes
0
answers
88
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Antipode action on quantum minors
Let ${\cal O}(SU_q(n))$ be the standard $q$-deformed coordinate algebra of $SU(n)$, with the canonical generators $x_{i,j}$. For $I = \{i_1,\ldots, i_r\}, J=\{j_1,\ldots,j_r\}\subseteq \{1, \ldots,n\}$...
3
votes
0
answers
245
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Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)
To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post).
A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together ...
2
votes
0
answers
59
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CP maps or states on the matrix quantum group $C_q[SU_2]$
This question is about the states on the matrix quantum group $C_q[SU_2]$ (generators $a,b,c,d$ with relations...), or possibly about the representations of the $C^*$ algebra $C_q[SU_2]$ - not about ...
6
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0
answers
259
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Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?
A fusion ring is a finite dimensional $\mathbb{Z}$-module
$\mathbb{Z}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^...
2
votes
0
answers
69
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Comodules of the $B,C$ and $D$ series quantum groups
In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
4
votes
2
answers
461
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Does a nonabelian Picard group exist?
Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=...
7
votes
0
answers
161
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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 ...
5
votes
1
answer
413
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Crystal basis for quantum groups and Lie algebras
Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...
9
votes
0
answers
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Are the quantum groups $C_q[SU_{1,1}]$ and $C_q[SL_{2}(R)]$ isomorphic?
Classically the group of Moebius transformations of the unit disk and Moebius transformations of the upper half plane are isomorphic, as the unit disk and upper half plane are transformed into each ...
2
votes
0
answers
92
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Simple modules of quantum toroidal algebras
Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...
14
votes
0
answers
525
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What is the (quasi-) classical limit of categorified quantum groups?
$\newcommand{\g}{\mathfrak g}$
Let $G$ be a reductive group and $U_q(\g)$ the associated quantum group. One can argue that the classical limit of $U_q(\g)$ is $G$ or $\g$, with some Poisson structure, ...
21
votes
0
answers
463
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What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?
Let $\mathfrak g=\mathfrak{sl}(2)$.
Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness.
Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
5
votes
0
answers
270
views
Deformation quantization of Poisson bracket without star-product
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
6
votes
1
answer
264
views
Bialgebraic structure of Sklyanin algebra
Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
5
votes
2
answers
247
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Comodules of Cosemisimple Hopf Algebras
A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
2
votes
1
answer
191
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When are Morita classes represented by certain structured algebra objects?
Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...
10
votes
2
answers
814
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Computing in quantum groups
I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by ...
6
votes
2
answers
275
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When are the braid relations in a quasitriangular Hopf algebra equivalent?
Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
6
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
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...
2
votes
2
answers
545
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Definition of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...
5
votes
0
answers
185
views
Modular double of elliptic quantum group
By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...
2
votes
0
answers
65
views
How to value $\Omega$ in T-system for twisted quantum affine algebras?
Let us proceed to the unrestricted T-systems. Choose $h\in {\mathbb{C}\backslash 2\pi \sqrt{-1} \mathbb{Q}}$ arbitrarily.
The unrestricted T-system for $U_{q}(X_{N}^{(\mathfrak{k})})$ is the following ...
3
votes
0
answers
170
views
Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
6
votes
1
answer
910
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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 ...
7
votes
1
answer
406
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Bounding $p$-adic characters and Jacquet-Langlands transfert
I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_\pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on ...
3
votes
1
answer
160
views
How to obtain an quantization of the algebra of functions of a given space?
My question is inspired by quantum spheres and quantum projective spaces (studied by Dijkhuizen & Noumi, Vaksman & Soibelman, etc., i.e. "A FAMILY OF QUANTUM PROJECTIVE SPACES AND RELATED $q$-...
5
votes
0
answers
165
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Distinguishing the Duflo star product
$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$
For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of ...
4
votes
0
answers
319
views
Quantization of $S^2$ as $C^*$-algebra?
The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...
9
votes
0
answers
256
views
Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
8
votes
1
answer
259
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Classifying Low Dimensional Solutions of the Yang--Baxter Equation
What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions?
To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
4
votes
1
answer
265
views
Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?
Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...
7
votes
0
answers
314
views
Flat connection from gauged WZW model
$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...
4
votes
0
answers
121
views
Cosemi-simple FRT Hopf Algebras
This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
7
votes
2
answers
549
views
How to make a premodular category a modular tensor category?
A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...
21
votes
4
answers
2k
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Why are quantum groups so called?
I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
6
votes
1
answer
348
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Open questions on (finite) tensor categories
I would like to know about problems on (finite) tensor categories. I have read Etingof´s notes from his course at MIT. I have a question:
There exists any reference where I can find an open problem ...
6
votes
1
answer
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 ...
8
votes
0
answers
198
views
When are the categories of algebras over props (co)complete?
Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...
7
votes
1
answer
212
views
6j symbols of SU(4) at level 4
Does anybody know of a reference that gives the (quantum) 6j symbols of SU(4) at level 4?
Alternatively, I know the S-matrix and the fusion rules, in the form
$a \times b = \sum_i N^{ab}_{c_i} c_i$
...
7
votes
2
answers
402
views
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,...
7
votes
0
answers
329
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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 ...
3
votes
2
answers
927
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Appropriate Recursion relations for Wigner 3j Symbols
I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...
5
votes
1
answer
200
views
What are the applications of the depth 2 reduction to the subfactors theory?
Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth ...
2
votes
0
answers
151
views
The role of the Vandermonde determinant in representations of affine Lie algebras
I am reading a paper 'Yangians and R-matrices' by Chari & Pressley (1990) and to classify representations for particular quantum groups, they define a "quantum Vandermonde determinant". They also ...
4
votes
0
answers
88
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Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?
This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...
0
votes
0
answers
89
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Deforming the category of representations of the Yangian of a simple Lie algebra?
Since I got a very good answer to my previous question,
217585,
I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...