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.
34 questions from the last 365 days
13
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Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
0
votes
0
answers
32
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Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
0
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0
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92
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Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
13
votes
0
answers
332
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Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
3
votes
0
answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
6
votes
0
answers
349
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Quantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum ...
3
votes
0
answers
86
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Tilting modules for quantum groups in non-regular blocks
Let $U_q(\mathfrak{g})$ denote Lusztig's quantum group associated to $\mathfrak{g}$ and $q$, where $\mathfrak{g}$ is a finite dimensional simple Lie algebra over $\mathbb{C}$ and $q \in \mathbb{C}^{\...
- 10.5k
4
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0
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183
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Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}...
- 255
0
votes
0
answers
44
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Categorical duals for Yetter-Drinfeld modules [duplicate]
Yetter-Drinfeld (YD) modules appear naturally in the theory of Hopf algebras. They are both modules and comodules at the same time, satisfying a certain compatibility condition, as presented here. The ...
1
vote
1
answer
64
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What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 137
2
votes
0
answers
132
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A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
- 137
1
vote
0
answers
54
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Coproduct on $U_q(sl_2)$
Recall that $U_q(sl_2)$ is the quotient of the free associative $\mathbb{Q}(q)$-algebra on generators $E$, $F$, $K^{\pm 1}$ such that $KE = q^2 EK$, $KF = q^{-2} FK$, and $[E,F] = \frac{K - K^{-1}}{q -...
- 11
9
votes
0
answers
144
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Ringel's interpretation of quantum groups as Hall algebras at $q=1$
Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
- 3,446
0
votes
0
answers
124
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Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
2
votes
0
answers
49
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Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
- 141
3
votes
0
answers
60
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$G$-crossed (braided) fusion categories and Tannaka duality
Many important concepts in tensor category theory have their counterpart in Hopf algebra theory under Tannaka duality. They have the general form: let $A$ be an XX-algebra, and let Rep$(A)$ denote the ...
- 727
3
votes
0
answers
41
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Tools or software for calculations in Quantum Affine Algebras
I am curious if there's a tool/math software to help with calculations in Quantum Affine Algebras, specially using the Drinfeld realization, i.e. the realization in Theorem 4.7 of Beck. For example I ...
- 31
2
votes
1
answer
78
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Does there exist a nontrivial triangular weak Hopf algebra?
Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair
($H,\mathcal{R}$) where $H$ is a WHA and
$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...
- 727
3
votes
0
answers
74
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Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.
Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
- 143
4
votes
0
answers
68
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Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
- 727
7
votes
2
answers
201
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Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?
Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let
$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
- 727
1
vote
0
answers
70
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Affiliating the whole algebra of 'coordinates' with a locally compact quantum group
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
- 143
6
votes
2
answers
198
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Proof that every commutative locally compact quantum group arises from a locally compact group
It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof ...
- 143
3
votes
1
answer
165
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quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras
In a nutshell, my question is:
Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?
I will now make it more precise. One could define a ...
- 213
4
votes
0
answers
82
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Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?
Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined as
$$...
- 727
4
votes
0
answers
95
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Ribbon fusion categories for quantum $\mathfrak{sl}_2$ at odd roots of unity
I will work over $\mathbb{C}$. Let $q=e^{2\pi i/N}$, and write $U_{q}(\mathfrak{sl}_2)$ for Lusztig's divided power quantum group for $\mathfrak{sl}_2$.
One can associate to $U_{q}(\mathfrak{sl}_2)$ a ...
3
votes
0
answers
107
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How to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac Fermions using $U(N \cdot k)_1 / SU(k)_N$?
It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A ...
- 545
13
votes
0
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573
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Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
4
votes
0
answers
165
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Coloured Jones polynomial at 4th root of unity and Arf invariant
Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
- 81
8
votes
1
answer
331
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Some fusion rings/categories I don't recognize
Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
- 293
5
votes
1
answer
154
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Explicit correspondence between classical double and quantum double
Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim.
Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be ...
- 255
2
votes
0
answers
78
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Quantum Schubert cell algebra and quantum odd-dimensional euclidean space
De Concini, Kac, Procesi introduced quantum Schubert cell algebra associated to a complex Lie algebra $\mathfrak{g}$ which is denoted by $\mathcal{U}^{w}_{\epsilon}$ where $w$ is an element of Weyl ...
4
votes
0
answers
168
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Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
- 73
7
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0
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164
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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?
- 3,816