Questions tagged [qa.quantum-algebra]

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

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6
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0answers
106 views

Why does the inverse Alexander polynomial appear in the MMR conjecture?

In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
3
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1answer
126 views

Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way. The axioms of Hopf algebras imply that $$ G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1. $$ ...
3
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0answers
97 views

Is there a fusion category not Grothendieck equivalent to a unitary one?

We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) for the notion of (unitary) fusion category. Two fusion categories are Grothendieck equivalent if they have the ...
3
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0answers
60 views

Hopf algebras structure and quantum affine algebras

I'm looking for some information about the Hopf algebras structure and the quantum groups. In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
3
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0answers
45 views

Are the tangle functors based off Khovanov homology braided monoidal functors?

I was wondering if the tangle functors constructed in "A functor-valued invariant of tangles" https://arxiv.org/pdf/math/0103190.pdf "An invariant of tangle cobordisms via subquotients of arc rings" ...
0
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1answer
107 views

Showing a product on a character space is continuous

Quoting from Timmermann's An invitation to quantum groups and duality: Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact quantum group. Then there exists a compact group $G$ ...
8
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1answer
218 views

DW, state sum models, and fully extended TQFTs

I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
4
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0answers
68 views

Is there a coproduct on the Weyl algebra which gives the coproduct on $\mathcal{U}_q(\mathfrak{gl}_2)$?

In the paper Modular Double of Quantum Group, Fadeev gives a presentation of $\mathcal{U}_q(\mathfrak{gl}_2)$ in terms of a Weyl algebra $\mathcal{C}_q$ with generators $w_i, i \in \mathbb{Z}/4$ and ...
4
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2answers
229 views

Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
3
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1answer
87 views

Irreducibility of product bicomodules

Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right, $H$-comodule respectively. The tensor product $$ V \otimes W $$ has an obvious $H$-$H$-bicomodule structure. If $V$ and $W$ are ...
2
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1answer
122 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 ...
3
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0answers
80 views

Quantum dimension in the Drinfeld center

Let $\mathcal{C}$ be a spherical tensor category. It is known that the Drinfeld center of $\mathcal{C}$ is modular (and therefore also spherical), see for example, Corollary 8.20.14 in [1]. Recall the ...
2
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0answers
91 views

Extended cyclotomic criterion for unitary categorification

According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object $a$ of a fusion category over $\mathbb{C}$ is a cyclotomic integer. Now, FPdim($a$) is the maximal ...
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87 views

The simple unitary fusion categories of multiplicity one

Here are two families of simple unitary fusion categories of multiplicity one: $Vec(C_p)$ with $C_p$ the cyclic group of order $p$ (one or prime), The even part of Temperley-Lieb $A_{2n}$ with $n \...
3
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0answers
108 views

Why is the RT invariant from $\mathcal Z(\mathcal C)$ the (norm) square of the one from $\mathcal C$?

The relationship between Turaev-Viro/state-sum invariants and Reshetikhin-Turaev/surgery invariants is roughly that $$\tau_{TV, \mathcal C}(M) = |\tau_{RT, \mathcal C}(M)|^2.$$ Here $\mathcal C$ is a ...
2
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1answer
159 views

Rigidity for the category of comodules over a Hopf algebra

On this page https://ncatlab.org/nlab/show/rigid+monoidal+category there is a discussion of rigidity (left-right duality) for the catagory of modules over a Hopf algebra. What happens if we look at ...
0
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1answer
119 views

Quantum entropy Venn diagrams

We know that in classical information theory the relation between different entropies can be depicted by Venn Diagram as given below. Can we create such Venn-diagrams for the quantum information ...
5
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1answer
175 views

Classification of $\operatorname{Rep}D(H)$

Question Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
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143 views

A fusion ring identity

Fusion rings I'll more or less stick to the presentation given in this question: [1] We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
4
votes
1answer
101 views

Quantum Hamiltonian reduction and tensor products

Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras. Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...
5
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1answer
130 views

Applications of quantum representations of the mapping class group to quantum computers

Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and 1 and 2. The following sources 3 ...
3
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0answers
101 views

Is there a non-irreducible maximal subfactor other than two-sided TLJ?

A subfactor $N \subseteq M$ is called: irreducible if $N' \cap M = \mathbb{C}$, maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$. The two-sided ...
2
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0answers
56 views

Existence of a unitary fusion category with this relation ruled out on finite groups

In this answer, Geoff ruled out the existence of a finite group $G$ such that the fusion category $\mathrm{Rep}(G)$ has simple objects $5_1$ and $7_1$ of FPdim $5$ and $7$ resp., with (for some object ...
5
votes
2answers
293 views

Characters on Hopf algebras

For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
4
votes
1answer
251 views

Drinfeld center of a braided category

Suppose I have a braided monoidal category $\mathcal{C}$. I specifically am interested in the case where $\mathcal{C}$ is the category of finite-dimensional modules of a quantum group, say $\mathcal{U}...
7
votes
2answers
168 views

Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
5
votes
1answer
227 views

What is the smallest rank for a noncommutative fusion ring?

A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying ...
1
vote
2answers
174 views

Is there a noncommutative simple fusion ring?

A fusion ring $\mathcal{F}$ is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying axioms slightly ...
8
votes
3answers
342 views

Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
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0answers
110 views

Are there skein relations for the colored Alexander/ADO invariants?

The Alexander polynomial/Conway potential can be computed as the quantum invariant associated to (a certain quotient of) $\mathcal U_q(\mathfrak{sl}_2)$ for $q = i$. More generally this works for $q = ...
2
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0answers
59 views

Embedding problems on quantum groups?

We work over the field of complex numbers. We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
2
votes
2answers
383 views

Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
3
votes
1answer
114 views

F-symbols for compact Lie groups

Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, ...
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0answers
96 views

Pants algebra $M_n$ as a dagger-special symmetric Frobenius algebra and $CP^*$

I'm looking at the paper Categorical Quantum Mechanics II: Classical-Quantum interaction by Coecke and Kissinger (arxiv link), and I'm having difficulty with one particular aspect. Throughout the ...
5
votes
1answer
383 views

Is there an integral fusion ring which is not of Frobenius type?

Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...
24
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0answers
382 views

What algebraic structure characterizes all natural operations between differential operators and differential forms?

On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms: the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$; ...
3
votes
1answer
297 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 $\...
4
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0answers
96 views

Star product on functions of a Poisson-Lie group

Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected). We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to ...
2
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0answers
223 views

Tracking down an elusive book

A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...
5
votes
1answer
185 views

Jack polynomial and Selberg integral

I am studying the generalisation of the Selberg's integral by using Jack polynomial as outlined by Forrester and Warnaar (arXiv: 0710.3981). They write down the symmetric Jack polynomial as \begin{...
4
votes
0answers
158 views

A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras

The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties: maximal, it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
5
votes
0answers
97 views

Homotopy transfer of cyclic L-infinity algebras

Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
5
votes
1answer
220 views

Non-associative deformation quantization

Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
4
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0answers
228 views

Are vertex operator algebras ever conspiratorial?

I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
8
votes
1answer
324 views

Image of Comultiplication on Finite Quantum Groups/Hopf Algebras

Let $A=:F(G)$ be the algebra of functions on a finite quantum groups aka a finite dimensional C*-Hopf Algebra. Suppose that $F(G)$ is neither commutative nor cocommutative. In their 1966 paper Kac ...
4
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0answers
106 views

Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$

I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial. The results was ...
10
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0answers
767 views

Conjectures of Peter Scholze about q-de Rham complex: examples

Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
11
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0answers
247 views

Is every finite quantum group a quantum symmetry group?

This post is basically a quantum extension of Is every finite group a group of “symmetries”? Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem ...
8
votes
2answers
308 views

q-difference equations and quantum mechanics

I have been trying to understand why the term quantum is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...). ...
17
votes
2answers
2k views

Quantum corrections to geometry

In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...

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