# Questions tagged [qa.quantum-algebra]

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

649
questions

**6**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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-...

**3**

votes

**0**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**3**answers

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,...

**1**

vote

**0**answers

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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, ...

**1**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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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**0**answers

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

**2**answers

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

**2**answers

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 ...