**5**

votes

**0**answers

114 views

### Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra?
In other words, of the following form ...

**11**

votes

**3**answers

481 views

### Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$.
In which cases the conjecture is known ...

**6**

votes

**4**answers

892 views

### Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...

**5**

votes

**2**answers

780 views

### What is a pointed Hopf algebra?

Hi,
I would like to know what pointed Hopf algebras are and why it is that they are important.
Thank you.

**4**

votes

**0**answers

190 views

### Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., ...

**5**

votes

**1**answer

125 views

### 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 ...

**0**

votes

**1**answer

122 views

### Gradings Induced by Coactions?

A "well-known" fact is that a ${C}_q[U(1)]$-coaction on a vector space $V$ induces a $Z$-grading. I don't see why this is so. Clearly, a ${C}_q[U(1)]$-grading will induce a $Z$-grading. But why does ...

**40**

votes

**15**answers

5k views

### What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...

**24**

votes

**2**answers

856 views

### quantum groups… not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...

**22**

votes

**6**answers

940 views

### How to recognize a Hopf algebra?

Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?
I guess in ...

**8**

votes

**1**answer

374 views

### Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question:
Is there some duality known between the quasi Hopf algebra
$D^\omega(H)$ of a finite group $H$ to an orbifold model (such as
SU(2)/$G$ or SO(3)/$G$ orbifold of some ...

**6**

votes

**2**answers

433 views

### If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll give my definitions before ...

**5**

votes

**1**answer

119 views

### 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 ...

**5**

votes

**2**answers

320 views

### Free cocommutative commutative Hopf monoids

I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories.
1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the ...

**3**

votes

**1**answer

317 views

### Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...

**2**

votes

**0**answers

181 views

### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

**3**

votes

**0**answers

278 views

### Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
A subalgebra $I$ of $H$ is called a left coideal subalgebra if $\Delta(I) \subset H \otimes I$.
$H$ is ...

**5**

votes

**1**answer

242 views

### Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...

**1**

vote

**0**answers

74 views

### Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...