**4**

votes

**0**answers

178 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}, ..., ...

**1**

vote

**0**answers

114 views

### How simplify the pentagonal equation from two fusion rings?

A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...

**5**

votes

**1**answer

161 views

### Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions.
Let me illustrate by giving this kind of description for the Hopf algebra of symmetric ...

**6**

votes

**0**answers

162 views

### Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning.
Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...

**3**

votes

**0**answers

73 views

### Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...

**5**

votes

**0**answers

223 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

**4**

votes

**1**answer

125 views

### How to make Schubert calculus into a Hopf (actually a PCH) algbera?

The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must ...

**14**

votes

**0**answers

158 views

### Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...

**0**

votes

**0**answers

181 views

### polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
...

**1**

vote

**0**answers

189 views

### universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$,
$U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$
and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...

**11**

votes

**2**answers

596 views

### Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?

Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...

**3**

votes

**0**answers

230 views

### Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible ...

**2**

votes

**1**answer

190 views

### Hopf Duals and Matrix Coefficients

One defines the finite dual of a Hopf algebra $A$ as
$$
H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}.
$$
As is well-known, $H^o$ has a ...

**1**

vote

**1**answer

123 views

### Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra ...

**4**

votes

**2**answers

326 views

### Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has ...

**0**

votes

**0**answers

79 views

### Degree of a commutator in a hyperalgebra or enveloping algebra

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...

**3**

votes

**0**answers

125 views

### Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...

**1**

vote

**0**answers

143 views

### Classification of Hopf algebra with exactly two 1-dimensional modules

Is there a classification of indecomposable non-semisimple finite dimensional Hopf algebras with exactly two 1 dimensional modules? If not, is there one when all simple modules are 1 dimensional and ...

**3**

votes

**0**answers

111 views

### Hopf-Borel theorem over polynomial rings

There is a theorem of Borel saying that:
For a connected cocommutative Hopf algebra $A$ over the perfect field $k$ such that underlying graded vector space is finite-dimensional over $k$ we have that ...

**1**

vote

**0**answers

243 views

### Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by
$$
\Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...

**4**

votes

**1**answer

223 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 ...

**2**

votes

**2**answers

209 views

### Different Hopf algebra structures on same graded algebra

I'm reading Hatcher's book on algebraic topology. In Section 3.C, he proves as Theorem 3C.4 that if $A$ is a graded commutative associative Hopf algebra over a field of characteristic $0$ and $A^n$ ...

**5**

votes

**2**answers

303 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 ...

**12**

votes

**2**answers

841 views

### Hopf Algebra for a physicist

Hello,
for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...

**3**

votes

**2**answers

167 views

### On local parameters at the origin in an algebraic group

Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ ...

**3**

votes

**1**answer

298 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

**1**answer

287 views

### Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$?

When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation ...

**1**

vote

**0**answers

90 views

### Equivalence between Algebraic Semi-group Structures and Coalgebra Structures for an Algebraic Variety?

I was looking at this old question
Hopf algebra and group structure correspondence for algebraic varieties
which says that there exists an equivalence between algebraic group structures on
an ...

**10**

votes

**3**answers

406 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 ...

**8**

votes

**1**answer

258 views

### Hopf algebras and bijective antipodes

By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective.
My question is the following:
Is it true that in every noetherian Hopf algebra the antipode ...

**12**

votes

**3**answers

792 views

### Classification of Hopf algebras (state of the art)

I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right?
My question is the following: What is the current state of the art? What is known about ...

**2**

votes

**1**answer

90 views

### Zero Actions on a Hopf Module Preserved Under the AntiPode?

Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that
$$
m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0?
...

**6**

votes

**1**answer

366 views

### Fusion category and Hopf algebra

Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed ...

**2**

votes

**1**answer

189 views

### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...

**22**

votes

**2**answers

740 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 ...

**2**

votes

**1**answer

227 views

### Non-Faithfully Flat Quantum Homogeneous Spaces

Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form
$$
M:= \lbrace m \in G ~ | ~ m_{(1)} \otimes \pi(m_{(2)}) = m \otimes 1 \rbrace
$$
a ...

**1**

vote

**1**answer

105 views

### $H$-Hopf modules equal the tensor products of their coinvariants with H

In a comment for this old question, it was said that
>
There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with H. (This theorem is usually worded ...

**1**

vote

**1**answer

252 views

### Category of Hopf algebras.

Can you tell me, where I can find a proof of the following fact: Let $R$ be a commutative ring. Consider the category of commutative Hopf algebras over $R$. Then this category is equivalent to the ...

**4**

votes

**0**answers

201 views

### Is the “Toeplitz algebra” the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...

**4**

votes

**3**answers

1k views

### Mistake in Wikipedia Entry “Coalgebra”

Consider the following quote from the Wikipedia entry Coalgebra:
The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$.
I can't ...

**1**

vote

**1**answer

204 views

### Trivial Hopf Coinvariant Subspace Example

For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that
$$
(G \otimes V)^{\text{co}H} = \lbrace g_{(1)} ...

**2**

votes

**0**answers

58 views

### 2-cocycles/Bigalois-objects over nontrivial liftings

It is easy to extend group-2-cocycles to smash-products with Nichols algebras over the group (just trivially). The same certainly doesn't work for nontrivial liftings.
As I would like to check a ...

**0**

votes

**1**answer

119 views

### Coinvariant Subalgebras of Hopf Comodules and Quotients

For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that ...

**16**

votes

**1**answer

1k views

### Connes-Kreimer Hopf algebra and cosmic Galois group

Hi,
I'm interested in the relation between the two following constructions motivated by renormalization:
Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...

**2**

votes

**1**answer

237 views

### Convolution inverse

I'm currently reading some stuff on Hopf algebras, specifically Hopf Algebras: An Introduction by Sorin Dascalescu, Constantin Nastasescu and Serban Raianu. One proof involves showing that for a Hopf ...

**4**

votes

**1**answer

188 views

### Dimension formula for Cartan-type abelian.group Nichols algebra?

Existence of a root system has been established for Nichols algebras $B(V)$ of a Yetter-Drinfel'd-module $V$ (resp. braided vectorspaces $V$) over abelian groups (resp. with diagonal braiding ...

**4**

votes

**3**answers

788 views

### Tensor product of linear mappings versus chain complexes

A chain complex of vector spaces $X_k$ is a sequence of linear mappings
$\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} ...

**5**

votes

**1**answer

187 views

### Liftings of Nichols algebras over racks via Doi twist

As a more nontrivial example for my Dissertation thesis, I'd require some example of the following type (of course I'll "cite" ;-) ), so thanx in advance:
Andruskiewitsch/Grana have by a new ...

**6**

votes

**2**answers

379 views

### A quantum Grothendieck group?

Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendick group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf ...

**1**

vote

**0**answers

122 views

### Hopf Comodule Decompositions and Coinvariant Elements

For $H$ an Hopf algebra, and $V$ a right $H$-comodule with coaction $\Delta_R$, for which there exists a vector space decomposition $V=V_+ \oplus V_-$. Are the following two statements equivalent?
...