**3**

votes

**0**answers

112 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

141 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 ...

**1**

vote

**0**answers

229 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

204 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

196 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

284 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 ...

**11**

votes

**2**answers

779 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

276 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

275 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

89 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 ...

**7**

votes

**2**answers

322 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

233 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

712 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

89 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

355 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

162 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 ...

**21**

votes

**2**answers

667 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

211 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

102 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

240 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

181 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

202 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

56 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

108 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

900 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

222 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

182 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 ...

**5**

votes

**1**answer

180 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

371 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

116 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?
...

**21**

votes

**6**answers

850 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 ...

**4**

votes

**1**answer

178 views

### Symmetrization for hyperalgebras in positive characteristic

Let $G$ be an algebraic group over an algebraically closed field $k$ of arbitrary characteristic and let $U$ be the hyperalgebra of $G$. Recall that $U$ is defined as the subspace of the full linear ...

**4**

votes

**0**answers

206 views

### $U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...

**1**

vote

**0**answers

96 views

### Group-like Elements in a Coquasitriangular Bialgebra

What do we know about group-like elements in coquasitriangular bialgebras?
In particular, we know that a commutative bialgebra in which all group-like elements are invertible is a Hopf algebra (see ...

**9**

votes

**1**answer

538 views

### The coproduct on the cohomology of a Hopf algebra

If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra.
Is there a reference ...

**3**

votes

**2**answers

615 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.

**0**

votes

**1**answer

311 views

### Frobenius isomorphism for Hopf algebras

It is known that a finite dimensional Hopf algebra $H$ over a field $k$ is a Frobenius algebra. Thus there is an isomorphism $H \cong H^\ast$ of left $H$-modules.
Question: Is it possible to write ...

**5**

votes

**2**answers

464 views

### Primitive elements of a tensor product of bialgebras

Given a field $k$ of characteristic $0$. For every $k$-bialgebra $A$, let $\mathrm{Prim} A$ denote the $k$-vector subspace of $A$ consisting of all primitive elements of $A$.
What conditions can we ...

**8**

votes

**2**answers

606 views

### Is a bialgebra with all group-like elements invertible a Hopf algebra?

We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question :
Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g ...

**8**

votes

**1**answer

681 views

### Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking)

Again, there is a general and a concrete question:
Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it ...

**10**

votes

**7**answers

2k views

### Hopf algebras examples

Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are ...

**4**

votes

**1**answer

320 views

### Is $A$ coflat over $A//B$?

Let $A$ be a Hopf algebra over a field $k$. Let $B$ be a normal subHopf algebra of $A$. Is $A$ coflat over $A//B$? An explanation would be greatly appreciated.
(A novice to Hopf algebras, I am ...

**13**

votes

**1**answer

828 views

### Lagrange's theorem for Hopf algebras

Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras?
I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...

**3**

votes

**6**answers

249 views

### Do stunted exponential series give projections of a cocommutative bialgebra on its coradical filtration?

Let $k$ be a field of characteristic $0$.
Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an ...

**3**

votes

**2**answers

218 views

### Does there exist a canonical “degree” filtration on quantum groups?

For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of ...

**3**

votes

**1**answer

625 views

### Finite connected groups over a perfect field of characteristic p

In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...

**1**

vote

**1**answer

140 views

### Linear functional kills all primitives of a connected filtered coalgebra => it lies in m^2?

Let $C$ be a connected filtered coalgebra over a field $k$. Maybe $k$ has characteristic $0$ (though I don't know where this can be of use). Let $1$ denote the unique element of $C_0$ mapping to $1\in ...

**7**

votes

**0**answers

297 views

### Quantum Braid Group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...