A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a ...

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8
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1answer
221 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
3answers
671 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
1answer
88 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
1answer
348 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 ...
1
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1answer
135 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 ...
0
votes
0answers
68 views

When is a Surjective Comodule Endomorphism an Automorphism?

Given a Hopf algebra $H$, a left $H$-comodule $V$, and a surjective comodule endomorphism $f: V \to V$. Can somebody give: (i) a set of neccessary, or sufficient, or both neccessary and sufficient, ...
17
votes
2answers
574 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
1answer
209 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
1answer
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
1answer
239 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
0answers
169 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
3answers
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
1answer
201 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
0answers
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
1answer
105 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
1answer
826 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
1answer
218 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
1answer
180 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
1answer
174 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
2answers
360 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
0answers
112 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
6answers
810 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
1answer
173 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
0answers
204 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
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0answers
95 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
1answer
490 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
2answers
569 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
1answer
304 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
2answers
454 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
2answers
574 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
1answer
670 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
7answers
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
1answer
316 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
1answer
811 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
6answers
247 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
2answers
213 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
1answer
623 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
1answer
138 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
0answers
277 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 ...
1
vote
0answers
131 views

There is nonzero primitive element in finite dimensional pointed hopf algebra over C???

I'm huge confused! There is nonzero primitive element in finite dimensional pointed hopf algebra over complex field??? I find in several articles,it is said that a is primitive,so a=0. I will ...
0
votes
2answers
257 views

Dimension of a Hopf algebra == sum of squares of its simple modules? [closed]

when I read an article,I find it seems there is a conclusion like the followings. $H$ is an Hopf algebra(or an abstract group). Then $dimH=\sum_{V:simple ~module ~of ~H}(dimV)^2$. who can tell me ...
2
votes
0answers
186 views

dimension of induced comodule

Let $\pi : G --> H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
1
vote
1answer
298 views

References For Important Hopf Algebras

Where can I find references that discuss important classes of Infinite Hopf Algebras. By important classes, I mean heavily used in research and of relevance to Hopf Algebraist(s),Physicists, ...
7
votes
4answers
2k views

Hopf Algebras and Quantum Groups

I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first ...
1
vote
0answers
247 views

Coideals of Hopf algebra coming from right (left) coideals K->K^+

If $H$ is a Hopf algebra over a field and $K$ is a a right or left coideal of $H$ then $K^+=K\cap\ker\epsilon$ is a coideal of $H$. Does this hold when $k$ is a commutative ring? If not what is a ...
4
votes
1answer
367 views

Coderivations of S(V) correspond to linear maps S(V) -> V. Only over characteristic 0?

Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication $\Delta ...
10
votes
1answer
537 views

2-cocycle twists of braided Hopf algebras

2-cocycle twists of Hopf algebras Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map $$ f: H \otimes H \to k$$ such that $$ f(x_{(1)},y_{(1)})f(x_{(2)} ...
7
votes
1answer
588 views

Yetter--Drinfeld Modules and Braidings

Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory ...
7
votes
1answer
292 views

What are the primitives of Lusztig's twisted bialgebra $\mathbf{f}$?

I'm trying to understand the coproduct on Lusztig's $\mathbf{f}$ and, apart from the Chevalley generators, I don't know of any more primitive elements. In the $q=1$ case, the Weyl group acts as a Hopf ...
9
votes
1answer
406 views

Geometric interpretation of integrals of coordinate rings

If $X$ is an affine scheme over the field $k$ than algebraic invariants of the coordinate ring $k[X]$ usually have a geometric interpretation in terms of $X$ (and vice versa). As an example, the ...