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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2answers
645 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
695 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
323 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
854 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
257 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
243 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
633 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
143 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
321 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
137 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
260 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
203 views

dimension of induced comodule

Let $\pi : G \to H$ 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
326 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
257 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 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
463 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
630 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
600 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
313 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 ...
0
votes
1answer
328 views

pointed Hopf algebra

Is that true that any pointed Hopf algebra is quantum group? Thanks!
9
votes
1answer
417 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 ...
6
votes
1answer
644 views

Distributions on product spaces

I hope this is suitable to MO. Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
1
vote
0answers
333 views

Complement to the Kernel of a Hopf Algebra Map

Given two Hopf algebras $A,B$ over a field $k$, and a Hopf algebra map $\pi:A \to B$, are there any general tricks for finding a complement to the kernel of $\pi$. That is, how can one find a subspace ...
3
votes
1answer
228 views

Image of the Coproduct of a Hopf Algebra

Let $H$ ba a Hopf algebra with coaction $\Delta: H \to H \otimes H$. Denote the action of $\Delta$ by $\Delta (h) = h_{(1)} \otimes h_{(2)}$. I was wondering if every element of $H$ can arise as a ...
11
votes
0answers
512 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in ...
1
vote
1answer
270 views

Compatibility of adjoint action with comultiplication in a Hopf algebra

I'm not especially well-versed in Hopf algebra theory, so apologies in advance if the following question has a very easy answer. Given a Hopf algebra $H$, let $\Delta$ denote the comultiplication, ...
10
votes
1answer
743 views

When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...
2
votes
2answers
330 views

a question about finite dimensional representation of a Hopf algebra

Let $H$ be a Hopf algebra over a field $k$ and $V$ a finite dimensional left $H$-module. Then $End_{k}(V)$ is a right $H$-module via $(f\cdot h)(v)=S(h_{1})f(h_{2}\cdot v)$. We set ...
0
votes
0answers
228 views

Annulator of Tensor Power in a Quantum Group

There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it. Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the ...
2
votes
1answer
634 views

a simple problems about Yetter-Drinfeld-Module

I will be appreciated if anyone can give me some clue for the following simple question, Let $H$,$A$ are both hopf algebras,$\pi :A \rightarrow H$,$\quad f:H\hookrightarrow A$ are both hopf morphism ...
16
votes
7answers
2k views

Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$. We define a ...
5
votes
1answer
432 views

Associated graded of filtered module-algebra over a Hopf algebra

I ran across the following statement in a paper, and it seems fishy to me: Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} ...
0
votes
0answers
251 views

Pure submodules

Is the dual of a pure module also pure? Suppose $H$ is a Hopf algebra over a field $K$ and $S$ is a ring which has a right $H$-action. If ${}_HA$ is a pure left $H$-submodule, will $S\otimes_H A$ be ...
5
votes
1answer
338 views

Is there an interpretation of the “anticommutative” symmetric monoidal structure on $\mathbb{Z}$-graded abelian groups in terms of $\mathbb{G}_m$ actions?

The category of $\mathbb{Z}$-graded abelian groups is equivalent to the category of comodules over the commutative Hopf algebra (over $\mathbb{Z}$) $A=\mathbb{Z}[t,t^{-1}]$, with comultiplication ...
2
votes
0answers
328 views

What happens geometrically when you take associated-graded (or complete, …) of a group ring at its augmentation ideal?

I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
1
vote
1answer
204 views

Dual of a Basis for a Hopf Algebra Conatined in all Dually Paired Hopf Algebras

For an infinite dimensional Hopf algebra $H$, a non-degenerate dually pairing Hopf algebra $H'$, and a choice of basis $e_i$ of $H$, is the dual basis $e^i$ (defined of course by $e^i(e_j) = ...
8
votes
0answers
316 views

Bicommutative Hopf algebras have internal hom objects. What are they?

Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
10
votes
2answers
455 views

An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?

Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...
3
votes
0answers
137 views

Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$. Question: is the graded Hopf algebra ...
10
votes
1answer
464 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
6
votes
1answer
327 views

Is a biideal of a Noetherian Hopf algebra automatically Hopf?

Let $H$ be a Hopf algebra over a field $k$, and $I$ be a biideal of $H$. I am looking for conditions that guarantee that $I$ is a Hopf ideal (that means $S\left(I\right)\subseteq I$). One condition ...
7
votes
4answers
650 views

Twist of a group Hopf-algebra

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
5
votes
1answer
855 views

What kind of structures allow Galois descent?

EDIT: Question solved. Let me explain what I mean. The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion: ...
3
votes
1answer
258 views

Cocyles for abelian extensions

Suppose we have an abelian extension of Hopf algebras, $$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$ According to the general theory there is a left action of $F$ on $G$ and a ...
0
votes
1answer
412 views

How is the coradical filtration defined?

I have seen the coradical filtration of a coalgebra $C$ defined as follows: $C_0 = \text{sum of all simple subcoalgebras of }C$; for any $n\geq 1$, let $C_n$ be $\Delta^{-1}\left(C\otimes ...
1
vote
1answer
167 views

Generators of the Augmentation Ideal (Counit Kernel)

For the Hopf algebra $SL_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i_i-1)$ and $u^i_j$, for $i \neq j$. However, I cannot seem to arrive at at ...
3
votes
2answers
231 views

A Triangular (Non trivial) quasi-Hopf algebra

What would be a good (as "easy" as possible) example of a Triangular (Non trivial) quasi-Hopf algebra? By trivial I mean the quasi strcture not to be trivial, but if the triangular structure is ...
0
votes
1answer
115 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 ...
2
votes
1answer
221 views

Hopf Algebras/Rings, A Question of Terminology

I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in ...