**7**

votes

**4**answers

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

**0**answers

253 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

**1**answer

383 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

**1**answer

548 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

**1**answer

589 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

**1**answer

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 ...

**0**

votes

**1**answer

326 views

**9**

votes

**1**answer

409 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

**1**answer

566 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

**0**answers

328 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

**1**answer

224 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

**0**answers

495 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

**1**answer

233 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

**1**answer

684 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

**2**answers

324 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

**0**answers

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

**1**answer

615 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

**7**answers

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

**1**answer

413 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

**0**answers

245 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 ...

**4**

votes

**1**answer

304 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

**0**answers

276 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

**1**answer

201 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

**0**answers

291 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

**2**answers

432 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

**0**answers

136 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

**1**answer

444 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

**1**answer

314 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

**4**answers

612 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

**1**answer

766 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

**1**answer

250 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

**1**answer

346 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

**1**answer

151 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

**2**answers

228 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

**1**answer

113 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

**1**answer

218 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 ...

**3**

votes

**1**answer

279 views

### Does There Exists a General Quantum Casimir Extending the $U_q({\mathfrak sl}_2$ Case?

As is well known (see Kassel), when $q$ is not a root of unity, the centre or the quantum enveloping algebra $U_q({\mathfrak sl}_2)$ of ${\mathfrak sl}_2$ is generated by the element
$$
C_q = EF + ...

**13**

votes

**1**answer

608 views

### The Major Families of Quantum Groups

If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups?
Of couse, to start with we have the h-adic completions ...

**3**

votes

**1**answer

445 views

### Prove that $U^*U=UU^*=1$ for $U_q(N,C)$

Let $u^i_j$, $i,j = 1, . . . N$, and det$_q^{-1}$ be the standard generators of the quantum group $U_q(N,C)$, and define the matrices $U$ and $U^{\ast}$ by setting $U_{ij} := u^i_j$ and ...

**2**

votes

**1**answer

239 views

### Hopf algebra and group structure correspondence for algebraic varieties

Let $V$ be a real algebraic variety and let ${\cal O}(V)$ denote its algebra of regular functions. If we put a group structure on $V$ (not necessarily an algebraic group structure) it will induce a ...

**4**

votes

**2**answers

262 views

### Sub-Hopf algebras of group algebras

Let $k$ be a field and $G$ a finite group. Is every sub-Hopf algebra over $k$ of the group algebra $k[G]$ of the form $k[U]$ for a subgroup $U$ of $G$ ?

**2**

votes

**2**answers

336 views

### The relation $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$

In the Hopf algebra $SL_q(N)$, it can be shown, using direct calculations, that $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$. Can anyone see a more elegant way of establishing this?
Moreover, does anyone ...

**8**

votes

**1**answer

461 views

### Hopf algebras as cohomology of $\mathbb{CP}^\infty$, $\Omega S^3$ and related $H$-spaces

Let me begin by a couple of questions :
Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra ...

**5**

votes

**1**answer

254 views

### Do dualizable Hopf algebras in braided categories have invertible antipodes?

A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided ...

**1**

vote

**1**answer

1k views

### A question about the smash product

Let $K$ be an algebraically closed field and $H$ a finite dimensional semisimple-cosemisimple Hopf $K$-algebra, and let $A$ be a finite dimensional $H$-module algebra and $H^{\ast}= Hom_{K}(H, K)$.
...

**10**

votes

**1**answer

482 views

### Is there a “correct” general setting for the principle: “tensoring any object with a projective object yields another projective”?

Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...

**1**

vote

**0**answers

134 views

### Duflot-type theorem for Hopf algebras ?

In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.
Is there a ...

**3**

votes

**3**answers

671 views

### Innocent question on tensor products of modular representations

Let $K$ be a field (of course, of positive characteristic, unless you want a trivial question). Let $G$ be a finite group, and $V$ and $W$ be two completely reducible (finite-dimensional) ...

**1**

vote

**1**answer

409 views

### The Killing Form for Co-Quasi-Triangular Hopf Algebras

For a co-quasi-triangular Hopf algebra $H$, with universal $r$-form $r$, there exists an important map $Q$ defined by
$$
Q:H \otimes H \to k, ~~~~~~h \otimes g \mapsto r(g_{(1)}\otimes ...

**0**

votes

**1**answer

120 views

### Action of Co-quasi-triangular Universal r-form on $a \otimes 1$

A very easy question I can't seem to answer: For a universal r-form on a co-quasi-triangular Hopf algebra why is $r(a \otimes 1) = r(1 \otimes a) = \epsilon(a)$?