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 structure exists on the group algebra $k G$ of a finite group $G$.
Questions tagged [hopf-algebras]
567 questions
11
votes
1 answer
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 is bijective?
warren
- 275
8
votes
2 answers
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.
Albert
- 99
5
votes
1 answer
If a Hopf Algebra has a nontrivial, finite-dimensional right ideal, then it is finite dimensional
Let $H$ be a Hopf Algebra (over a field $K$), with comultiplication $\Delta$, counit $\varepsilon$, and antipode $S$.
A $K$-subspace V is said to be:
A right ideal if $VH \subseteq V$
A right coideal if $\Delta (V) \subseteq V \otimes H$ (some…
de wa nai
- 53
4
votes
2 answers
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 trivial it's ok (it's actally better for me)
4
votes
1 answer
Cobraided and coquasitriangular Hopf algebras
In a Hopf algebra, are the notions "cobraided" and "coquasitriangular" the same? Kassel (Hopf algebras) uses "cobraided" and Montgomery (HAs and their actions on rings) uses the other -- both on page 184. See the note at the top of Kassel page 174.…
Paddychut
- 43
3
votes
0 answers
connected Hopf algebra of infinite Gelfand-Kirillov dimension but of finite dimensional primitive space
I would like to know some examples of connected Hopf algebras which has infinite Gelfand-Kirillov dimension but with primitive space finite dimensional. Any commments are welcome!
G.-S. Zhou
- 554
- 2
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3
votes
1 answer
Twists of commutative Hopf algebras
I have a dumb question.
Given a Hopf algebra $H$, take an invertible element $J\in H\otimes H$ and define $\Delta^J=J^{-1} \Delta J$. This becomes a new coproduct when $J$ satisfies a certain condition. Such $J$ are called twists, and let's denote…
Yuji Tachikawa
- 6,043
3
votes
1 answer
Bases of free Hopf algebras over Hopf subalgebras
Let $H$ be a Hopf algebra and $K$ a Hopf subalgebra of $H$. If $H$ is finite-dimensional, then by the Nichols-Zoeller Theorem $H$ is free as a left (and right) module over $K$.
Moreover, the same conclusion holds when $H$ is pointed, by the main…
Rocky Smith
- 618
3
votes
2 answers
Identities that connect antipode with multiplication and comultiplication
I asked this initially in math.stackexchange:
The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities:
$$
S\otimes S\circ \Delta=\sigma\circ\Delta\circ S
$$
$$
\nabla\circ S\otimes…
Sergei Akbarov
- 7,274
2
votes
1 answer
Convolution inverse
I'm currently reading some stuff on Hopf algebras, specifically Hopf Algebras: An Introduction by Sorin Dascalescu, Constantin Nastasescu and Serban Raianu. One proof involves showing that for a Hopf algebra $H$ with antipode $S$ that…
Ryan
- 545
2
votes
0 answers
Looking for an automorphism constructed from Hopf algebraic data
I am in an automorphism quest.
Let $H$ be a quasitrianglar Hopf Algebra with R-matrix $\mathcal{R} \in H \otimes H$. I know that $\mathcal{R}_{21}^{-1 }$ is a solution of the Yang Baxter equation and
$$\mathcal{R}_{21}^{-1} \Delta \mathcal{R}_{21}…
KraKeN
- 21
1
vote
1 answer
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 down such an isomorphism entirely in terms of the…
user18951
- 173
1
vote
0 answers
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 appreciate if someone can give me some clue.
X---
- 153
1
vote
0 answers
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 counterexample?
Thanks!
1
vote
1 answer
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, $\sigma$ the coinverse, and $*$ the adjoint action…
Chuck Hague
- 3,627