All Questions
25 questions
7
votes
0
answers
385
views
How to define $U_q \mathfrak{g}$ without generators and relations?
I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
- 700
5
votes
2
answers
343
views
Classifying Hopf algebras that admit a single irreducible comodule
Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k ...
4
votes
1
answer
569
views
Name for a Hopf algebra whose only grouplike element is the identity?
For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in ...
7
votes
2
answers
469
views
Low dimensional noncommutative non-cocommutative Hopf algebras
Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
- 141
18
votes
2
answers
1k
views
Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
2
votes
1
answer
205
views
A comodule algebra map from a Hopf algebra to itself
Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
- 1,144
5
votes
1
answer
129
views
Covariant splittings of Hopf algebra projections
What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
- 1,144
6
votes
2
answers
543
views
Confusion around the reflection equation algebra
I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
- 437
2
votes
1
answer
127
views
Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
- 219
12
votes
3
answers
832
views
Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
- 3,318
4
votes
3
answers
344
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
3
votes
1
answer
104
views
Irreducibility of product bicomodules
Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...
- 1,144
5
votes
2
answers
680
views
Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
8
votes
1
answer
821
views
A "concrete" example of a one-sided Hopf algebra
I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more ...
- 718
10
votes
1
answer
518
views
Functoriality of the Hopf dual
Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
- 835
2
votes
0
answers
312
views
Module algebras and comodule algebras
Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-...
- 6,201
5
votes
3
answers
487
views
On a dual of Kaplansky's $2^{nd}$ conjecture: admissible algebras?
Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states ...
- 7,746
10
votes
3
answers
1k
views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
- 7,746
1
vote
1
answer
349
views
What are all primitive elements in a tensor algebra?
Let $H$ be a Hopf algebra and $V$ a Yetter-Drinfeld module over $H$. Then there is a braiding $\Psi: V \otimes V \to V \otimes V$ given by $\Psi(x \otimes y) = x_{(-1)}.y \otimes x_{(0)}$, where $x_{(-...
- 6,201
2
votes
1
answer
96
views
Are braided commutators primitive elements of a braided Hopf algebra?
Let $H$ be a braided Hopf algebra. The multiplication on $H \otimes H$ is defined by $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$, $a,b,c,d \in H$.
Let $H = T(V)$. There is a algebra map $\...
- 6,201
4
votes
1
answer
689
views
Primitive elements in group hopf algebras over fields of non-zero characteristic
An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:
"If $H$ is a $\...
- 7,746
5
votes
2
answers
281
views
Comodules of Cosemisimple Hopf Algebras
A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
- 1,479
2
votes
2
answers
566
views
Definition of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...
- 449
4
votes
0
answers
122
views
Cosemi-simple FRT Hopf Algebras
This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
- 159
2
votes
1
answer
201
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 ...
- 1,776