All Questions
10 questions
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
$$
k \to k \otimes H, ~~ k \mapsto k ...
- 7,746
2
votes
1
answer
127
views
Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
- 7,746
9
votes
1
answer
342
views
An inner product approach to Hopf algebras
We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$.
Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^...
- 375
2
votes
1
answer
408
views
Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional equation
What is an example of a coalgebra $C$ which admit a linear operator $T$, different from scalar operators $T=\lambda Id$, which satisfy $$(T\otimes T)\circ \Delta=\Delta \circ T^2 $$ but $C$ is not ...
- 7,746
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 ...
- 2,160
3
votes
1
answer
344
views
What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?
Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...
- 7,746
5
votes
1
answer
347
views
Category of bicomodules of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
- 7,746
4
votes
2
answers
661
views
Cocommutativity, comultiplication and coalgebra maps
Given a coalgebra $(C,\Delta,\varepsilon)$, over a field, the following is a well-known property:
the comultiplication $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is ...
- 7,746
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 ...
- 7,746
4
votes
3
answers
2k
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 ...
- 33.8k