All Questions
13 questions
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Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
CommunityBot
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1
vote
1
answer
110
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On reflexive bialgebras
Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
2
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1
answer
152
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Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
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7
votes
2
answers
360
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Different Bialgebra/Hopf algebra structures on coalgebras
Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
- 2,160
1
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1
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652
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What is a coalgebra?
A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
- 28.1k
5
votes
2
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343
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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
3
votes
1
answer
171
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Show that a certain element is a linear combination of tensors
I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here.
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
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2
votes
1
answer
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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 $\...
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1
vote
0
answers
49
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Weakly reflexive algebra vs proper (residually finite-dimensional) algebra
Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...
- 12.9k
3
votes
0
answers
99
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Simple coalgebra under base change
Let $C$ be a simple coalgebra over a field of characteristic $0$. Let $K$ be a field extension of $k$. Is the coalgebra $C\otimes_k K$ over $K$ simple?
- 63.9k
5
votes
3
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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
2
votes
2
answers
566
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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
1
vote
1
answer
171
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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 ...
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