All Questions
Tagged with coalgebras ra.rings-and-algebras
30 questions
0
votes
1
answer
294
views
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 ...
user525442
1
vote
0
answers
110
views
Reference for cocommutative coalgebras
I'm looking for references on cocommutative coalgebras where I can see them as kind of infinitesimal spaces. I'm trying to understand this post Why do Lie algebras pop up, from a categorical point of ...
- 35
5
votes
1
answer
246
views
Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
- 985
3
votes
1
answer
123
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Equivalent definitions of pro-unipotent coalgebras
I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras.
Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \...
2
votes
1
answer
152
views
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 ...
1
vote
1
answer
140
views
Morphism of conilpotent coalgebras
I have a stupid question about morphisms of two conilpotent coalgebras $\phi:X\to Y$. Is it a morphism of coaugmented coalgebras such that $\phi(F_n(X))\subseteq F_n(Y)$? Here $F_n$ denotes the ...
- 359
5
votes
0
answers
114
views
Conilpotent coalgebras as pushouts of trivial coalgebras
Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$
whose underlying $K$-vector space is finite dimensional.
Question: Can one obtain $C$ by iterately taking ...
- 1,361
7
votes
2
answers
360
views
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.
- 305
1
vote
1
answer
651
views
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. ...
- 123
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
$$
k \to k \otimes H, ~~ k \mapsto k ...
3
votes
1
answer
171
views
Show that a certain element is a linear combination of tensors
I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here.
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
user167952
1
vote
1
answer
232
views
$M$ comodule if and only if $N$ and $L$ comodules
Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence
$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$
of vector spaces, we have that $N$...
- 113
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
1
vote
1
answer
110
views
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 ...
- 1,066
1
vote
0
answers
49
views
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 ...
- 519
4
votes
0
answers
155
views
Do comodules form an exact category?
Let $R$ be a commutative ring, $C$ a coalgebra over $R$. I am asking about the category of $C$-comodules $C$-Comod.
It is clear that if $C$ is a flat $R$-module, then $C$-Comod is abelian. Hence, is ...
- 12.3k
3
votes
0
answers
99
views
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?
- 564
9
votes
0
answers
371
views
Duality between coalgebras and (pseudocompact) algebras - uniqueness
The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...
- 2,450
3
votes
0
answers
178
views
Completion of coalgebras
Is it possible to complete commutative dg associative (conilpotent) coalgebras over $\mathbb{Q}$ in a way so that when we complete the symmetric coalgebra $Sym(V)$ it becomes completed with respect to ...
- 523
2
votes
1
answer
169
views
Variant of co-Tor in a bimodule category
Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a coalgebra object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion ...
- 7,952
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
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
1
vote
0
answers
163
views
An identity satisfied by "Differentiation"
I asked this question in MSE but I did not received any answer. So I repeat it here:
Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...
- 356
5
votes
0
answers
246
views
Injectivity criterion for surjective coalgebra maps: does it hold in full generality?
Let $\mathbf{k}$ be a commutative ring.
Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \...
- 33.8k
1
vote
0
answers
165
views
Coinduction and corestriction are quasi-inverse equivalences for comodules?
I'm reading http://arxiv.org/abs/math/0310337.
There the following statement is given without proof:
Let $k$ be a field. Let $C$ be a counitary coaugmented coalgebra, i.e. there is $\eta: C\to k$ ...
- 2,450
1
vote
1
answer
435
views
(Co)Universal Property of Quotients/Subs
I'm not completely sure if this bunch of questions is the appropriate Level of MO. However at the same time I think that it is at least slightly above the level of stackex. ...
The tensor algebra ...
- 2,074
2
votes
1
answer
626
views
Reference for the fact that a coderivation of the (non reduced) tensor coalgebra is determined by its corestrictions
If $V$ is a vector space, let us consider the tensor coalgebra $TV=\bigoplus\limits_{k=0}^\infty V^{\otimes^k}$ with coproduct given by
$$\Delta (x_1\otimes \dots \otimes x_n):= \sum\limits_{i=0}^{n}(...
- 261
1
vote
1
answer
171
views
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 ...
- 33.8k
13
votes
1
answer
1k
views
Why not _co_free modules?
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...
- 54.6k
39
votes
5
answers
4k
views
Is there an explicit construction of a free coalgebra?
I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...
- 9,132