# Questions tagged [coalgebras]

For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...

135 questions
Filter by
Sorted by
Tagged with
31 views

### Sweedler Notation: limtations and left comodule version.

I have two questions: In the nlab entry about the sweedler notation it reads "One can formalize in fact which manipulations are allowed with such a reduced notation." Has anyone done that/an idea ...
34 views

### $\times$ versus $\otimes$ in Definition of $K$-Algebra

While reading about algebras and coalgebras, I keep coming across two definitions of an algebra $A$. One definition uses the Cartesian product $\times$, while another uses the tensor product $\otimes$....
36 views

35 views

### if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
47 views

### final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
31 views

### Reference about the proof of this proposition?

Let $M$ be a finite dimensional Hopf algebra with antipode $S$ and let $M^*$ be the dual bialgebra. Recalls that $x^* \in M^*$ is a primitive in $M^*$, then $x^*$ is a derivation of $M$. '' I am ...
21 views

### Convolution in Bialgebras

On the wikipedia page on convolution, there is a section on convolution in bialgebras. It's completely mysterious to me. If it has something to do with the regular concept of convolution, can some one ...
129 views

### Is the category of finite-dimensional $k[x]$-modules a comodule category?

Fix a field $k$, denote by $k[x]$ the polynomial algebra. The category of finite-dimensional modules over $k[x]$ is precisely the category $\mathcal{C}$ consisting of pairs $(V, T_V: V \to V)$ of ...
26 views

86 views

### Whether we can define the finitely generated coideal?

In the question How to understand the coidea of a colagebra?, we had posed the definition of coideals. Let $R$ be a unitary commutative ring and $X$ be a finite subset of $R$. Then the ideal ...
74 views

223 views

### Example of $V^* \otimes V^*$ not isomorphic to $(V \otimes V)^*$

There is always an injection between $V^* \otimes V^*$ and $(V \otimes V)^*$ given by $$f(v^* \otimes w^*)(x \otimes y)=v^*(x)w^*(y),$$ where $x,y \in V$. I've been given to understand that in ...
84 views

### Product in the category of cocommutative coalgebras

Let $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ be two coalgebras. Consider their tensor product $C\otimes C'$ and the two coalgebra homomorphisms \begin{align*} \pi:C\otimes C'\to C, \quad c\...
80 views

### $n$-fold product is a morphism of coalgebras

Let $(H,\mu,\eta,\Delta,\varepsilon)$ be a bialgebra with antipode $S$ which is cocommutative. On $\text{End}(H)$ we have the product $$f\ast g:=\mu\circ(f\otimes g)\circ\Delta\in\text{End}(H).$$ ...
29 views

### Intersection of coideals

"Let $C$ be a coalgebra. If $I$ and $J$ are two coideals of $C$, show that $I\cap J$ is a coideal of $C$." This is an exercise on page 45 of the book "Hopf Algebra" by Sweedler, Moss E(1969). But I ...
39 views

### coalgebras are right or left vector spaces

Following the definition of a coalgebra found here https://en.wikipedia.org/wiki/Coalgebra, I was wondering if it is a right or a left vector space or both? Indeed, when we use the Sweedler notation, ...
86 views

Starting with a graded vector space $V$, you can construct the tensor algebra $\operatorname{T}(V) := \bigoplus_{n>0} V^{\otimes n}$ and you can construct the symmetric algebra $\operatorname{S}(V) ... 2answers 201 views ### Exercises to help a student become accustomed to Sweedler notation For a coassociative coalgebra$A$, we have a comultiplication map$\Delta \colon A \to A \otimes A$. An element$c \in A$is sent to a sum of simple tensors, which can be a mess of indices, so we can ... 0answers 68 views ### Leibniz rule and Alexander-Whitney coproduct Is there anything more than a superficial similarity between the following? The Alexander-Whitney coproduct$\Delta$on the tensor algebra$\bigotimes^\bullet V$of a vector space$V$is defined by ... 1answer 101 views ### What are the primitive elements in a polynomial hopf algebra with primitive indeterminates? Is there a result that says that in any polynomial Hopf algebra$K[X_1, X_2, ...]$over a field$K$with indeterminates primitive, the primitive elements are precisely the linear homogeneous ... 0answers 106 views ### Is the algebra dual to a graded coalgebra graded? Given a graded coalgebra$C = \bigoplus_{n\geq 0} C_n $with coproduct $$\Delta : C_n \to \bigoplus_{i=0}^n C_i\otimes C_{n-i}$$ must we have that the dual$C^* = \bigotimes_{n\geq 0}C_n^*$is a ... 0answers 59 views ### The associated graded of a filtered coalgebra Given a coalgebra$C$with a filtration$F$such that$\Delta(F^n C)\subset \sum_{i=0}^n F^i C\otimes F^{n-i} C$, how does the coproduct manifest in the associated graded? Do we get something to the ... 1answer 150 views ### Why doesn't the functor$\bar{\mathcal{P}}\bar{\mathcal{P}}$preserve pullbacks? I've tried finding examples on my own but the sizes of the sets is a bit hard to manage. In the litterature I've seen this fact referenced in a few places but they all point to Rutten: Universal ... 1answer 196 views ### algebra vs Dual of a coalgebra Let$(A,m,u, \Delta, \varepsilon)$be a bialgebra. Taking dual,$(A^\star, \Delta^\star,\varepsilon^\star)$is a algebra. What is the relationship between the two algebras$(A, m, u)$and$(A^\star, \...
Let $R$ be a commutative ring and let $M$ be some $R$-module. Is there a coalgebra $A$ such that $A$-comodule structures on $M$ (i.e. the $R$-linear maps $M \to M \otimes_R A$ satisfying the two usual ...
### Difference between (co)algebras and $F$-(co)algebras
I was reading the page on coalgebras and it made a mention to $F$-coalgebras in the first paragraph as though $F$-coalgebras are just a specific type of coalgebras. However, I am having a difficulty ...