# Questions tagged [coalgebras]

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

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### Modules over the dual of an infinite dimensional coalgebra

Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of ...
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### Proving uniqueness of antipodes in Hopf algebras

Let $(H,\mu,\nu,\Delta,\epsilon)$ be a Bialgebra where H is the vector space, $\mu, \nu$ are the product and unit whilst $\Delta, \epsilon$ are the coproduct and counit. Now, for $f,g \in end(H)$ ...
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### A monic coalgebra morphism whose underlying $\text{Set}$ morphism is not injective

Let $F:\mathscr A\to \mathscr A$ be a functor. Consider the following category $\mathscr C$. The objects are arrows $A\to F(A)$. If $\alpha:A\to F(A)$ and $\beta:B\to F(B)$ are two objects, then a ...
33 views

### Confusing with definition of (co)algebra

I began to read the book « Hopf algebras » by Sweedler. He gave the definitions of algebra and coalgebra using commutative diagrams. These diagrams just show (co)associativity and (co)unitary ...
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### A question about infinitesimal bialgebra or Newtonian coalgebra

Recently, I am studing an interesting coalgebra which was called the infinitesimal bialgebra by Joni and Rota. It can be regarded as an algebraic framework for the calculus of divided differences. ...
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### Subcoalgebra generated by an element

Let $(A, \Delta, \epsilon)$ be a coalgebra and $f\in A$. What is the subcoalgebra generated by $f$ like? For example, if $A$ is the dual of the quaternions $\mathbb{H}$ (which is $\mathbb{R}$-algebra ...
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### Wedge product of coalgebras

I need to prove the following: "Let $U$ and $V$ be subspaces of a coalgebra $(C,\Delta, \epsilon)$. Suppose that $U\subseteq\ker(\epsilon)$. Show that $U\wedge V\subseteq V$ and $V\wedge U\subseteq V$...
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### Every coalgebra is the quotient of a cosemisimple.

I am trying to show that every coalgebra is the quotient of a cosemisimple coalgebra (I'm actually not sure if it's true or not). Here is my attempted solution: Let $C$ be a coalgebra. We know ...
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### Is a coalgebra comodule cosemisimple if and only if every subcomodule is a direct summand?

It is well known that if $R$ is a ring, then every $R$-module $M$ is semisimple (that is, $M$ is the direct sum of simple $R$-modules) if and only if every submodule of $M$ is a direct ...
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### Explicit formula for the equalizer of coalgebras

The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is ...
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### The proof of $S(gh)=S(h)S(g)$ of an antipode in Hopf algebras

Let $H=(H, m, \Delta, \mu, \epsilon, S)$ be a Hopf algebra. Then there is a property of antipde $S$: $$S(gh)=S(h)S(g)$$ where $g,h \in H$. I have seen in some materials the proof of this property ...
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### How is the differential induced by $d_C$ on $\Omega C$ defined for $(C,d_C)$ is a dga coalgebra?

Again I am confused about something regarding the cobarconstruction of a dga coalgebra $(C,d_C)$. The cobar construction of $C$ is the dga algebra $(T(s^{-1}\bar{C}),d_1+d_2)$ where $d_2$ is induced ...
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### Does the differential of an augmented dga algebra fix the augmentation ideal?

I am reading about the bar/cobar construction in the book Algebraic Operads. The differential on the bar construction of a augmented dga algebra $A$ is a sum of two differentials $d_1+d_2$ where $d_1$ ...