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Results tagged with coalgebras
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user 85967
For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...
6
votes
Cocommutativity, comultiplication and coalgebra maps
The comultiplication $\Delta:C\rightarrow C\otimes C$ being a morphism of coalgebras, or a coalgebra map, by definition means that for an arbitrary $c\in C$ we have:
$\varepsilon_C(c)=\varepsilon_{C\otimes … Thus, we have shown that: If the comultiplication $\Delta:C\rightarrow C\otimes C$ is a morphism of coalgebras, or a coalgebra map then this implies the cocommutativity of $C$. …
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2
votes
Accepted
What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?
No, in general the claim is not true:
To see why, consider a field $k$ of characteristic $p$ and take the polynomial hopf algebra $k[x]$ (in a single variable). Then $x$ is primitive and so is $x^p$ …
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6
votes
Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional eq...
From a more general point of view, since $(f\otimes f)\circ \Delta=\Delta\circ f$ is -by definition- satisfied for any morphism of coalgebras, then your functional equation should be satisfied for any …
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3
votes
Accepted
Bialgebra maps and Hopf algebra maps
Yes it is.
It is actually a standard result, for any hopf algebra, that under the circumstances you are describing any bialgebra map $\phi$ commutes with the antipode, i.e. we have: $$S_{H'}\circ \phi …
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3
votes
On a dual of Kaplansky's $2^{nd}$ conjecture: admissible algebras?
A couple of examples of non-admissible algebras:
$\bullet$ It is known (see here, p.222) that a semisimple Hopf algebra is also separable (as an algebra). Consequently, a semisimple but non-separabl …
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4
votes
Definition of subcoalgebra over a commutative ring
The definition mentioned in your notes is valid (as you have already mentioned in the OP) for vector spaces and special cases for $R$-coalgebras (when $D$ is flat or pure as an $R$-submodule for example …
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5
votes
Category of bicomodules of a cosemisimple Hopf algebra
The answer is yes, if we are talking about finite dimensional, Hopf algebras over a field:
$\bullet$ $H$ being cosemisimple (as a coalgebra) is equivalent to the dual hopf algebra $H^*$ being semi …
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2
votes
Accepted
Non-counital coalgebras
The finite dual of a non-unital algebra has been introduced in Semiperfect and coreflexive coalgebras, S. Dăscălescu, M. C. Iovanov, Forum Math. 27 (2015), No. 5, 2587--2608. …
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7
votes
Accepted
Classifying Hopf algebras that admit a single irreducible comodule
The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element …
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2
votes
Definition of a cosemisimple Hopf algebra
The above isomorphism should be interpreted as an isomorphism of $k$-coalgebras. … Also $kg$, for all $g\in G$ is simple (since it is $1$-dim) and $kg_1\cong kg_2$ (as $k$-coalgebras) for all $g_1,g_2\in G$. …
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