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Results tagged with hopf-algebras
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user 85967
A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.
5
votes
Tannaka-Krein duality in Chari-Pressley's book
I do not know much about the Tannaka-Krein duality itself. But regarding the last part of your question
Also if somebody could cast some light on possible generalizations of this proposition (to the …
- 5,711
10
votes
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
This is a question on an active area of research, with lots of work on it (for the general case of algebraically closed fields of char zero). It is historically and conceptually closely connected to K …
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5
votes
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
Since you mention classification results for $R$-matrices:
For finite abelian groups, there is a bijection between the set of universal $R$-matrices of the group hopf algebra $\mathbb C[G]$, the set o …
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2
votes
Primitive elements in the universal enveloping algebra of Lie superalgebra
$\DeclareMathOperator\chr{char}$Yes this is true: Under your assumptions $\mathcal{P}(U(g))=g$.
Also, since for any primitive element $x$ we have $\epsilon(x)=0$, for any grouplike element $y$ we have …
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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 …
- 12.9k
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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3
votes
Accepted
Name for a Hopf algebra whose only grouplike element is the identity?
There is a one-to-one correspondence between the grouplike elements and the simple, $1$-dim subcoalgebras. So if the only grouplike element is $1_H$, then there is a unique $1$-dim simple subcoalgebra …
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1
vote
Cosemisimple pointed Hopf algebras
Assuming that we are speaking about finite dimensional hopf algebras, the answer is yes:
Since $H$ is cosemisimple if and only if $H\cong Corad(H)$ (as coalgebras) and $H$ is pointed if and only if $C …
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2
votes
Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
I am not really sure if this what the OP is looking for but i guess that a closely relevant notion here is that of connected Hopf algebras (i.e HAs which are connected as coalgebras). These are Hopf a …
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2
votes
The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
Well, i am not sure if this is what the OP is looking for but here is an heuristic method for computing the limit, avoiding the use of another algebra defined at $q=1$ and thus bypassing the "double c …
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9
votes
Accepted
Low dimensional noncommutative non-cocommutative Hopf algebras
By standard results (in fin dim, over an alg closed field of zero char),
all cocommutative HAs are group algebras (for some finite group),
all commutative HAs are duals of group HAs (for some finite …
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8
votes
Accepted
$\mathbb{Z}$-graded algebras and tensor products
No it cannot happen.
And not only for strongly $\mathbb{Z}$-graded rings; this is always the case for any strongly $G$-graded ring, where $G$ is a group. $A_k \otimes_{A_0} A_l \simeq A_{k+l}$ is an i …
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1
vote
Comodule Morita equivalence for Hopf algebras
There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops t …
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4
votes
Definition of subcoalgebra over a commutative ring
You are right.
In the case of an $R$-submodule $D$ of an $R$-coalgebra $C$, the correct definition for $D$ being a subcoalgebra of $C$ is your definition (2) and not the one posted in your notes. This …
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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. See also: arXiv:1512.0934 …
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