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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 $G$.
10
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Name for the action of a bialgebra on an algebra
According to nLab, such an action is called a Hopf action and your data specify a left $B$-module algebra. Such a structure is also referred to in the literature as an algebra in the category (of left …
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10
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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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10
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3
answers
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About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f. …
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9
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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
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$\mathbb{Z}$-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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8
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Inner automorphisms of Hopf algebras
I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad_l h)\triangleright k=\sum h_1 kS(h_2)$ of a hopf algebra $H$ on itself.
(It is known tha …
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Examples of representations of quantum groups
If i have correctly understood your question, there are various such examples, arising from mathematical physics contexts (where some of the original motivations for the study of quantum groups first …
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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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7
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1
answer
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Classification of quasitriangular Hopf algebras
The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)commutat …
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Cocommutativity, comultiplication and coalgebra maps
Given a (coassociative and counital) coalgebra $(C,\Delta,\varepsilon)$, over a field $k$, we can form the tensor product coalgebra $(C\otimes C,\Delta_{C\otimes C},\varepsilon_{C\otimes C})$ through: …
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Cartier-Kostant-Milnor-Moore theorem
The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).
…
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Abelian category from the category of Hopf algebras
$\DeclareMathOperator\Hker{Hker}\DeclareMathOperator\Hcoker{Hcoker}\DeclareMathOperator\Im{Im}\DeclareMathOperator\coIm{coIm}\DeclareMathOperator\Id{Id}$The category $\mathcal{H}$ of finite dimension …
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When is this map of Hopf algebras Surjective?
Some thoughts, regarding question (a):
In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classifica …
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6
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Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional eq...
About your first question:
Since you are asking for an example, take any group hopf algebra $k\mathbb{G}$, pick some subset $S\subset \mathbb{G}$ and denote $kS$ the linear subspace of $k\mathbb{G}$ …
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Classification of $\operatorname{Rep} D(G)$
There are some classic results on the classification of the irreducible $D(G)$-modules:
If the field is the complex numbers $\mathbb{C}$, it has been shown that a representation of the finite group $G …
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