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Results tagged with quantum-groups
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user 85967
Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
23
votes
Accepted
Is there any published physics article where $q$-mathematics is applied?
There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformatio …
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16
votes
Accepted
Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do …
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12
votes
What is quantum algebra?
I think that a modern realistic perception of the term "quantum algebra" has to be understood in its historical context, that is, the algebraic/geometric methods, originating from the study of the qua …
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11
votes
Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$
This is a very interesting question. I have also made some search but i have not found this result explicitly mentioned somewhere in the literature. However, i remember i have heard such a claim in th …
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10
votes
Accepted
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
votes
3
answers
1k
views
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
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
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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7
votes
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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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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6
votes
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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6
votes
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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6
votes
Accepted
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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6
votes
Accepted
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
votes
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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