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Results tagged with qa.quantum-algebra
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user 85967
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
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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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
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
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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5
votes
2
answers
399
views
Indecomposable, non-simple, modules of quantum groups at roots of unity
Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for whi …
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5
votes
Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
The following is not really an answer but a rather too-long comment, with respect to your second question:
Does there exist deformations of the monoidal category of $U(\frak{g})$-modules which a …
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4
votes
Comodules of Cosemisimple Hopf Algebras
Regarding your first question, I think the following definition and theorem settles the answer to the affirmative:
Definition: A coalgebra $C$ is called right cosemisimple (or right completely reduc …
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4
votes
Can one define quantized universal enveloping algebras in a basis-free way?
I do not know the answer in general. But towards the end of the OP you say:
"Even a construction that still involves generators and relations but avoids choosing a basis of the root system would …
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