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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
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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2
votes
q-difference equations and quantum mechanics
Regarding the first part of the question:
Have there ever been actual uses of q-calculus and quantum groups to computing or understanding solutions of Schrodinger equations, or functions actually …
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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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1
vote
Accepted
quantum affine $gl_2$
I guess you mean the following presentation in terms of generators and relations:
The excerpt is from:
Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras, arXiv:1709.01592v4 [math. …
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1
vote
Commutative and Cocommutative Quantum Groups
If the definition of a finite quantum group, you use, is a pair $(A,\Phi)$ of a finite dimensional $C^*$-algebra $A$, with a comultiplication $\Phi$, such that $(A,\Phi)$ is a Hopf $*$-algebra, then t …
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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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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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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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4
votes
Hopf Subalgebras of Quantized Algebras
Since the OP is asking for examples of
sub-Hopf algebras which are not generated by the standard generators
i.e. the Chevalley generators (which are actually the generators of the Cartan–Weyl b …
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4
votes
Accepted
Is there another quantum deformation of sl(2)?
Regarding your second question, on other possible deformations of $sl(2)$:
There have been various studies on (multi-parametric) deformations of Lie algebras -as has already been mentioned in the co …
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4
votes
Accepted
Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
Although i have some doubts as to what the OP is exactly looking for (see my comments above), i hope that the following will be of some interest for its purposes. In:
$U_q(sl(n))$ Difference Operator …
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2
votes
Hopf dual of the Hopf dual
Regarding your first question: the answer is generally no, the restricted dual of the restricted dual of $A$ is generally not isomorphic to $A$:
$$
(A^{\circ})^\circ\ncong A
$$
as has already been ind …
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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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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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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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