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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.

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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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. …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
2 votes

Comultiplication of elements of partition of unity

This answer was given before the edit (with the understanding that under the stated assumptions the comultiplication described in the OP is cocommutative for the $d$ idempotents $p_i$ and $k$ is an a …
Konstantinos Kanakoglou's user avatar
4 votes
Accepted

Characters on Hopf algebras

I think that a general example is the so-called Larson's character, which in a sense ties together the trace and determinant functions. To make the long story short: Let $C$ be a cocommutative bial …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar
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 …
Konstantinos Kanakoglou's user avatar

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