Search Results

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 …

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 …

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 …

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 …

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. …

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 …

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 …

The category of the finite dimensional comodules of a hopf algebra over a field, is a rigid, monoidal category. (just as the category of the finite dimensional modules). If we take a fin dim, righ …

$\DeclareMathOperator\Hker{Hker}\DeclareMathOperator\Hcoker{Hcoker}\DeclareMathOperator\Im{Im}\DeclareMathOperator\coIm{coIm}\DeclareMathOperator\Id{Id}$The category $\mathcal{H}$ of finite dimension …

To any skew-pairing $\lambda:U\otimes H\rightarrow k$, one can associate a hopf algebra $D(U, H)$ (built on $U\otimes H$) which is called the generalized quantum double of $U$ and $H$. (If $H$ is fini …

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 …

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 …

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 …

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 …

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 …