Rigged Hilbert space and function of ultra-rapid decay
Consider the Hilbert space $\ell^2(\mathbb{Z})$ and unbounded operators $u$ and $v$ defined as $u: f_k \to f_{k+1}$ and $v: f_k \to q^k f_k$ for some fixed $0<q<1$. Together with $u^{-1}$ and...
View ArticleMotivation for twisted Yangians?
I am asking this question, hoping to understand the motivation behind the construction of the so-called twisted Yangians. This will be a long question, so my thanks in advance for any comment and/or...
View ArticleIntuition behind the definition of quantum groups
Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to...
View ArticleHausdorff dimension and von Neumann dimension
Non-integral dimensions arise in two key areas:fractal geometry: This includes the well-known Hausdorff dimension of fractals,von Neumann algebra: In this context, consider a type ${\rm II_1}$ factor...
View ArticleEquivalent definitions of the simple $\ell$-roots $A_{i,a}$ of quantum affine...
Let $U_q(\widehat{\mathfrak g})$ be the quantum affine algebra over a simple Lie algebra $\mathfrak g$. I am trying to understand and compare the so called simple $\ell$-roots $A_{i,a}$ seen in both...
View ArticleSubalgebra of quantum groups from extended Dynkin diagrams
Given a semisimple Lie algebra, we can construct subalgebras corresponding to removing nodes from the extended Dynkin diagram. For example $\mathfrak{su}_3 \subseteq \mathfrak{g}_2$. I'm interested in...
View ArticleHighest-weight modules for RTT-quantum groups
First of all, my question is motivated by the following phenomenon. Suppose that $Y$ is a quantisation of a Lie bialgebra (say $y$), and suppose that $Y' \subset Y$ be a coideal subalgebra. In the...
View ArticleIs the topological dual of a quantized enveloping algebra a Hopf algebra?
I am looking for a proof of Proposition 10.2 in Etingof and Schiffmann’s Lectures on Quantum Groups. In particular, I would like to understand the proof of the following statement in an elementary...
View ArticleSecond Quantization with Coulomb potential
I am trying to understand how one can perform second quantization in the case of the Hamiltonian of the hydrogen atom, i.e. when the one particle Hamiltonian acquires an external coulomb potential. The...
View ArticleExplicit description of the Poisson–Lie group associated with the Drinfeld...
Let $\mathfrak{b}_+$ be the Borel subalgebra of $\mathfrak{sl}_2$. It is equipped with a Lie bialgebra structure given by$$[H, E] = 2E, \quad \delta(E) = E \otimes H - H \otimes E, \quad \delta(H) =...
View ArticleThe other classical limit of a quantum enveloping algebra?
Let $\mathbb K$ be a field (of characteristic 0, say), $\mathfrak g$ a Lie bialgebra over $\mathbb K$, and $\mathcal U \mathfrak g$ its usual universal enveloping algebra. Then the coalgebra structure...
View ArticleIs every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem states...
View ArticlePractical Ways to get Skew-Schur Functions
The Schur polynomials satisfy many, many identities and there is a whole book about them.I think the easiest way is with the Vandermonde Determinant.$$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc}...
View ArticleClassification of commutative and co-commutative super Hopf algebras
I've read (for example here on mathoverflow) that finite-dimensional cocommutative Hopf algebras over $\mathbb C$ are always isomorphic to group Hopf algebras. If the algebra is also commutative, the...
View ArticleIs there a version of Ringel's Theorem that works for $O_q^+(SL_2)$ instead...
(crossposted from mse, where it didn't get much traction)It's a famous theorem of Ringel (later extended by Green and others) that the hall algebra of the $\mathbb{F}_q$-valued representations of an...
View ArticleExplicit computation of the Heisenberg double Poisson structure in Theorem...
In Jiang-Hua Lu's paper "Moment Maps at the Quantum Level" (Comm. Math. Phys. 157, 1993), Theorem 4.4 states that the smash product algebra$$\mathcal{O}_\hbar(P) \# U_\hbar(\mathfrak{g})$$provides a...
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