Skip to main content

Questions tagged [quantum-groups]

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.

Filter by
Sorted by
Tagged with
0 votes
0 answers
26 views

A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
fusheng's user avatar
  • 1,159
1 vote
0 answers
55 views

Are there any useful convexity properties of quantum dynamical semigroups?

I'm am wondering if there are any useful properties of quantum dynamical semigroups I can exploit for convex/concave optimization with respect to the semigroup parameter. A proper definition of a ...
nlupugla's user avatar
0 votes
0 answers
36 views

On the orthogonality relations for quantum Clebsch-Gordan coefficients

I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients. To make this question self-contained, let me introduce the ...
richrow's user avatar
  • 4,232
0 votes
0 answers
33 views

Best Introductory Textbooks on Compact & Locally Compact Quantum Groups [duplicate]

This is a soft question / reference request. I’ve been recently interested in starting research in the area of compact and locally compact quantum groups. However, I’ve noticed the subject is really ...
Isochron's user avatar
  • 1,399
2 votes
0 answers
45 views

Normality of the Lebesgue integral with respect to a Haar measure

Let $G$ be a locally compact second countable topological group and let $\mu$ be a Haar measure on $G$. I want to show that the weight $\varphi : {L^\infty(G)}^+ \to \overline{\mathbb R_+}$, $f \...
Valentin Massicot's user avatar
3 votes
1 answer
61 views

Tensor product and extension of $\sigma$-weakly continuous linear map.

Let $M$ be a Von Neumann algebra and let $\Delta$ be a $\sigma$-weakly continuous unital $*$-morphism. We say that $\Delta$ is a comultiplication if $\Delta$ satisfies $(\Delta \otimes \iota)\Delta = (...
Valentin Massicot's user avatar
5 votes
2 answers
119 views

Normal character on a group von Neumann algebra

For a locally compact group $G$, I will denote by $L(G)$ its group von Neumann algebra, which is the von Neumann algebra acting on $L^2(G)$, generated by the image of left regular representation $\...
Mogget's user avatar
  • 641
2 votes
0 answers
43 views

How to lower bound the quantum conditional entropy?

I am trying to lower bound the quantum conditional entropy $H(X|Y)$ when $X$ and $Y$ are two quantum systems. Classically, it can be done as follows: $$ H(X|Y) = \sum_{y}P_Y(y) H(X|Y=y) \geq \sum_{Y \...
Jaswanthi Mandalapu ee19d700's user avatar
1 vote
1 answer
34 views

Corepresentations of quantum subgroups

If $H$ is a Hopf subalgebra of the Hopf algebra $G$ and $\alpha:V\longrightarrow V\otimes H$ is right corepresentation of $H$ on the vector space $V$ (finite--dimensional), is there a way to induce a ...
AmSa's user avatar
  • 139
1 vote
0 answers
44 views

Morphisms between modules over a Hopf algebra

Let $U$ be a Hopf algebra over a field $K$ and $M,N$ be $U$-modules. Then the set of all $K$-linear maps $\textrm{Hom}_{K}(M,N)$ has a canonical $U$-module structure given by $$(u.f)(m) = \sum_{(u)} ...
Luka's user avatar
  • 126
5 votes
1 answer
104 views

How to check Pentagon axiom with induced associator of skeletal category?

In the book Tensor Categorties by EGNO, there are Exercise 2.8 Show that any monoidal category $\mathcal{C}$ is monoidally equivalent to a skeletal monoidal category $\bar{\mathcal{C}}$. In the hint,...
MatrixBi's user avatar
  • 137
1 vote
1 answer
42 views

How to define tensor product of operators on $B(H \otimes H)$

Let $H$ be a Hilbert space. Working on locally compact quantum groups, I have met with operator of the form $\iota \otimes \omega_{\xi, \eta}$ with $\xi,\eta \in H$ and $\omega_{\xi,\eta}(T) = (T\xi,\...
Valentin Massicot's user avatar
1 vote
1 answer
34 views

Some properties about the equation $P(z)\phi(z)=0$

Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$. If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get ...
fusheng's user avatar
  • 1,159
0 votes
0 answers
30 views

Zeta functions on q-deformed compact Lie groups

I’m reading some of the recent works on representation zeta functions for groups. Along the way, I have also explored some of the remarkable properties of Witten zeta function. I’m wondering if there ...
TMRiddle's user avatar
0 votes
0 answers
34 views

Solution of the Yang-Baxter equation not coming from quasi-triangular structure

Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}...
Minkowski's user avatar
  • 1,540
0 votes
0 answers
46 views

How to compute braidings from modular data

I'm trying to compute the braiding morphisms between simple objects of the so-called metaplectic categories, i.e. the modular categories with fusion rules equivalent to those of $\operatorname{SO}(N)...
nanowillis's user avatar
0 votes
0 answers
28 views

Non-reversible time-dependent wave function - which type of PDE?

Is anyone aware of a time-dependent wave-function that is non-reversible? With non reversible, I mean that the initial state of some wavefunction as solution to some PDE, $\psi_0(x_1,\dots,x_n,t)$ in ...
Superunknown's user avatar
  • 2,973
3 votes
0 answers
70 views

$(p,q)$-Weyl Algebra

In this Introduction to representation theory they define the '$q$-Weyl algebra by the primary defining relation $$xy = qyx$$ This seems appropriate in $q$-deformations based on the basic building ...
Mako's user avatar
  • 690
1 vote
0 answers
30 views

${[n]}_{q,q^{-1}}$ $q$-deformation

It seems that in some $q$-deformations the following definition of a $q$-number is used: $$(n)_q = \frac{q^n-q^{-n}}{q-q^{-1}}$$ If we define $${[n]}_q = \frac{1-q^n}{1-q}$$ as the 'conventional' $q$-...
Mako's user avatar
  • 690
0 votes
0 answers
11 views

Does the antipode leave Peter-Weyl blocks invariant.

Let $\mathcal U_q(\mathfrak{g})$ be a algebraic quantum group over $\mathbf{Q}(q)$, and $A_q(G)$ be its linear dual of matrix entires. Suppose I have a matrix entry $c_{f,v}\in L^r(\lambda)\otimes L(\...
esteban's user avatar
  • 571
2 votes
2 answers
127 views

Restricting and extending *-homomorphisms with norm-dense subalgebras

Let $G$ and $H$ be discrete groups, and suppose that I have a surjective *-homomorphism: $$\pi:\mathrm{C}^*(G)\to C_{\text{r}}(H).$$ Is it true that the restriction to the group algebra $\pi_{|_{\...
JP McCarthy's user avatar
  • 7,789
0 votes
1 answer
23 views

Properties of Bicrossed Product multiplcation

I am reading Kassel's Quantum Groups book (the chapter on Drinfeld doubles). In it, there is the following claim: If $H,K\subseteq G$ are groups such that $\forall g\in G$, $\exists!(y,z)\in H\times K$...
Wyatt Kuehster's user avatar
1 vote
1 answer
70 views

Haar integral of a finite dimensional Hopf algebra: an explicit expression

Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
Lagrenge's user avatar
  • 836
0 votes
1 answer
40 views

What do Kassel, Rosso, and Turaev mean by "duality"?

In their book "Quantum Groups and Knot Invariants", Kassel, Rosso, and Turaev prove that $U_q\mathfrak{sl}(N+1)$ has a PBW basis. I'm having trouble following the last step, though. In ...
Petra's user avatar
  • 303
2 votes
0 answers
106 views

Is the preimage of a Hopf subalgebra a Hopf subalgebra?

The following must be simple, but I have no intuition here, so excuse me. Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
Sergei Akbarov's user avatar
4 votes
0 answers
119 views

"The" Universal R Matrix in Quantum Groups

For a quantum group (a quasitriangular Hopf algebra) $A$, it has a distinguished element in $A \otimes A$ called the universal R matrix in many texts (e.g. Kassel). This confuses me, because nowhere ...
Aaron's user avatar
  • 195
0 votes
1 answer
41 views

Verifying Yang Baxter Equation

$V$ be a 3 dimensional space with basis $\{e_1,e_2,e_3\}$ over $\mathbb{C}$. Let $q\in \mathbb{C}^\ast$. $\beta: V\otimes V \to V\otimes V$ defined by \begin{equation} \beta(e_i\otimes e_j) = \...
Learner's user avatar
  • 369
1 vote
0 answers
46 views

If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.

Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
Anacardium's user avatar
  • 2,522
6 votes
1 answer
184 views

Reference request: unusual expansion of product of binomial coefficients

There is a well-known formula for the product of the binomial coefficients: $$\binom{n}{a}\binom{n}{b}=\sum_{i=0}^{min(a,b)}\binom{a+b-i}{i,a-i,b-i}\binom{n}{a+b-i}$$ I'm interested in a different ...
Alvaro Martinez's user avatar
2 votes
1 answer
107 views

How to calculate $\text{End}(V^{\otimes n})$

Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles. For example, in the case ...
William Leynoid's user avatar
2 votes
0 answers
47 views

Are all module categories over the category $\mathrm{Rep}_k(G)$ given by $\mathrm{Rep}_k(H)$ for a subgroup $H\subset G$?

I am reading this article, where they classify the semisimple module categories with finitely many irreducibles over the category of representations of quantum $SL(2)$. They say that, in the classical ...
shin chan's user avatar
  • 249
0 votes
1 answer
19 views

Compute $S(E^i F^j K^l)$ in $U_q$

Here is the question I am trying to solve: Compute $S(E^i F^j K^l)$ in $U_q.$ Here is my thoughts: Definition: We define $U_q = U_q(\mathfrak{sl}(2))$ as the algebra generated by the four variables $E ...
Emptymind's user avatar
  • 2,087
0 votes
2 answers
326 views

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.) Here is the exercise before it: Let $L$ ...
Emptymind's user avatar
  • 2,087
3 votes
2 answers
230 views

Determining the grouplike elements of a Hopf algebra

Here is the question I am trying to solve: For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$ Some definitions: 1-To any Lie Algebra $L$ we assign an (...
Emptymind's user avatar
  • 2,087
0 votes
1 answer
135 views

Proving that a symmetric bilinear form on $L$ is invariant

Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$ ...
user avatar
0 votes
0 answers
46 views

Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?

Here is the question I am trying to solve: (Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
user avatar
3 votes
1 answer
117 views

Proving the uniqueness of a map

Here is the question I am trying to solve: (Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
user avatar
2 votes
1 answer
88 views

What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?

Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra: $\require{AMScd}$ $$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\...
Emptymind's user avatar
  • 2,087
1 vote
1 answer
65 views

Do multi-parameter unitary subgroups exist?

I'm working with an $N$-dimensional quantum system that is defined by the following Hamiltonian $$ H = H_{\text{drift}} + \sum_{j=1}^n a^j H_{\text{drive}}^j $$ Where $n \ll N$ (in my case $n = 4$ ...
Aaron Trowbridge's user avatar
2 votes
0 answers
49 views

q analogue of a number is a polynomial in $[2]_q$

$[m]_q = \frac{q^m−q^{-m}}{q-q^{-1}}$ is the q- analogue of the number $m\in\mathbb{Z}_{\geq0}$. $[0]_q=0$, $[1]_q=1$ and $[2]_q=q+q^{-1}$. I don't know how to prove Every $[m]_q$ can be expressed as ...
Learner's user avatar
  • 369
0 votes
1 answer
60 views

Why we need an orthonormal basis?

Here is the question I am trying to solve: Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
Emptymind's user avatar
  • 2,087
1 vote
1 answer
151 views

What happens if $(f_1\otimes g_1)(u_1 \otimes v_1) = (f_2 \otimes g_2)(u_2 \otimes v_2)$?

Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their tensor product $f \otimes g: U \otimes V \to ...
Emptymind's user avatar
  • 2,087
1 vote
1 answer
106 views

Poincare Series of a graded algebra (revisited)

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Intuition's user avatar
  • 3,181
2 votes
1 answer
124 views

Poincare series of a graded algebra

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Intuition's user avatar
  • 3,181
1 vote
1 answer
81 views

Finding expression for $([[e_r^{\ast},e_k], e_s^{\ast}], e_l).$

Let $(\mathfrak {g}, \mathfrak {g}_+, \mathfrak {g}_-)$ be a finite dimensional Manin triple i.e. $\mathfrak g$ is a finite dimensional Lie algebra endowed with a non-degenerate invariant bilinear ...
Anil Bagchi.'s user avatar
  • 2,912
1 vote
1 answer
73 views

When is a triangular structure called non-degenerate?

Let $(\mathfrak {g}, [\cdot, \cdot], \delta)$ be a triangular Lie bialgebra with classical $r$-matrix $r \in \bigwedge^2 \mathfrak {g}.$ In such a case we say that $r$ is a triangular structure on $\...
Anil Bagchi.'s user avatar
  • 2,912
0 votes
0 answers
212 views

Does no one know where the canonical commutation relations come from?

It's been quite a long time, reading about the origins of the mathematical formulation of quantum mechanics, absolutely no one gives a justified derivation of the canonical commutation relations, even ...
user536450's user avatar
0 votes
0 answers
187 views

Any symplectic manifold is a Poisson manifold.

Let $M$ be a symplectic manifold with a non-degenerate closed $2$-form $\omega.$ Then for any $f \in C^{\infty} (M)$ and for any vector field $u \in \mathfrak {X} (M)$ there exists a vector field $V_{...
Anil Bagchi.'s user avatar
  • 2,912
1 vote
1 answer
75 views

Quantum plane is a bialgebra

I am reading ‘Hopf algebras and their actions on rings’. Susan wrote the quantum plane as an example at 1.3.9 Example. He said $B = k \langle x,y \mid xy = qyx \rangle$, $0 \neq q \in k$ with ...
Z.B. Zuo's user avatar
  • 510
4 votes
0 answers
38 views

Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?

It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure,...
Chenqi Meng's user avatar

1
2 3 4 5
7