Questions tagged [quantum-groups]

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

1
vote
0answers
60 views

group von Neumann algebra and its Plancherel weight

Lat $G$ be a locally compact group and $\mathcal{L}(G)$ its von Neumann algebra equipped with the Plancherel weight $\omega_G$. If $G$ is discrete then $\omega_G$ is a finite trace. Suppose that $\...
4
votes
0answers
40 views

Where does the condition $\pi(X_-)+\pi(X_+)=0$ in dual Lie algebra come from?

I am trying to understand the dual Lie algebra in a Lie bialgebra. In the above article, it is said that: "Let ${\displaystyle {\mathfrak {g}}}$ be any semisimple Lie algebra. To specify a Lie ...
1
vote
1answer
138 views

Generators Relations of the Weyl Group

Suppose we have a Root System and call $s_{\alpha}$ the transformation $$s_{\alpha}\left(\beta\right)=\beta-\frac{2\left\langle \beta,\,\alpha\right\rangle }{\left\langle \alpha,\,\alpha\right\rangle }...
1
vote
0answers
49 views

Question on simplification of terms in a Hopf algebra

Let $H=(H,\mu,\eta,\Delta,\epsilon,S)$ be a Hopf algebra and $S$ is the antipode in $H$. By the definition of antipode $S*id_H=id_H*S=\eta \circ\epsilon $. If I denote the Sweedler's notation by $ \...
1
vote
1answer
107 views

Dual Jacobi identity for Lie bialgebra

I'm studying Lie bialgebras: https://en.wikipedia.org/wiki/Lie_bialgebra. I'm a bit confused about the way of writing the so called "dual Jacobi identity". On Majid's book "a Quantum group Primer" ...
2
votes
1answer
67 views

Finding an explicit isomorphism between $\mathbb{C}_q[x^{\pm 1},y^{\pm 1}]/\langle x^k-\alpha, y^k-\beta \rangle$ and $M_k(\mathbb{C})$

Consider the quantum torus \begin{align*} \mathbb{C}_q[x^{\pm 1},y^{\pm 1}] = \frac{\mathbb{C}\langle x,y \rangle}{\langle yx-qxy \rangle }[x^{-1}, y^{-1}] \end{align*} where $q \in \mathbb{C}^\times$...
1
vote
1answer
32 views

The antipode and the compact quantum group von Neumann algebra

I came across a technical question (at least I believe so) of the following nature. Let $(\mathbb{G},\Delta)$ be a compact quantum group, where $\mathbb{G}$ is a $C^*$-algebra and $\Delta$ is a co-...
1
vote
1answer
94 views

Haar state on nonseparable, noncocommutative compact quantum group

I am trying to understand what seems to be a common knowledge, that every compact quantum group has a Haar state. However, each approach that I found on the internet is a bit hard for me to grasp. I ...
0
votes
1answer
56 views

States uniquely defined by operators

In this article authors define quantum states $\left|\psi\right\rangle$ (elements of some Hilbert space H - in most simple case a $2\times 1$ complex vector such that $|\psi| = 1$) by a set of ...
6
votes
0answers
93 views

Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
2
votes
1answer
539 views

Frobenius’ Method to Solve an ODE (Hydrogen Atom - Radial Equation)

My goal is to find two linearly independent solutions to the ODE $$ r^2\frac{d^2R}{dr^2}+2r\frac{dR}{dr}+[r^2+\lambda r-l(l+1)]R=0 $$ in the interval $[0,\infty)\ni r$, where $R=R(r)$, and $\lambda\...
1
vote
0answers
39 views

Dual Pairing between $U_q(\mathfrak{su}(n))$ and $\mathcal{O}(SU_q(n))$

In page 327 of Klymik and Schmudgen`s book Quantum Groups and their representations, theorem 18 reads: There exists unique dual pairings of the pairs: $U_q(gl_n)$ and $\mathcal{O}(GL_q(n))$, SL_q(n)....
3
votes
0answers
68 views

On root vectors

I don't fully understand the definition of root vectors in the construction of the so called Andruskiewitsch-Schneider Hopf-algebras. See for example section 2 of the paper 'On the Classification of ...
0
votes
0answers
50 views

Trivial representation, sign representation of $gl_n$.

How to write down the action of trivial representation and sign representation of $U_q(gl_n)$? The algebra $U_q(gl_n)$ is generated by $E_i, F_i, K_i$, $i=1,2,\ldots, n-1$. Let $\mathbb{C} \cong \...
2
votes
0answers
83 views

$VN^{\infty}$ as an example of a quantum group

I'm trying to learn quantum groups and I have the following problem: Suppose that $G$ is a locally compact group and $$VN^{\infty}(G) = \bigg\{L_f \in \mathcal{B}(L^2(G))\ :\ f \in L^{\infty}\bigg\},$...
1
vote
1answer
37 views

How to interpret a braiding as a natural transformation?

A braided monoidal category is a monoidal category with a braiding (a commutativity constraint $\gamma_{A,B}: A \otimes B \to B \otimes A$) which satisfies the hexagon identities: $$ \gamma_{A, B \...
3
votes
0answers
39 views

Grothendieck ring of finite dimensional modules over quantum affine algebras

Let $\mathcal{C}$ be the category of all finite dimensional modules over $U_q(\hat{g})$, where $g$ is a simple Lie algebra over $\mathbb{C}$. Let $M$ be a module in $\mathcal{C}$. Suppose that $M$ has ...
6
votes
1answer
257 views

Duality between universal enveloping algebra and algebras of functions

There is a well known duality (of Hopf algebras) between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous ...
1
vote
1answer
45 views

In non-semisimple category $M \otimes N \not\cong N \otimes M$?

Let $\mathcal{C}$ be the category of all finite dimensional $U_q(\hat{g})$-modules, where $U_q(\hat{g})$ is an quantum affine algebra. In the category $\mathcal{C}$, usually $M \otimes N \not\cong N \...
4
votes
2answers
198 views

If $H$ is a Hopf algebra, do we have $H^{cop}$ is a Hopf algebra?

Let $H=(H, m, u, \Delta, \epsilon, S)$ be a Hopf algebra, see for example the lecture notes, where $m$ is the multiplication, $u$ is the unit, $\Delta$ is the comultiplication, $\epsilon$ is the ...
2
votes
0answers
41 views

The relation between the central element in an affine Lie algebra and the corresponding quantum affine algebra.

Let $g$ be a Lie algebra. Then we have a corresponding affine Lie algebra \begin{align} \hat{g} = g \otimes \mathbb{C}[t,t^{-1}] \oplus \mathbb{C}c, \end{align} where $c$ is the central element. The ...
3
votes
1answer
105 views

Filtrations of quasi-Hopf algebras

What is a quasi-Hopf algebra filtration? I know what a Hopf-algebra filtration is, but what is the extra condition needed on the Drinfeld associator $\Phi$? And given such a filtration how does one ...
2
votes
0answers
47 views

$SU(2) \otimes SU(2)$ and Compact Matrix Quantum Groups

$SU_{\mu}(2)$ (where $\mu \in \mathbb{R}^{+}$) is an example of a compact matrix quantum group, as defined by Woronowicz, but is $SU(2) \otimes SU(2) = Spin (4)$ also a compact matrix quantum group?
1
vote
1answer
52 views

Cotensoring by a Hopf Algebra

For $H$ a Hopf algebra, with bijective antipode. For a right, and a left, $H$-comodule $(V,\alpha_R)$, and $(W,\alpha_L)$ respectively, the cotensor product of $V$ and $W$ is $$ V \square_H W := \ker(...
0
votes
1answer
66 views

What are the differences between submodules and subfactors.

In non-semisimple category, for example the category of representations of a quantum affine algebra $U_q(\hat{g})$, where $g$ is a simple Lie algebra over $\mathbb{C}$. What are the differences ...
1
vote
2answers
91 views

Quantum Serre relations and braided commutator.

I am reading the lecture notes. On page 21, it is said that when $a_{ij}=-1$, we have \begin{align} ad_c(x_i)^{1-a_{ij}}(x_j)=x_i^2x_j - (q+q^{-1})x_ix_jx_i+x_jx_i^2. \quad (1) \end{align} Here $ad_c(...
3
votes
1answer
141 views

Primitive elements of finite dimensional Hopf algebras

An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that: "If $H$ is a $\...
0
votes
1answer
63 views

Looking for a good name for this “Quantisation Regime”

This is an attempt to salvage the wreckage of my hope to motivate (finite) quantum groups a lá this question. Let $\{S_i\}_{i=0}^n$ be a family of finite sets and let $\varphi$ be a map $$\varphi:...
0
votes
0answers
61 views

Cosemisimple coalgebras and uniqueness of their cosimple decompositions [duplicate]

A coalgebra $C$ is called cosimple if it has no subcoalgebras. It is called cosemisimple if it is a direct sum of simple coalgebras. Is this direct sum decomposition unique? Explicitly, can there ...
2
votes
1answer
72 views

How to prove that the cross product is associative directly?

The following result is proved using diagrams in Lemma 14.2 on page 85 in the book a quantum groups primer by S. Majid. Let $B, C$ be two algebras in a braided monoidal category. Then the braided ...
0
votes
2answers
108 views

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
-2
votes
1answer
28 views

induction mathematics in bracket of quantum group

By the induction mathematics . How can prove the following [fm,f1m]=0,where f is generators in quantum group and m greter than 3.
2
votes
0answers
40 views

How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
2
votes
1answer
144 views

States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
1
vote
0answers
32 views

What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
0
votes
0answers
393 views

Relation between integrable representations and highest weight representations.

Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest ...
2
votes
0answers
38 views

Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
4
votes
1answer
53 views

Endomorphism ring of an irreducible comodule

Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that ...
2
votes
2answers
82 views

Quantum group notation

I was jumping into the deep end and reading a few papers and lectures on quantum groups. My knowledge on Lie algebras is a bit thin but I was just wondering the notation used in the starting of this ...
3
votes
1answer
99 views

Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
1
vote
1answer
160 views

How $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$

My question is how $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ if we set $t=q^h$ and $q$ tends to 1.
1
vote
0answers
45 views

A question of the book “a guide to quantum groups”

I am reading this book "a guide to quantum groups" written by V.C. and A.P. But the proof of propersition 4.2.3 on page 121 confused me. Just this place " Applying $id \bigotimes S \bigotimes S^2$ to ...
9
votes
1answer
124 views

Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
1
vote
0answers
62 views

Why the coproduct of quantum groups are defined in this way?

Let $U_q(g)$ be a quantum group generated by $e_i, f_i, k_{\lambda}$, $\lambda \in Q$, $Q$ is the weight lattice of the Lie algebra $g$. The coproduct of $U_q(g)$ is defined as follows (I only write ...
1
vote
0answers
71 views

An equation about the Sklyanin bracket

I am reading the lecture http://www.math.uiuc.edu/~ruiloja/Poisson2014/EtingofLectures.pdf. I have a question on page 25. I do not know how to calculate the equation (3.2). Thank you very much for any ...
1
vote
1answer
86 views

How calculate the cartan matrix of the twisted quantum affine algebras?

the cartan matrix of the type $A_{2}^{(2)}$, $A_{2r-1}^{(2)}$, $A_{2r-1}^{(2)}$, $D_{r+1}^{(2)}$, $E_{6}^{(2)}$. I know the cartan matrix of the type $A_{2}^{(2)}$ is \begin{align} \left( \begin{...
3
votes
1answer
93 views

Alternative Cotensor Definition

Let $H$ be a Hopf algebra, and $(M,\rho_r)$ and $(N,\rho_l)$ right and left $H$-comodules respectively. As usual, we define their cotensor product to be $$ M \square_H N := \text{ker}\{(\rho_r \...
0
votes
1answer
253 views

How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed?

I am reading the book a guide to quantum groups. I have a question on page 18. How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed? Any help will be greatly appreciated!...
1
vote
0answers
26 views

Representations of $SU_q(n)$

I am searching for a classification of all irreducibel representations of the quantum group $SU_q(n)$ for general $n$. Can someone give referenced or some statements about this? Moreover does one has ...
2
votes
0answers
62 views

What is $q^{\frac{1}{2}h \otimes h}$?

One can find in some sources (e.g. Pavel Etingof, Lectures on representation theory and KZ equation, page 91) formula for $R$ matrix (for $\mathfrak{sl}_2$ case) $$R = q^{\frac{1}{2} h \otimes h} \...