Questions tagged [quantum-groups]

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

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142 views

What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one. First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. ...
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1answer
75 views

Structure of simple modules of $U_q(\mathfrak{sl}_2)$ at a root of unity.

This is Theorem VI.5.5 in Kassel's Quantum Groups: Let $q$ be a root of unity. Prove: Any simple $U_q$-module of dimension $e$ is isomorphic to a module of the following list: $V(\lambda, a,...
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Verma module $V(\lambda)$ of $U_q(\mathfrak{sl}_2)$ is not simple $\implies$ $\lambda = \pm q^n$

This is exercise $VI.6.2$ in Kassel's Quantum Groups. My observations are the following: If $V(\lambda)$ is not simple, then there is a surjective module homomorphism $\psi:V(L)\rightarrow V$ to some ...
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3answers
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Do there exist $2 \times 2$ matrices satisfying $XY = q YX$, $YZ = q ZY$ and $ZX = q XZ$

Let $q = e^{2\pi i /3}$. Do there exist (distinct) $2 \times 2$ matrices with coefficients in $\mathbb{C}$ satisfying the commutator relations: $ XY = qYX $ $ YZ = qZY $ $ ZX = qXZ $ Even thinking ...
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1answer
75 views

Usefulness of representations of quantum groups

For compact quantum groups there exists a rich representation theory. What I still not well understand, how this helps in various calculations. My problem is somehow, that these representations are ...
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194 views

A question on coalgebras(2)

Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order comultiplication can be defined inductively as follows(with some abuse of notations we denote them by ...
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39 views

Uniqueness of quantizations

When studying quantum groups, in particular quantized universal enveloping algebras, people will tell you that such a quantization is in some sense unique. More specifically, you might hear that a ...
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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 $\...
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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 ...
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1answer
142 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 }...
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1answer
89 views

Relation between Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$.

What is the relation between the definitions of Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$? There are two definitions of $U_q(\widehat{sl_2})$. The following is Jimbo ...
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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 $ \...
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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 $q \in \mathbb{C}^\times$...
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1answer
33 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-...
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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 ...
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1answer
95 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 ...
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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 ...
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1answer
542 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\...
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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 ...
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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.
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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)....
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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 $\mathbb{C} \cong \...
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$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\},$...
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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 \...
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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 ...
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1answer
114 views

Quantum planes and quantum matrices.

Let $A = \mathbb{C}_q\langle x,y\rangle/(xy-qyx)$ be a quantum plane. Let $M = \left( \begin{matrix} a & b \\ c & d \end{matrix} \right)$. If we require that $x'y' = qy'x'$, where $\left( \...
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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 ...
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199 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 ...
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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 \...
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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(...
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1answer
176 views

Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the dual-...
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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 ...
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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 ...
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1answer
351 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...
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0answers
48 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?
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Trace of Product of Powers of $A$ and $A^\ast$

Let $n$ be odd, $\displaystyle v=1,...,\frac{n-1}{2}$ and $\displaystyle \zeta=e^{2\pi i/n}$. Define the following matrices: $$A(0,v)=\left(\begin{array}{cc}1+\zeta^{-v} & \zeta^v+\zeta^{2v}\\ \...
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1answer
68 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 ...
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2answers
168 views

Converse of Hopf Algebra Theorem

There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode. May I know if the converse is true? (i.e. if $S^2=id_H$, does it ...
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2answers
92 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(...
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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:...
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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 ...
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1answer
73 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 ...
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2answers
122 views

$U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$

In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations \begin{...
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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{...
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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, $...
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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.
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1answer
145 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 ...
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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. ...
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34 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 ...
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1answer
56 views

Reference request: product in $\mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$.

Let $a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where $X, Y$ are two algebraic varieties. Suppose that algebraic group $T$ acts on $X, Y$. Then there are coactions $\delta:...