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7 views

Is it possible to obtain a Hermitian, positive semidefinite matrix as some sum of non-commuting matrices?

I am working with generalized Pauli matrices given by $X \vert j \rangle = \vert (j+1)mod~p \rangle$, where $p$ is a prime number. $Z = \vert j \rangle = \omega \vert j \rangle$, where $\omega = ...
0
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2answers
27 views

Reference request for a special kind of numbers.

Let $q$ be an element of a field $k$ (possibly $\mathbb{C}$), different from $-1$ and $1$. We have $$[n]=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-2}+\dots+q^{-n+1}$$ Where $n$ is a natural number. ...
2
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0answers
26 views

How to understand the algebra $U_A(Lg)$?

Let $g$ be a complex simple Lie algebra and $Lg = g \otimes \mathbb{C}[t, t^{-1}]$. Let $q$ be a non-zero complex number and $U_q(Lg)$ the quantum loop algebra corresponding to $g$. Let $A = ...
2
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1answer
29 views

Motivation behind Quasitriangular Hopf algebra

I would like to know why it is interesting to define the quasi-triangular structure on a Hop algebra. I understand that the pseudo-co-commutative (the existence of an intertwining operator between the ...
1
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0answers
18 views

A triangular Hopf algebra and its unitary R-matrix

Why is the R-matrix of a Hopf algebra called unitary when it satisfies the relation $$R^{-1}=R_{12},$$ I would say invertible, why then call it unitary? Is that a nomenclature that maybe comes from ...
4
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0answers
44 views

Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
0
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0answers
21 views

Binomial-like expansion for non-commuting operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
1
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2answers
30 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
2
votes
1answer
40 views

What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
0
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0answers
19 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 ...
3
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0answers
40 views

What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
1
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0answers
36 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
2
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0answers
39 views

Quantum flag manifolds

What are quantum flag manifolds? Please introduce me a good reference for to learn them. I no nothing about flag manifolds and I need an introductory text book related to this subject. If someone ...
0
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0answers
26 views

Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
2
votes
2answers
56 views

What is $\Delta(1)$ for $1$ in $U(\mathfrak{g})$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra. Then $U(\mathfrak{g})$ is a hopf algebra. Is $\Delta(1) = 1 \otimes 1$ or $\Delta(1) = 1 \otimes ...
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0answers
28 views

An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
2
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1answer
61 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
2
votes
1answer
57 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
1
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1answer
54 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
1
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1answer
22 views

Questions about root operators.

I am reading the notes. On line 13 in the section Root operators, it is said that The operator $f_1$ maps from the space $V(\mu)$ to $V(\mu-(1,-1,0))$. I don't know why. We have $$ f_i V (\mu) ...
4
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0answers
58 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
0
votes
1answer
35 views

How do I compute the specific map between two isomorphic finite C* algebras?

Starting with a finite C* algebra $\mathcal{A} \subset M_{n}\left({\mathbb C}\right)$ (complex $n\times n$ matrices), $\mathcal{A}$ is known to be isomorphic to a canonical algebra of the form ...
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0answers
122 views

A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ...
1
vote
1answer
50 views

Quantum: Toffoli gates

How do I prove that a toffoli gate is a controlled CNOT gate. i.e, $G_{Toffoli} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes G_{CNOT}$. I am not sure how to approach this, thoughts? ...
2
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0answers
47 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
0
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1answer
74 views

Is there a Relationship between Quantum Groups and Lie Groups?

I know that the Lie Group is all about continuous transformation groups. I know that the quantum group denotes various kinds of noncommutative algebra with additional structure. Transformation group ...
5
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1answer
106 views

The “$\mathbb{Z}_n$-theta function” - what is it? Is it being studied somewhere?

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a ...
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0answers
51 views

Quantum $SU(2)$ and $3$-sphere

There is a dipheomorphism of $SU(2)$ and $3$-sphere. What happens when you construct quantum $SU(2)$? Do we have an lifting of homeomorphism (or some other morphism) of quantum group to some other ...
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0answers
64 views

An $SU(3)$ isomorph in Clifford $G(5,0)$?

I am a computer scientist using the geometric (Clifford) algebras $G(n,0)$ over $\mathbb{Z}_3 = \{0,1,-1\}$ to describe distributed computations. My question concerns $G(5,0)$ with generators ...
3
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1answer
66 views

$f:C\longrightarrow D$ coalgebra homomorphism $\Rightarrow$ $f^*:D^*\longrightarrow C^*$ algebra homomorphism

I'm trying to prove the next problem using only arrows (avoiding Sweedler's notation): If $f:C\longrightarrow D$ is a coalgebra homomorphism, then $f^*:D^*\longrightarrow C^*$ is an algebra ...
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2answers
871 views

What are Quantum Groups?

I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. ...
2
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0answers
54 views

A characterisation of the cyclic subfactors by the existence of a cyclic vector?

A cyclic subfactor is a subfactor admitting a distributive intermediate subfactors lattice. Let's start with the finite index irreducible depth 2 subfactors, i.e. the class of subfactors of the form ...
1
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0answers
38 views

The spectrum of an element of the convolution algebra of a nonabelian group

Let $G$ be a locally compact group and $L^1(G)$ its convolution algebra. If $G$ is Abelian, then the spectrum of an element $f \in L^1(G)$ is equal to the image of $\hat{f}$, the Fourier transform of ...
5
votes
1answer
129 views

Towards a Quantum Peter Weyl Theorem

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Let $(A,\Delta)$ be a *-Hopf algebra and let $\chi:V\rightarrow V\otimes A$ be a corepresentation of $(A,\Delta)$ on a vector ...
0
votes
1answer
41 views

Pauli Spin Operator General Rotation

I would like to calculate the Pauli spin operator rotation $$ U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ...
1
vote
1answer
92 views

Fusion rules and comultiplication for a Kac algebra

Let $\mathbb{A}$ be a Kac algebra (i.e. a Hopf C*-algebra) of finite dimension, and $\mathbb{A}^{*}$ its dual : Is there a formula revealing the fusion rules for the irreducible ...
1
vote
1answer
71 views

$C(G) \otimes C(G) \simeq C(G \times G)$

Let be $G$ a compact group and $C(G)$ the vector space of continue complex functions defined over $G$, $C(G):=\{f:G \to \mathbb{C}\}$, where $f$ is a continue function. How can I prove that $$ C(G) ...
5
votes
1answer
239 views

Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
0
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1answer
51 views

A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example ...
2
votes
1answer
64 views

Construction of a Manin triple

Let $\mathfrak{g}$ be a Lie bialgebra and $\mathfrak{g}^*$ be its dual. My question is how to construct a bracket on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^*$ such that we obtain a Manin ...
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vote
2answers
87 views

What is an Example of a Co-commutative but not Commutative Quantum Group?

I am looking for 'a' right candidate for an "abelian" quantum group. In a comment to another question it was suggested that the correct candidate was co-commutative. It is straightforward to show ...
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0answers
157 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
4
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0answers
123 views

Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
2
votes
1answer
96 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
2
votes
1answer
79 views

A question on a Hopf algebra

Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode. I am reading a proof of the fact $S(xy)=S(y)S(x)$. First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by ...
3
votes
1answer
66 views

Center of a quantum matrix algebra

Let $p \in k^\times$ be a nonroot of unity. It seems to be a well-known fact that the center of the quantum matrix algebra $\mathcal{O}_p(M_n(k))$ is generated by the quantum determinant $D_p$. It is ...
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1answer
94 views

Different definition of antipode for $SL_q(2)$?

In Majid's book (A Quantum Groups Primer) (pg11), the antipode for the Hopf algebra $SL_q(2)$ is defined as $Sd=a$, $Sa=d$, $Sb=-qb$, $Sc=-q^{-1}c$. However, in Kassel's book (Quantum Groups) (pg ...
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1answer
135 views

Quantum Groups: prove $U_1'\cong U[K]/(K^2-1)$ and $U\cong U_1'/(K-1)$

Regarding this theorem, which is in Kassel pg 126, I have two questions. I have typed in the relevant material for reference. 1) How does $$U_1'\cong U[K]/(K^2-1)$$ imply $$U\cong U_1'/(K-1)$$? ...
2
votes
1answer
94 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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0answers
171 views

Basis of $SL_q(2)$

While trying to show that $SL_q(2)$ is noncocommutative, I needed to prove the following fact: Show that the set $\{a^ib^jc^k\}_{i,j,k\geq 0}\cup\{b^ic^jd^k\}_{i,j\geq 0,k>0}$ is a basis of ...