All Questions
Tagged with qa.quantum-algebra reference-request
58 questions
3
votes
0
answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
1
vote
1
answer
64
views
What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 137
5
votes
1
answer
505
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
2
votes
0
answers
94
views
Automorphism group of the quantum Weyl field
Let $\mathsf{k}$ be a field with zero characteristic, and $q \in \mathsf{k}$ a non-zero elemento which is not a root of unit. The quantum plane $\mathsf{k}_q[x,y]$ is the algebra given by generators $...
- 3,318
8
votes
0
answers
138
views
Relation between "homotopical" and "representation-theoretic" categorifications
This might be a bit of a soft question, and I apologize in advance for this. Here it is:
What is the relationship between the "homotopical" categorification (e.g. we consider every category ...
- 1,374
5
votes
0
answers
207
views
parameter of a quantum group
I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
- 323
3
votes
0
answers
109
views
Noncommutative group schemes corresponding to quantum groups
I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
- 1,516
6
votes
1
answer
217
views
Reference request : table of quantum Clebsch-Gordan coefficient
From a quick Google search, one can find a table of the first Clebsch-Gordan coefficient. For example this table. Those are used to pass between the tensor product bases and the bases as sum of ...
- 437
6
votes
1
answer
172
views
Norm of contragredient of unitary representations of compact quantum groups
Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a ...
- 960
5
votes
0
answers
255
views
Is the category of topologically free $k[[h]]$-modules locally presentable?
$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...
- 8,524
6
votes
1
answer
226
views
Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
7
votes
0
answers
244
views
Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?
If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary ...
- 784
3
votes
0
answers
752
views
Where can I find Drinfeld's original papers on quantum groups?
Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
- 1,066
5
votes
1
answer
165
views
Reference requests : Presentation of the braided dual of $U_q(\frak{sl_2})$
I am interested in the braided dual of the quantum group $U_q(\frak{sl_2})$. This is the algebra generated by the matrix coefficients but where the multiplication is twisted by an action of the $R$-...
- 51
5
votes
1
answer
215
views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
- 5,230
9
votes
0
answers
445
views
Lagrangian subgroups/submanifolds, 2d topological boundary and 3d "non-abelian" Chern–Simons theory
This post is meant to ask for proper references to fill a gap in the literature.
My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
- 10.5k
5
votes
0
answers
481
views
Reference Request: Vertex Algebras
I am currently a graduate student in mathematics with an interest in vertex algebras. I am comfortable with the algebraic aspects and would like to learn more about the geometric aspects. The issue is ...
- 223
0
votes
0
answers
167
views
Negative $q$-binomial series: reference request
There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$
$$
\sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots
(1+q^{r}x)^{-1}
$$
where the $q$...
- 1,143
1
vote
0
answers
76
views
Reference request: classical polarization argument
I am reading the article ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS,
SOLOMON IDEMPOTENTS AND THE MAGNUS FORMULA of Frédéric Chapoton and Frédéric Patras. Many definitions and results used in this article ...
- 153
6
votes
3
answers
442
views
Commutative and Cocommutative Quantum Groups
I am using this definition:
An algebra of functions on a finite quantum group $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
I have the following (very well known --...
- 1,037
7
votes
0
answers
323
views
Flat connection from gauged WZW model
$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...
- 8,524
6
votes
1
answer
357
views
Open questions on (finite) tensor categories
I would like to know about problems on (finite) tensor categories. I have read Etingof´s notes from his course at MIT. I have a question:
There exists any reference where I can find an open problem ...
- 63
7
votes
2
answers
405
views
The Irreducible Representations of the Sekine Quantum Groups
Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
- 1,037
4
votes
0
answers
220
views
What does "control of a deformation problem" mean?
Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
- 3,880
3
votes
0
answers
216
views
A "nice" Orthogonal Basis for Translation Invariant Symmetric Polynomials
It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
- 613
11
votes
2
answers
1k
views
Yang–Baxter explanation
What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...
- 12.3k
2
votes
0
answers
193
views
How to write BRST-BV for dg-Lie?
The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?
- 3,880
1
vote
0
answers
115
views
Are the Standard Quantum Groups Coordinate Rings Noetherian?
Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?
- 163
12
votes
1
answer
688
views
Comparing a Chevalley basis with the canonical basis of the adjoint module?
First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \...
- 52.9k
8
votes
1
answer
358
views
Abel's five terms relation from Yang-Baxter equation?
Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.
- 838
2
votes
2
answers
385
views
Reference to complete derivation of Kossakowski–Lindblad equation and its steady solutions
Are there recommended textbook or good intro-reference to explain with complete stretch of Kossakowski–Lindblad equation especially how is the idea to derive it from ground?
$$\dot\rho=-{i\over\hbar}[...
- 23
4
votes
0
answers
205
views
Where is the Courant operad discussed?
Where is the Courant operad discussed? And hopefully defined precisely.
By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
- 3,880
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
1
vote
1
answer
168
views
$q$-differential equation for the Rodgers polynomials?
The Rodgers polynomials $C_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on Wikipedia. For the special case of $q=1$, we ...
- 167
7
votes
1
answer
672
views
The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...
- 1,037
10
votes
1
answer
410
views
Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.
Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links?
I'm assuming there does exist such ...
- 115
1
vote
0
answers
94
views
A list of infinite dimensional coalgebras over a field
I'm looking for a vast list of infinite list of coalgebras of infinite dimension, I'm familiar with the standard ones, any example is well received. I'm currently writing a paper on coalgebras, so the ...
- 307
7
votes
1
answer
321
views
Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...
- 8,559
13
votes
7
answers
2k
views
Open problems in the theory of compact quantum groups
What are the important open problems in the theory of compact quantum groups? Or conjectures?
Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
24
votes
9
answers
3k
views
expository papers related to quantum groups
Hello all,
I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related ...
- 1,719
4
votes
0
answers
218
views
FRT construction in the case of super algebras
I'm looking on papers which are talking about the super quantum algebra osp(2|1). I want to understand how one applies the FRT construction in the case of osp(2|1).
Of course there is a super ...
- 113
5
votes
2
answers
923
views
Status of a conjectural definition of H. Nakajima
In his paper '$t$-analogue of $q$-characters of finite dimensional representations of quantum affine algebras' - http://arxiv.org/abs/math/0009231 - H. Nakajima states a conjectural definition of the $...
- 1,201
10
votes
2
answers
664
views
$6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials
Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin.
My Question:
How much are known about quantum $6j$-symbolos ...
- 2,273
3
votes
1
answer
374
views
On expressions of colored Jones polynomials
In the paper by Masbaum, it was shown that the colored Jones polynomials for a twist knot $K_p$ can be written as
\begin{eqnarray}
J_{n}(K_p;q)&=&\sum_{k=0}^{\infty} {\cal C}_{K_p}(k)
\...
- 2,273
8
votes
2
answers
1k
views
AJ conjecture for links
Garoufalidis proposed a conjecture on $q$-difference equations for the colored Jones polynomials of knots.
\begin{equation}
\hat{A}_K(\hat{l},\hat{m};q)J_n(K;q)=0
\end{equation}
where the actions of ...
- 2,273
8
votes
1
answer
1k
views
Closed formula for colored Jones polynomial of the trefoil? (reference request)
(EDIT: Powers of $q$ in the formula corrected.)
I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil:
$\frac{1}...
- 2,869
12
votes
2
answers
789
views
Knot Invariants from Twisted Quantum Doubles
In "Topological Gauge Theories and Group Cohomology", Dijkgraaf and Witten construct a 3-manifold invariant from a finite group $G$ and 3-cocycle $\omega$. I would think there is also an associated ...
- 121
15
votes
0
answers
1k
views
Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
- 18.7k
13
votes
0
answers
591
views
Generalization of Witten's computation of the volume of moduli space
Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.
There ...
- 18.7k
12
votes
1
answer
834
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
- 8,559