Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

**7**

votes

**1**answer

187 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 ...

**2**

votes

**1**answer

190 views

### A question on Lusztig's `graph with automorphism' construction?

Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix ...

**4**

votes

**2**answers

533 views

### $q$-Deforming Woronowicz's Leibniz Rule

The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and
$$
\text{d}:A \to \Omega,
$$
is a bimodule map, ...

**17**

votes

**3**answers

2k views

### What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...

**13**

votes

**1**answer

884 views

### On Tamarkin's proof of Etingof-Kazhdan quantization of Lie bialgebra

This question is motivated by the desire to understand better the interplay between Drinfeld associators, algebraic structures up to homotopy and related results.
Recall that a Lie bialgebra is a Lie ...

**4**

votes

**1**answer

314 views

### Classical limit and Drinfelds realization of quantum groups

Let
$$\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\oplus\mathbb{C}d$$
be untwisted affine Lie algebra (as defined in V.G.Kac, Infinite-Dimensional Lie Algebras, 3d ed. ...

**1**

vote

**1**answer

112 views

### Zero Sums in a $q$-Deformation Remain Zero for $q=1$

Looking at previous question, I begun to think and came upon the following more solid question: Let $SU_q(N)$ be the usual quantized coordinated algebra of $SU(N)$. Now consider, for some fixed value ...

**7**

votes

**0**answers

168 views

### Does the braid group act faithfully on the quantized enveloping algebra?

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where ...

**0**

votes

**1**answer

117 views

### Deformations and Dimensions: $q$-Deform Finite to Infinite?

Let $A$ be a (complex) (finitely generated) algebra with some type of $q$-deformation $A_q$, where $A_1 = A$. Moreover, let $V_q$ be a vector subspace of $A_q$, such that $V_1$ is a ...

**8**

votes

**2**answers

509 views

### Is the quantum algebra unique (up to isomorphism) in deformation quantization ?

Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).
Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and ...

**1**

vote

**0**answers

65 views

### Coproduct of Weak Bialgebras

Hi,
I have two questions concerning the coproduct of weak bialgebras.
First, I would like to know if there is a proof of the existence of the coproduct in the category of weak bialgebras?
Second, ...

**11**

votes

**7**answers

1k 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 ...

**8**

votes

**1**answer

414 views

### Eigenvalues of the free sphere

Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x_1,\ldots,x_n:\mathbb R^n\to\mathbb R$, and in fact we have the ...

**3**

votes

**0**answers

161 views

### AS Cohen Macaulay algebras and dualizing complexes

Let $A$ be an $\mathbb N$-graded algebra such that $A_0 = k$ is a field. This are usually called graded connected algebras.
One can define a torsion functor with respect to the ideal $\mathfrak m = ...

**3**

votes

**0**answers

300 views

### det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

Context: some probably know that there are Capelli identities which state
det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ...

**18**

votes

**9**answers

1k 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 ...

**4**

votes

**0**answers

136 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 ...

**8**

votes

**1**answer

180 views

### Are annihilation modules in the quantum torus necessarily principal?

I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but ...

**5**

votes

**0**answers

319 views

### Is there a version of the 2d cobordism hypothesis for surfaces with non-empty incoming and outgoing boundary?

Question: Is there a condition on an object $x$ of an $(\infty,2)$-category $\mathcal C$ which is equivalent to $x = Z(pt_+)$ for a unique TFT $Z$ from the $(\infty,2)$-category of framed bordisms ...

**6**

votes

**0**answers

296 views

### Paving conjecture for Toeplitz matrices

Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...

**2**

votes

**1**answer

289 views

### Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$?

When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation ...

**5**

votes

**3**answers

554 views

### Links with same Jones polynomial

Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing I would try is to compute the (formal - you would have ...

**5**

votes

**0**answers

438 views

### Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...

**7**

votes

**1**answer

325 views

### Reference request: the “Kauffman bracket skein category”?

There should be a category $3\text{CobTang}$ whose
objects are some kind of surfaces with a finite set of marked points
morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms ...

**0**

votes

**0**answers

116 views

### Schur's Di-Lemma: finite and Lie groups different?

For a finite group it's nothing special if two one-dimensional irreps pop up in a product, e.g. for $C_{3v}$ symmetry, $E\bigotimes{E}=A_1\bigoplus{A_2}\bigoplus{E}$ or in dimensions, $2*2=1+1+2$. ...

**2**

votes

**1**answer

95 views

### Zero Actions on a Hopf Module Preserved Under the AntiPode?

Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that
$$
m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0?
...

**5**

votes

**2**answers

691 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**

vote

**1**answer

163 views

### Generalizing the Reshitikhin-Turaev construction possible?

OK, I have to ask a dumb question again: Where do Lie groups enter in the
construction of the Reshitikhin-Turaev invariant? The parts of the proof I
understand are that 6j symbols take care of ...

**0**

votes

**2**answers

131 views

### Structure of Homomorphisms of commutative C^* algebra

Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the ...

**2**

votes

**1**answer

197 views

### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...

**7**

votes

**0**answers

150 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

**3**

votes

**1**answer

313 views

### Kazhdan Lusztig Map and conjugacy classes of Weyl groups.

The Kazhdan Lusztig map gives a correspondence between conjugacy classes of Weyl groups and nilpotent orbits. So, let $w$ be a conjugacy class and
$N$ be the nilpotent orbit it gets mapped to under ...

**0**

votes

**1**answer

187 views

### Phase choice for 6j symbols

If you define 6j symbols completely formally via trivalent graphs
(take http://math.ucr.edu/home/baez/qg-fall2000/qg10.2.html for a start,
but be careful - looks like Racah coefficients to me...well, ...

**12**

votes

**3**answers

922 views

### Isomorphisms of quantum planes

Let $k$ be a field and $q\in k^{*}$. The quantum plane $k_{q}[x,y]$ is the algebra $k\langle x,y\rangle/\langle xy=qyx \rangle$ (i.e. the quotient of the free non-commutative $k$-algebra on two ...

**17**

votes

**1**answer

1k views

### Question about the Yangian

I've a slightly technical question about the Yangian which I'm hoping an expert out there can answer.
Recall that the Yangian $Y(\mathfrak{g})$ is a Hopf algebra quantizing $U(\mathfrak{g}[z])$. ...

**0**

votes

**1**answer

155 views

### “Fictive” irreps of the enveloping general Lie algebra

Notation abuse warning: I will use the E7 series irrep names. You'll soon see why.
In the general Lie algebra, the irrep you "start" with is $J$, the
adjoint. From the series for ...

**24**

votes

**2**answers

822 views

### quantum groups… not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...

**2**

votes

**2**answers

246 views

### “Mini” fusion categories via 6j symbols

Just for fun, I set up the following scheme:
- A 6j symbol is everything that fulfils Biedenharn-Elliott. (Plus symmetry, orthogonality etc. if that doesn't follow from it anyway.)
- There are only a ...

**9**

votes

**2**answers

385 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 ...

**4**

votes

**0**answers

148 views

### The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...

**4**

votes

**0**answers

302 views

### On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials.
Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...

**1**

vote

**1**answer

243 views

### Simple 6j symbol question

Consider a Clebsch-Gordan expansion $R_i\bigotimes{R_j}=\bigoplus_p{R_p}$. Assume the irrep $R_k$ does NOT appear in the sum on the right side. Does it now follow that the "triangle" ${R_i,R_j,R_k}$ ...

**7**

votes

**1**answer

200 views

### Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$?

Let $n\in\mathbb N$. Let $k$ be a commutative ring in which $1,2,3,\ldots$ are invertible. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., ...

**4**

votes

**1**answer

222 views

### About the term “tangential derivation” on a free Lie algebra.

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, ...

**2**

votes

**1**answer

228 views

### Non-Faithfully Flat Quantum Homogeneous Spaces

Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form
$$
M:= \lbrace m \in G ~ | ~ m_{(1)} \otimes \pi(m_{(2)}) = m \otimes 1 \rbrace
$$
a ...

**1**

vote

**1**answer

108 views

### $H$-Hopf modules equal the tensor products of their coinvariants with H

In a comment for this old question, it was said that
>
There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with H. (This theorem is usually worded ...

**4**

votes

**2**answers

396 views

### When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?

I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a ...

**3**

votes

**1**answer

339 views

### Pictorial explanation of Dynkin index and quadratic Casimir?

Draw a colored loop. You explained quantum dimension. :-) OK, now you may use trivalent nodes and knot crossings too (with edges colored by irreps). You can now pictorialize 6j symbols, writhe ...

**25**

votes

**4**answers

2k views

### A mysterious Heisenberg algebra identity from Sylvester, 1867

I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...

**3**

votes

**1**answer

268 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)
...