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

**5**

votes

**2**answers

658 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

160 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

128 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

186 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

143 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

298 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

182 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

742 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

915 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

154 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 ...

**22**

votes

**2**answers

731 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

236 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

356 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

143 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

290 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

229 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" ${R_i,R_j,R_k}$ ...

**5**

votes

**0**answers

131 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. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...

**4**

votes

**1**answer

209 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

222 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

105 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

385 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

311 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

263 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)
...

**4**

votes

**1**answer

146 views

### Bigalois Groupoid Of Drinfel'd Group Double

2-cocycles of a given Hopf algebras $H$ no longer form a group, but a groupoid between different Doi twists of the Hopf algebra $H,L$. The subgroup of "lazy" 2-cocycles precisely preserve the ...

**4**

votes

**1**answer

257 views

### braids and dynamics of roots of a polynomial

The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time ...

**4**

votes

**1**answer

243 views

### Vertex embeddings of quantum groups via quivers

Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion ...

**7**

votes

**2**answers

677 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 ...

**0**

votes

**0**answers

136 views

### Which is the configuration space of a finite dimensional Hilbert space ?

A quantum particle on the real line R has as configuration space this real line R, while its state space is the infinite dimensional complex Hilbert space of square integrable complex valued functions ...

**5**

votes

**0**answers

188 views

### deformed Gauss Bonnet formula?

I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss ...

**0**

votes

**1**answer

82 views

### Pseudo-dimensions of quantum Lie groups

In my hunt for spurious "alternatives" to the $E_7$ family I always encounter
"fake" solutions. They turn out to be mostly $E_7$ family solutions disguised
by $q\rightarrow{i*q}$. The effect is that ...

**4**

votes

**0**answers

296 views

### TQFT and Mapping Class Groups

It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference ...

**1**

vote

**1**answer

203 views

### Trivial Hopf Coinvariant Subspace Example

For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that
$$
(G \otimes V)^{\text{co}H} = \lbrace g_{(1)} ...

**2**

votes

**0**answers

57 views

### 2-cocycles/Bigalois-objects over nontrivial liftings

It is easy to extend group-2-cocycles to smash-products with Nichols algebras over the group (just trivially). The same certainly doesn't work for nontrivial liftings.
As I would like to check a ...

**4**

votes

**3**answers

434 views

### q-deformation of the permutation group?

The only definition of a quantum group I know of involves q-deforming the relation $EF-FE=H$ or for SL(2):
\[ \left[ \left( \begin{array}{cc} 0 & 1 \\\\ 0 & 0 \end{array} \right), \left( ...

**4**

votes

**0**answers

219 views

### Braid group action on canonical basis

This is a question about Lusztig's theory of based modules. This theory is elementary but far from easy and is developed in Chapter 27 of "Introduction to quantum groups".
Let $V$ be a highest weight ...

**0**

votes

**1**answer

118 views

### Coinvariant Subalgebras of Hopf Comodules and Quotients

For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that ...

**5**

votes

**1**answer

283 views

### Jones(unlink)=phi

Somewhat nebulous question: there are many well known "special" values of
the Jones polynomial, especially those at roots of unity. I always run into
one that has unlink value $\phi$ (golden mean) and ...

**6**

votes

**4**answers

526 views

### level 2,3 characters of affine su(2)

Does anyone know where I can find an explicit formula to compute the level 2 or level 3
characters of affine $su(2)$? I have found several sources that give a formula to compute the
level 1 ...

**1**

vote

**0**answers

140 views

### quantization of Poisson manifolds/ bialgebras

Can we apply the theorem of quantization of Lie bialgebras of Etingof-Kazdhan http://www.springerlink.com/content/h285401597138rg7/ to a Poisson manifold $\textbf{R}^{n}?$
Does it give something in ...

**2**

votes

**2**answers

253 views

### A property of quantum group R matrices?

Assume Q is a quantum Lie group which allows a R matrix (with the usual
quantum Yang-Baxter equation).
Take the Jordan normal form J of R. Does a Q exist with offdiagonal J elements
(i.e. R has ...

**2**

votes

**0**answers

301 views

### Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone!
Question
I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices ...

**10**

votes

**1**answer

445 views

### Unitary representations of Quantum Groups

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...

**1**

vote

**1**answer

204 views

### associative Yang-Baxter on U(g)

Consider $\mathfrak{g}$ a finite-dimensional Lie algebra over the field $\textbf{k}$.
If A is an associative algebra, we are searching for functions from $\textbf{C}\times \textbf{C}$ to $A\otimes A$ ...

**11**

votes

**0**answers

570 views

### Capelli determinant = Duflo ( determinant) - was it known ?

Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$).
I googled a lot ...

**13**

votes

**2**answers

786 views

### Intrinsic characterization of Soergel bimodules?

A Soergel bimodule (for $S_n$) is a bimodule over $R = \mathbb{Q}[x_1,\dots,x_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules
$$B_{i,i+1} = R \otimes_{i,i+1} R$$
...

**1**

vote

**0**answers

121 views

### Hopf Comodule Decompositions and Coinvariant Elements

For $H$ an Hopf algebra, and $V$ a right $H$-comodule with coaction $\Delta_R$, for which there exists a vector space decomposition $V=V_+ \oplus V_-$. Are the following two statements equivalent?
...

**9**

votes

**1**answer

646 views

### Compatibility of the KZ connection with operadic composition

In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}\_{0,n}$'s?
Here are (some) details, ...

**0**

votes

**1**answer

98 views

### Span of tangle vector space for different Lie groups

In his $G_2$ paper, Kuperberg gives the following numbers of acyclic freeways
for n=0...6: 1, 0, 1, 1, 4, 10, 35. (Which is identical to $dim Inv(V^{⊗n}_{1,0})$, the spanning size of the tangle vector ...

**4**

votes

**1**answer

634 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:
...