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

**2**

votes

**0**answers

58 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

437 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

221 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

119 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

285 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

559 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

142 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

258 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

305 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

459 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

205 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

577 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

797 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

123 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

652 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

99 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

654 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:
...

**2**

votes

**1**answer

394 views

### Divisibility Rules for Quantum Integers

I take a random but practical example direct from "R-matrices and the magic square"
by Bruce Westbury: The adjoint irrep $A$ of the $E_7$ family has quantum dimension
...

**1**

vote

**4**answers

324 views

### Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction

Shahn Majid presents in section 9.4 of "Foundatations of Quantum Group Theory" a Tannaka-type reconstruction theorem for producing a $k$-bialgebra out of its $k$-linear monoidal category of modules ...

**5**

votes

**0**answers

270 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

**-4**

votes

**1**answer

2k views

### Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the ...

**1**

vote

**0**answers

97 views

### Group-like Elements in a Coquasitriangular Bialgebra

What do we know about group-like elements in coquasitriangular bialgebras?
In particular, we know that a commutative bialgebra in which all group-like elements are invertible is a Hopf algebra (see ...

**2**

votes

**1**answer

271 views

### A diophantine equation for the E8 knot polynomial family

Let $x,y,z$ be dimensions that appear in the Clebsch-Gordan series
$x*x=1+t+u+y+z$.
(E8 family if $t=x$ (say), but there is at least another family. E.g. B4(R4) belongs to the latter.)
With ...

**6**

votes

**0**answers

202 views

### Category of modules over a coPoisson-bialgebra

Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
...

**9**

votes

**1**answer

746 views

### Coherent states vs quantization of Lagrangian submanifold

Coherent states http://en.wikipedia.org/wiki/Coherent_states
are vectors in the Hilbert space which in certain sense are strongly localized
and "corresponds" to points in classical phase space (see ...

**5**

votes

**1**answer

379 views

### Is there a fusion rule in positive characteristic?

Verlinde's fusion gives a certain "tensor product" of representations of loop groups. The category of representations of loop groups has (essentially equivalent) two incarnations. One is analytic, ...

**1**

vote

**0**answers

145 views

### Is there a two-variable E8 polynomial? (Conjectural or proven)

On MO I learnt about the two-variable E7 polynomial (status: conjectural).
What about a two-variable E8 polynomial? I have reasons to believe such a
thing exists too, but I do magic, not math, so my ...

**5**

votes

**1**answer

1k views

### Homotopic morphisms between curved A-infinity algebras

I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...

**48**

votes

**4**answers

4k views

### Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...

**8**

votes

**1**answer

536 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 ...

**2**

votes

**1**answer

356 views

### Why is the quantum Lorentz group not connected?

Podles and Woronowicz' construct the quantum Lorentz group, by which they mean $SL_q(2,\mathbb{C})$, as a quantum double of the compact quantum group $SU_q(2)$. More precisely, it is a bicrossed ...

**9**

votes

**0**answers

487 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 ...

**6**

votes

**1**answer

319 views

### Quantum coordinate ring at root of unity

Noah Snyder gave a great answer to this question about different versions of a quantum group $U_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring ...

**7**

votes

**2**answers

623 views

### Quantum E6/E7 knot polynomials

Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8?
I suspect these haven't been ...

**3**

votes

**1**answer

545 views

### Alternative Definition of the Quantum Determinant?

Let $M_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i_j$ for $i,j = 1, \ldots ,N$, and reations
$$
u^i_ju^k_j = qu^k_ju^i_j, \text{ for } i < k, ~~~~~~~ ...

**8**

votes

**2**answers

658 views

### Is a bialgebra with all group-like elements invertible a Hopf algebra?

We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question :
Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g ...

**10**

votes

**0**answers

365 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 $\pi_1(\Sigma)$. ...

**5**

votes

**1**answer

252 views

### What vector space does the Kauffman bracket skein algebra of FxI act on?

The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). ...

**11**

votes

**0**answers

381 views

### Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups?
This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain ...

**3**

votes

**1**answer

445 views

### Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ?

Please give suggestions about soft to make symbolic computations with NON-commutative variables.
Typical examples I am interesting - Capelli identities
...

**10**

votes

**7**answers

2k views

### Hopf algebras examples

Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are ...

**6**

votes

**0**answers

466 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
...

**9**

votes

**1**answer

562 views

### Automorphic forms and quantum groups

The paper Eisenstein series and quantum affine algebras by Kapranov makes contact between automorphic forms and quantum groups. I haven't found even one other paper devoted to this theme.
Have other ...

**12**

votes

**2**answers

587 views

### Projective modules over quantum groups

My question is short:
How can one calculate $\operatorname{Tor}_{U_q(\mathfrak g)}(k,k)$?
($k$ is the ground field of characteristic zero).
If we had a regular universal enveloping algebra ...

**3**

votes

**2**answers

250 views

### Does there exist a canonical “degree” filtration on quantum groups?

For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of ...

**4**

votes

**2**answers

305 views

### monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis

Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e_1,\ldots, e_n$ some basis of $L$. The formula $[e_i,e_j] = \sum_k C_{ij}^k e_k$ determines the structure ...

**10**

votes

**1**answer

453 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 ...

**7**

votes

**2**answers

637 views

### What is a twisted modular operad?

I find Getzler and Kapranov's article Modular Operads difficult to understand. Can anyone explain what a (twisted) modular operad is conceptually, or what the underlying idea behind the concept of a ...

**2**

votes

**2**answers

263 views

### 2-tangles and quantum groups and 2-groups

Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and ...

**35**

votes

**1**answer

1k views

### Invertible matrices over noncommutative rings

Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...