Tagged Questions

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

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3
votes
1answer
251 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
1answer
135 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
1answer
245 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
1answer
232 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
2answers
628 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
0answers
122 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
0answers
182 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
1answer
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
0answers
275 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
1answer
201 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
0answers
56 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
3answers
422 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
0answers
197 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
1answer
105 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
1answer
278 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
4answers
470 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
0answers
137 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
2answers
214 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
0answers
284 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 ...
9
votes
1answer
382 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
1answer
190 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
0answers
537 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
2answers
721 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
0answers
115 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
1answer
636 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
1answer
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
1answer
540 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
1answer
369 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
4answers
292 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
0answers
255 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
1answer
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
0answers
96 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
1answer
261 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
0answers
199 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 ...
8
votes
1answer
649 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
1answer
337 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
0answers
137 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 ...
4
votes
1answer
914 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 ...
42
votes
3answers
3k 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
1answer
486 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
1answer
346 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
0answers
413 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
1answer
307 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
2answers
586 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
1answer
465 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
2answers
592 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
0answers
328 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 $\pi_1(\Sigma)$. ...
4
votes
1answer
218 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
0answers
334 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
1answer
325 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 ...