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

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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 ...
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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 ...
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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 ...
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1answer
270 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 ...
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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 ...
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1answer
708 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
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1answer
360 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, ...
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143 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 ...
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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 ...
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4answers
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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 ...
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506 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 ...
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349 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 ...
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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 ...
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1answer
317 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 ...
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2answers
607 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 ...
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1answer
498 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, ~~~~~~~ ...
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627 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 ...
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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 $\pi_1(\Sigma)$. ...
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1answer
234 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). ...
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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 ...
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378 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 ...
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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 ...
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461 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
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1answer
538 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 ...
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578 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 ...
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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 ...
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2answers
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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 ...
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436 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 ...
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2answers
607 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 ...
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2answers
259 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 ...
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1answer
954 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 ...
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Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...
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Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
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1answer
681 views

Kontsevich Integral without associators?

Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
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1answer
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Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$

Consider the automorphism of the algebra $U_q(\widehat{\mathfrak{sl}}_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the ...
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Triangle condition for quantum 6j symbols?

For SU2 and even SU2(q) the triangle condition is, well, the triangle condition (conveniently, all irreps are described by (half)integer J completely). Additionally, all three J of a triple must add ...
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1answer
327 views

Knot polynomials: Skein>Matrix>Group?

OK, the heading was a bit tersely formulated... If you have a quantum group and an irrep, you theoretically know the R matrix (mathematicians are a notoriously idle lot, they give the general formula ...
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325 views

Open symplectic embeddings and deformation quantization

I would like to know whether there is a specific relationship between the deformation quantizations of the Poisson algebras of the functions on a symplectic manifold, say $M$, and of the functions on ...
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239 views

Oriented vs. unoriented knot polynomials

I only ask for the subset of Reshetikhine-Turaev invariants for now. Is the following sum up into cases a) and b) correct? - Dots in Dynkin diagrams of Lie groups G come as a) symmetry-equivalent ...
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Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
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Braidings for Comodules of Co-quasi-triangular Hopf algebra

Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a ...
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When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...
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1answer
396 views

How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?

The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
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Product knot invariants

Let $I_1(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_1$ and representation $\lambda_1$ having the S matrix $S_1$. Let $I_2(L)$ be a Reshetikhine-Turaev link ...
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Local Cohomology and Maximal-Cohen-Macaulay modules

Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course): Let $A$ be a graded connected noetherian algebra (not necessarily ...
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Is there a notion of partial trace in a ribbon category?

I've seen some definitions of "right partial trace" and "left partial trace" in http://arxiv.org/abs/1103.1660, but these don't seem canonical in any way. The motivation for this questions is that ...
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Can any Laurent polynomial symmetric under q->1/q and whose roots are all roots of unity be written as a ratio of products of quantum numbers?

Quantum dimensions are quantum integer fractions (or so I heard). Example: $G_2(\lambda_2)$ (technically it should read q^{1/6}, I know...) $q^{-10}+q^{-8}+q^{-2}+1+q^2+q^8+q^{10}$ = ...
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Hopf Algebras and Quantum Groups

I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first ...
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472 views

How can I see the “missing” Poisson center when the rank of the Poisson structure drops?

Recall that a Poisson algebra is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The ...
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Quantum knot invariants dummyfied

This construction should define an invariant for colored tangled trivalent graphs. Choose a quantum group G. (Without loss of geniality, G=A1(q) :-) Color the edges with representations of ...