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

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179 views

polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write $$X=\left( \begin{array}{ccc} 0 & 1\\ 0 & 0\\ \end{array} \right),~~ Y=\left( \begin{array}{ccc} 0 & 0\\ 1 & 0\\ ...
10
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1answer
210 views

Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.

Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links? I'm assuming there does exist ...
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2answers
284 views

Gerstenhaber bracket out of $L_\infty$ algebras

Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that ...
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0answers
103 views

When does the rank of a module behave sub-multiplicatively under tensoring?

Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product $ \cal{E} \otimes_A ...
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2answers
236 views

Existence of a projection operator onto a classical set of density matrices

I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as $$ \rho = \int ...
13
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2answers
343 views

How unique are extensions of TQFTs to lower dimension?

Say I have an "ordinary" TQFT $F$ of dimension $n$, assigning groups or vector spaces to closed $(n-1)$-manifolds and linear maps to cobordisms. Consider the different ways $F$ can be obtained from a ...
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1answer
94 views

Decomposition of C' Kazhdan-Lusztig basis element associated to longest word in S_n

I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S_n$, $C'_{w_0}$ into products and sums of elements $C'_w$ where $w < w_0$ in the Bruhat order. For ...
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75 views

A list of infinite dimensional coalgebras over a field

I'm looking for a vast list of infinite list of coalgebras of infinite dimension, I'm familiar with the standard ones, any example is well received. I'm currently writing a paper on coalgebras, so the ...
11
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2answers
561 views

Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?

Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...
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222 views

Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible ...
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1answer
183 views

Hopf Duals and Matrix Coefficients

One defines the finite dual of a Hopf algebra $A$ as $$ H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}. $$ As is well-known, $H^o$ has a ...
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1answer
119 views

Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra ...
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2answers
275 views

Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
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2answers
477 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
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117 views

Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...
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176 views

abstract algebra for component wise operations on “vectors” or what it might be called

I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations: - multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
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0answers
133 views

$h$-adic Completion of $U_q(\frak{sl}_2)$?

Consider the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as ...
4
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4answers
306 views

Non-Drinfeld--Jimbo Deformations and Finite Quantum Groups

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called ...
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174 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 ...
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1answer
183 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 ...
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2answers
504 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, ...
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1k 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 ...
11
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1answer
684 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 ...
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0answers
231 views

Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by $$ \Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...
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1answer
283 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. ...
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1answer
110 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
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0answers
157 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 ...
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1answer
112 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 ...
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2answers
460 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 ...
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0answers
60 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, ...
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7answers
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
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1answer
406 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 ...
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0answers
151 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
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0answers
260 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 ...
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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 ...
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0answers
120 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
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1answer
163 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
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0answers
289 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
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0answers
265 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
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1answer
283 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
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3answers
485 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
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0answers
388 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 ...
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1answer
299 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 ...
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0answers
113 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
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1answer
90 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
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2answers
647 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 ...
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1answer
156 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 ...
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2answers
127 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
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1answer
173 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 ...
3
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1answer
287 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 ...