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

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votes

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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**

votes

**2**answers

557 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 ...

**3**

votes

**0**answers

220 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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votes

**1**answer

182 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 ...

**1**

vote

**1**answer

116 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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votes

**2**answers

261 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 ...

**16**

votes

**2**answers

466 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 ...

**3**

votes

**0**answers

115 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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votes

**0**answers

175 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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votes

**0**answers

131 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**

votes

**4**answers

303 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 ...

**7**

votes

**1**answer

172 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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votes

**1**answer

181 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 ...

**4**

votes

**2**answers

502 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, ...

**17**

votes

**3**answers

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**

votes

**1**answer

657 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 ...

**1**

vote

**0**answers

230 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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votes

**1**answer

281 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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vote

**1**answer

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**

votes

**0**answers

155 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 ...

**0**

votes

**1**answer

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 ...

**8**

votes

**2**answers

454 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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vote

**0**answers

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, ...

**11**

votes

**7**answers

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

147 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**

votes

**0**answers

256 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 ...

**18**

votes

**9**answers

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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votes

**0**answers

117 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**

votes

**1**answer

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 ...

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votes

**0**answers

287 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 ...

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votes

**0**answers

260 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**

votes

**1**answer

276 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**

votes

**3**answers

482 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**

votes

**0**answers

386 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 ...

**7**

votes

**1**answer

297 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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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?
...

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votes

**2**answers

628 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

152 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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votes

**2**answers

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 ...

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votes

**1**answer

170 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**

votes

**1**answer

277 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 ...

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votes

**1**answer

177 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

675 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

835 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 $U(\mathfrak{g}[z])$. ...

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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 ...

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votes

**2**answers

688 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 ...

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votes

**2**answers

226 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 ...

**7**

votes

**1**answer

262 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 ...