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

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12
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0answers
330 views

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 ...
4
votes
1answer
342 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 ...
1
vote
0answers
109 views

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 ...
3
votes
1answer
398 views

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 ...
2
votes
1answer
271 views

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 ...
1
vote
1answer
156 views

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...) $q^{-10}+q^{-8}+q^{-2}+1+q^2+q^8+q^{10}$ = ...
7
votes
4answers
1k views

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 ...
5
votes
1answer
419 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 ...
0
votes
1answer
287 views

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 ...
7
votes
1answer
267 views

If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?

If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if ...
4
votes
1answer
349 views

What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants?

To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate ...
1
vote
0answers
190 views

Do the real polarization and Kahler polarization of the character varieties of closed surfaces give rise to equivalent representations of the Mappping Class Group?

This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$. These ...
10
votes
1answer
514 views

2-cocycle twists of braided Hopf algebras

2-cocycle twists of Hopf algebras Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map $$ f: H \otimes H \to k$$ such that $$ f(x_{(1)},y_{(1)})f(x_{(2)} ...
7
votes
1answer
223 views

Mnemonic for how left and right duals interact with Homs

Suppose you have a rigid monoidal category. This means you have a left dual and a right dual. By Frobenius reciprocity, you can move objects from one side of a Hom to the other side at the price of ...
1
vote
1answer
374 views

Are Quantum Clebsch-Gordan coeffients quantum group dependent?

Background: I'm a quantum chemist and even once wrote a proggie for 3j and 6j symbols. Imagine my gaping mouth when I read the paper (Reshetikhin/Turaev, I think) who let them pop up in my fave, knot ...
12
votes
1answer
320 views

Are there interesting semisimple algebras in non-semisimple categories?

Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple ...
1
vote
0answers
97 views

Haar Functionals and Coquasi-triangular Structures

In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
3
votes
0answers
262 views

What is the state in the WRT TQFT associated to a handlebody?

Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
1
vote
1answer
308 views

Classification of quantum Lie groups

Obviously, I tumbled over Classification of (compact) Lie groups - are the quantum Lie groups (or make that: algebras) easier to classify? Or does the whole q-deformation thingie make it even more ...
6
votes
0answers
248 views

Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, ...
6
votes
1answer
468 views

Is the complete functorial structure for Khovanov--Lee homology known?

I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$. The groups ...
7
votes
1answer
585 views

Yetter--Drinfeld Modules and Braidings

Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory ...
34
votes
3answers
2k views

Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?

For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions: • The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra ...
7
votes
1answer
290 views

What are the primitives of Lusztig's twisted bialgebra $\mathbf{f}$?

I'm trying to understand the coproduct on Lusztig's $\mathbf{f}$ and, apart from the Chevalley generators, I don't know of any more primitive elements. In the $q=1$ case, the Weyl group acts as a Hopf ...
0
votes
0answers
437 views

Knot Numerology

EDIT: Ok, I condense it to only that what is needed. Assume it's possible to use the method described here Matrix decomposition the other way to decompose a $S$ matrix from knot theory. Then each ...
6
votes
0answers
395 views

Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra. Now presumably this algorithm has been implemented ...
2
votes
1answer
416 views

What is the current status for Lusztig's positivity conjecture for symmetric Cartan datum?

This is related to the earlier question here In Conjecture 25.4.2 in his "Introduction to Quantum Groups," Lusztig conjectures that "If the Cartan datum is symmetric, then the structure constants ...
5
votes
1answer
519 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
9
votes
3answers
1k views

How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?

I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
3
votes
1answer
217 views

Image of the Coproduct of a Hopf Algebra

Let $H$ ba a Hopf algebra with coaction $\Delta: H \to H \otimes H$. Denote the action of $\Delta$ by $\Delta (h) = h_{(1)} \otimes h_{(2)}$. I was wondering if every element of $H$ can arise as a ...
12
votes
0answers
488 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in ...
15
votes
2answers
1k views

How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
6
votes
4answers
912 views

Compact Quantum Groups from Hopf Algebras

For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way ...
7
votes
2answers
672 views

Quantization of conjugacy classes in a Lie group

Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\\!/G$. Then let ...
2
votes
2answers
322 views

a question about finite dimensional representation of a Hopf algebra

Let $H$ be a Hopf algebra over a field $k$ and $V$ a finite dimensional left $H$-module. Then $End_{k}(V)$ is a right $H$-module via $(f\cdot h)(v)=S(h_{1})f(h_{2}\cdot v)$. We set ...
22
votes
3answers
1k views

Why is a 2d TQFT formulated as a functor?

Usual mathematical formulation of a 2d (closed) TQFT is as a functor from the category of 2-dim cobordisms between 1-dim manifolds to the category of vector spaces (satisfying various properties.) ...
6
votes
1answer
788 views

I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt

Question: I am talking about the proof given on pages 50-52 of Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten ...
7
votes
2answers
501 views

Drinfeld's equivalence of quantized function algebras and quantized universal enveloping algebras

In his 1986 ICM address, Drinfeld discusses a way of producing a quantized function algebra (or more precisely a quantized formal series Hopf algebra) from a quantized universal enveloping algebra -- ...
7
votes
1answer
167 views

Is there a fusion category with an object which does not commute with its dual?

Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)? Feel free to add adjectives such as pivotal, spherical, ...
0
votes
0answers
227 views

Annulator of Tensor Power in a Quantum Group

There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it. Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the ...
13
votes
4answers
827 views

2-TQFT are to Frobenius Algebras as ??? are to Hopf Algebras

The question arose this morning during a seminar about HAs. In a few words: can the equivalence $2-TQFT_k \leftrightarrow Frob_k$ be "modified" in a sensible way to give a similar one between the ...
3
votes
1answer
501 views

What if I change field in a Topological Quantum Field Theory?

Of course I'm talking about the algebraic notion of field. In a few words, if a TQFT consists of a functor $Z\colon Cob(n)\to \mathbf{Vec}_k$, I'm wondering if there are sensible relations among ...
1
vote
1answer
605 views

a simple problems about Yetter-Drinfeld-Module

I will be appreciated if anyone can give me some clue for the following simple question, Let $H$,$A$ are both hopf algebras,$\pi :A \rightarrow H$,$\quad f:H\hookrightarrow A$ are both hopf morphism ...
15
votes
0answers
514 views

Splitting of homomorphism from cactus group to permutation group

We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
2
votes
1answer
240 views

Continuity of Borel measurable Gleason frame functions

Gleason's theorem (Journal of Mathematics and Mechanics, Vol. 6, No. 6, 1957) classifies measures on the closed subspaces of a separable Hilbert space. A key lemma toward the proof of the theorem ...
7
votes
3answers
755 views

In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?

I work entirely over a field of characteristic $0$, in case it matters. Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...
13
votes
1answer
388 views

Why do sl(2) and so(3) correspond to different points on the Vogel plane?

Vogel assigns to every simple metric Lie algebra (and more generally to every simple metric Lie algebra object in a symmetric monoidal category) a point in the orbifold $\mathbb{P}^2/S_3$ (where $S_3$ ...
15
votes
7answers
2k views

Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$. We define a ...
24
votes
4answers
2k views

Invertible matrices of natural numbers are permutations… why?

I have heard the following statement several times and I suspect that there is an easy and elegant proof of this fact which I am just not seeing. Question: Why is it true that an invertible nxn ...
4
votes
0answers
306 views

Quantum Drinfeld-Sokolov reduction of a Whittaker module

Take a Whittaker module $Wh$ of a (finite or affine) semi-simple Lie algebra $\mathfrak{g}$ , and apply the quantum Drinfeld-Sokolov reduction $qDS$ with respect to an $sl(2)$ embedding $\rho:sl(2) ...