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

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4
votes
1answer
187 views

Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...
1
vote
0answers
90 views

Exact sequence of L-infinity-algebras

We call a sequence of $L_\infty$-algebras (weak) maps $$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$ is exact if it is exact on the the underlying chain complexes level. Thought I don't know ...
8
votes
1answer
184 views

When are Jones-Wenzl projectors defined?

(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.) Consider the Temperley-Lieb algebra ...
3
votes
0answers
136 views

Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...
1
vote
0answers
68 views

Classifying Equivariant Maps Between Fin-Dim Irreducible Modules

Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known ...
6
votes
1answer
236 views

Identifying the little disk operad with parenthesized braids

Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of ...
6
votes
2answers
277 views

Are subfactor planar algebras hard to classify at index 6?

Given a finite index inclusion, $N\subset M$, of $II_1$ factors we can construct two towers of finite dimensional algebras known as the $\textit{standard invariant}$. For low index, this has allowed ...
7
votes
3answers
245 views

What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors. Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
5
votes
1answer
248 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

I asked this question on Math.Stack but have not had any answers. Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$? The trivial ...
4
votes
1answer
119 views

Reference for the image of the adjoint to the differential in graph cohomology (which yields STU & IHX)?

One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a ...
5
votes
1answer
104 views

when is an algebra map conjugate to a star algebra map

Take a (unital) algebra map $f:A\to B$ between two unital C* algebras - not necessarily star preserving. Under what circumstances is there a $b\in B$ so that $g(a)=b\ f(a)\ b^{-1}$ is a star algebra ...
6
votes
1answer
263 views

Are Turaev--Viro invariants secretly a discretized path integral?

Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with ...
4
votes
0answers
102 views

Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $. It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. It's cyclic if its lattice of ...
4
votes
1answer
167 views

Quantum 9j symbols?

A formula for (SU2) quantum 6j symbols exists. A formula expressing ordinary (q=1) 9j symbols in terms of 6j symbols is long known. Unfortunately, combining both (I tried it myself) got tricky - the ...
2
votes
0answers
117 views

About the classification of infinite depth irreducible finite index maximal subfactors

The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group ...
5
votes
0answers
185 views

How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two. Version 1: Let $(C,\otimes)$ be any monoidal ...
7
votes
1answer
1k views

The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results: - Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980). - A Galois correspondence for depth 2 irreducible subfactors ...
5
votes
1answer
102 views

Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Is there an infinite depth irreducible finite index maximal subfactor (other than ...
14
votes
0answers
147 views

Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with. Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
8
votes
0answers
233 views

Are there only finitely many maximal subfactors of a fixed finite index ?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. Question: are there only finitely many maximal subfactors of a fixed finite ...
11
votes
0answers
591 views

Non-“weakly group theoretical” integral fusion categories?

Can you exclude integral fusion categories of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices (I don't write the trivial one) ...
0
votes
0answers
173 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\\ ...
9
votes
1answer
190 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 ...
1
vote
2answers
268 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 ...
0
votes
0answers
102 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 ...
2
votes
2answers
211 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 ...
12
votes
2answers
295 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 ...
0
votes
1answer
87 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 ...
0
votes
0answers
74 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
2answers
503 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
0answers
209 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 ...
2
votes
1answer
172 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
1answer
106 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 ...
4
votes
2answers
223 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
2answers
413 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
0answers
105 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 ...
0
votes
0answers
164 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 ...
0
votes
0answers
129 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
4answers
287 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
1answer
164 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 ...
2
votes
1answer
172 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
2answers
428 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, ...
8
votes
1answer
610 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
1answer
557 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
0answers
216 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, ...
4
votes
1answer
273 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. ...
1
vote
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
votes
0answers
148 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
1answer
110 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
2answers
410 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 ...