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

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On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
2
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114 views

Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of Are there workable algebraic geometry approaches for the pentagon equation? I've replaced "algebraic geometry" by "numerical" in its content, ...
6
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2answers
248 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
6
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1answer
525 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
6
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2answers
177 views

How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be? The reason I am interested in this is that my ...
6
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1answer
156 views

Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations? If yes, how? Thanks for any help.
2
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1answer
171 views

Universal ribbon category of ribbon graphs

I'm skimming through Turaev's "Quantum invariants of knots and 3-manifolds". One of the main results is Theorem 2.5. In my opinion, this Theorem is conceptually half-baked: 1) The ribbon structure on ...
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1answer
119 views

classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity. The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...
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38 views

Does the standard Podlés sphere have a quasitriangular Hopf algebra structure? Do quantum homogeneous spaces have one in general?

Function spaces on (classical) homogeneous spaces can have a bialgebra structure: Take $S^2$ to be the unital, associative algebra generated by $x, y, z$ with the relation $x^2 + y^2 + z^2 = 1$ and ...
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1answer
122 views

Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...
4
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2answers
326 views

Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings. Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$? Here, ...
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116 views

Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra? For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
2
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1answer
146 views

“Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
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182 views

Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
4
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1answer
133 views

Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...
2
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1answer
82 views

Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
3
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1answer
233 views

A little bit of Intuition for Corepresentations from Representations

I asked this question over on Math.Stack --- where it has a bounty --- but I didn't really get a helpful response so I am asking the question here. One commenter suggests that I am confusing left- ...
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1answer
82 views

Reference to complete derivation of Kossakowski–Lindblad equation and its steady solutions

Are there recommended textbook or good intro-reference to explain with complete stretch of Kossakowski–Lindblad equation especially how is the idea to derive it from ground? ...
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75 views

dual notion to hopf galois extension and properties thereof

Let $H$ be a finite dimensional hopf algebra and $B \subset A$ be an $H$-extension of algebras. We know that the following are equivelant 1) $A \cong B \times_\sigma H$ is a cocycle crossed product ...
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1answer
208 views

alternative to Kontsevich formality

Has anyone considered an alternative approach to Kontsevich formality in which the DGCA of poly-vector fields is deformed to an $L_\infty$-algebra?
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66 views

Is multiplication continuous in quantum-SU(2) with respect to the $L^2$-norm

For the Hopf $\ast$-algebra ${\cal O}_q(SU(2)$, with its unique Haar measure $h$, we have an inner product, and a norm, on ${\cal O}_q(SU(2))$ defined by $$ <x,y> : = h(xy^*), ...
4
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3answers
386 views

Jordan-Hölder theorem for subfactors?

All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors. First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
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67 views

Simplest (?) example of bicrossed product Hopf algebra

Suppose we have two Hopf algebras, H and A and additionally A is (left) H-module algebra and H is (right) A comodule coalgebra. This means that A is left module over H and moreover that ...
5
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1answer
180 views

Jordan-Hölder theorem for planar algebras?

First recall the Jordan-Hölder theorem for groups: Theorem (Jordan-Hölder): Let $G$ be a group, and let $$ G=G_1 \supset G_2 \supset \dots \supset G_r = \{ e \} $$ be a normal tower such that ...
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1answer
312 views

The category of subfactors extending the category of groups?

This post was inspired by this answer of Dave Penneys. In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
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178 views

Where is the Courant operad discussed?

Where is the Courant operad discussed? And hopefully defined precisely. By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
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185 views

Fusion categories with permutation “associativity matrices”

Let $\mathcal{C}$ be a fusion category and let $(H_1,...,H_r)$ be its simple objects. $\mathcal{C}$ is non-pointed if at least one of its simple object has Perron-Frobenius dimension $ \neq 1$. ...
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1answer
179 views

Tensor product of generator of SU(n)

I'm doing research in quantum mechanics and got some trouble. Any help would be very much appreciated. Let $\{\lambda_j\}$ be the set of generator of $SU(n)$. Consider the operator: $K=\sum_j ...
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24 views

Vanishing of non commutative ( Wodzicki) residue on pseudo differential projections

Its a known fact that the non-commutative (Wodzicki) residue of a pseudo-differential projection is always zero. My question is: Is it possible to get this result by looking at structure of the ...
2
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113 views

quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...
4
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1answer
151 views

K-Theory of Algebra of Zeroth Order Pseudo differential operators

Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators? Thanx!
12
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1answer
246 views

Associators, Grothendieck-Teichmüller group and monoidal categories

The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations. In other words, denoting by $ ...
4
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2answers
174 views

Reference request: “duality” relations between $U_q(\mathfrak{g})$, $O_q(G)$ and $O_q(G^*)$

Let $\mathfrak{g}$ be a bialgebra, $\mathfrak{g}^*$ its dual, and $G$ and $G^*$ the corresponding connected simply-connected Poisson-Lie groups. I have repeatedly heard claims of the following ...
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1answer
251 views

What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions: 1.) The naive approach to define a homotopy would be ...
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115 views

Is an integral simple fusion ring, categorifiable?

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum ...
2
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1answer
94 views

Understanding Sweedler's notation for the structure map of a comodule

I was hoping someone might be able to shed some light on the choice of indices for expressing the coaction using Sweedler notation. For example, in the paper of Andruskiewitsch About ...
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121 views

About the Lie algebra of polyvector fields

I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
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1answer
83 views

dual quantum plane

Let $k\left\{\phi, \gamma\right\}$ be free algebra and $I_{q}$ be the two sided ideal generated by the elements $\phi \gamma + q^{-1}\gamma \phi$, $\gamma^{2}$ and $\phi^{2}$, $k_{q}[\phi,\gamma]= ...
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79 views

Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...
3
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1answer
152 views

example of a compact quantum group at a root of unity?

In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some ...
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83 views

Quantum Algebras — Crystal Basis/Graph

Suppose I have a finite-dimensional irreducible $U_q(sl_2)$-module say $V$, and (L,B) is its crystal basis. How do you find the crystal basis of the evaluation $U'$-module $V_{x=1}$? And is there a ...
6
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1answer
281 views

What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...
3
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89 views

Intuition for pointed Hopf algebras

I have familiarized myself with various definitions (one-dimensionality of simple left comodules, generated as an algebra by group-like and skew-like elements...) and examples of pointed Hopf algebras ...
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1answer
488 views

Gauss linking integral and quadratic reciprocity

In the setting of Mazur's "primes and knots" analogy, prime ideals $\mathfrak p\subset\mathcal O_K$ correspond to "knots" $\operatorname{Spec}\mathcal O_K/\mathfrak p$ inside a "3-manifold" ...
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107 views

$q$-differential equation for the Rodgers polynomials?

The Rodgers polynomials $C_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on Wikipedia. For the special case of $q=1$, we ...
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907 views

A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form : $$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that : $d_{i}\vert n$ $1=d_{1}<d_{2} \le d_{3} \le ... \le ...
7
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1answer
140 views

realizing fusion categories as subfactors of the hyperfinite

Let R be the hyperfinite II_1 or the hyperfinite III_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules. This question is inspired by the recent ...
2
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1answer
138 views

Embedding e_n -> e_m

Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to ...
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160 views

An embedding theorem for a fusion ring planar algebra?

We first recall the embedding theorem for finite depth subfactor planar algebras: The planar algebra generated by a (finite depth) subfactor, is embeddable into the planar algebra generated by its ...
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Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., ...