Questions tagged [qa.quantum-algebra]

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

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What algebraic structure characterizes all natural operations between differential operators and differential forms?

On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms: the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$; ...
Dmitri Pavlov's user avatar
21 votes
0 answers
463 views

What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?

Let $\mathfrak g=\mathfrak{sl}(2)$. Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness. Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
André Henriques's user avatar
18 votes
0 answers
558 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 ...
Qiaochu Yuan's user avatar
16 votes
0 answers
418 views

Do TQFTs give a complete set of invariants of manifolds?

An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by ...
Henry's user avatar
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16 votes
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805 views

Capelli determinant = Duflo ( determinant) - was it known ?

Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$). I googled a lot ...
Alexander Chervov's user avatar
15 votes
0 answers
1k views

Representations of quantum groups at roots of unity

I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
John Pardon's user avatar
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14 votes
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What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?

Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
Theo Johnson-Freyd's user avatar
14 votes
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525 views

What is the (quasi-) classical limit of categorified quantum groups?

$\newcommand{\g}{\mathfrak g}$ Let $G$ be a reductive group and $U_q(\g)$ the associated quantum group. One can argue that the classical limit of $U_q(\g)$ is $G$ or $\g$, with some Poisson structure, ...
Adrien's user avatar
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14 votes
0 answers
784 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 ...
MTS's user avatar
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13 votes
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585 views

Generalization of Witten's computation of the volume of moduli space

Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$. There ...
John Pardon's user avatar
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13 votes
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571 views

Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups? This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain why ...
John Pardon's user avatar
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12 votes
0 answers
538 views

On a revised quantum Riemann hypothesis

This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...
Sebastien Palcoux's user avatar
12 votes
0 answers
330 views

Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
Theo Johnson-Freyd's user avatar
12 votes
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603 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 $q\in k$...
Julian Kuelshammer's user avatar
11 votes
0 answers
240 views

Quantum groups at small roots of 1

I wonder if there is any literature about representations of quantum groups at a root of 1 of small order. For example, I would like to understand the case of $\mathrm{SL}(2)$ and $q=-1$ (in the ...
Alexander Braverman's user avatar
11 votes
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314 views

Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
Sebastien Palcoux's user avatar
11 votes
0 answers
277 views

Is there a non-Kac complex finite dimensional semisimple Hopf algebra?

A complex (finite-dimensional) Hopf algebra is said to be a Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (...
Sebastien Palcoux's user avatar
11 votes
0 answers
247 views

Analogy between BV formalism and integration by residues

Domenico Fiorenza begins his description of the Batalin–Vilkovisky formalism by pointing out an analogy with integration by residues: Take a top form (density) on $\mathbf R$ resp. space of fields; ...
Alex Shpilkin's user avatar
11 votes
0 answers
411 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 ...
Chris Schommer-Pries's user avatar
10 votes
0 answers
283 views

What's the relation between half-twists, star structures and bar involutions on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
Manuel Bärenz's user avatar
9 votes
0 answers
402 views

Lagrangian subgroups/submanifolds, 2d topological boundary and 3d "non-abelian" Chern–Simons theory

This post is meant to ask for proper references to fill a gap in the literature. My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
wonderich's user avatar
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9 votes
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117 views

Are the quantum groups $C_q[SU_{1,1}]$ and $C_q[SL_{2}(R)]$ isomorphic?

Classically the group of Moebius transformations of the unit disk and Moebius transformations of the upper half plane are isomorphic, as the unit disk and upper half plane are transformed into each ...
Edwin Beggs's user avatar
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9 votes
0 answers
255 views

Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
Alexander Chervov's user avatar
9 votes
0 answers
605 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 ...
Christopher Beem's user avatar
8 votes
0 answers
216 views

$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
Pulcinella's user avatar
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8 votes
0 answers
131 views

Relation between "homotopical" and "representation-theoretic" categorifications

This might be a bit of a soft question, and I apologize in advance for this. Here it is: What is the relationship between the "homotopical" categorification (e.g. we consider every category ...
Grisha Taroyan's user avatar
8 votes
0 answers
342 views

Finite-dimensional representations of quantum $SU(2)$

The most famous of all the quantum groups is $SU_q(2)$ - the Quantum special unitary group. The irreducible comodules of this quantum group are very well understood - they are labelled by integers (or ...
Jake Wetlock's user avatar
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8 votes
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198 views

When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy ...
Dmitri Pavlov's user avatar
8 votes
0 answers
455 views

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 ...
Sinan Yalin's user avatar
  • 1,589
8 votes
0 answers
304 views

Are there only finitely many maximal irreducible amenable subfactors at 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 irreducible amenable subfactors at ...
Sebastien Palcoux's user avatar
8 votes
0 answers
488 views

det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$

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 the 19-th century, but ...
Alexander Chervov's user avatar
8 votes
0 answers
597 views

Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq 1$}...
John Pardon's user avatar
  • 18.3k
8 votes
0 answers
896 views

duality between universal enveloping and function algebra for GL(n)

Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...
Zoran Skoda's user avatar
  • 5,162
7 votes
0 answers
278 views

Does the pentagon axiom force the associativity constraint to be a natural isomorphism?

Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
Sebastien Palcoux's user avatar
7 votes
0 answers
354 views

How to define $U_q \mathfrak{g}$ without generators and relations?

I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
igorf's user avatar
  • 600
7 votes
0 answers
225 views

Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?

If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary ...
AlexArvanitakis's user avatar
7 votes
0 answers
114 views

Are there attempts to numerically finding algebraic structures over finite-dimensional vector spaces?

By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of ...
Andi Bauer's user avatar
  • 2,901
7 votes
0 answers
212 views

Representations of 2-groups and quantum double constructions

Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a ...
Student's user avatar
  • 5,008
7 votes
0 answers
168 views

How to translate connection on four graphs to quantum 6j symbols

I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...
Ying's user avatar
  • 437
7 votes
0 answers
217 views

Universal Enveloping algebra of a L$_\infty$ algebra

In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
mtraube's user avatar
  • 183
7 votes
0 answers
206 views

IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists

Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc. Let $G_{g,b}$ denote the set of finite ...
Dan Petersen's user avatar
  • 39.2k
7 votes
0 answers
170 views

Relationship between R-matrix and Casimir element?

Given a simple Lie algebra $\mathfrak{g}$, is there any relation between its Casimir element and the $R$-matrix of the related Yangian $Y(\mathfrak{g})$?
Zhihua Chang's user avatar
7 votes
0 answers
241 views

Is the Dimer Model a TQFT?

The answer to my question is "yes". Technically, it's a spin-TQFT but now I am trying to make sense of that answer. Dimers on surface graphs and spin structures. I David Cimasoni, Nicolai ...
john mangual's user avatar
  • 22.6k
7 votes
0 answers
137 views

Triviality of Semisimple Hopf Algebras of Cyclic Dimension

A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277 Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
Sebastien Palcoux's user avatar
7 votes
0 answers
238 views

Trace on a KLR algebra

The cyclotomic KLR algebra is isomorphic to the Ariki-Koike algebra over a field and so admits a trace (this is used in Hu-Mathas' paper to define bases for the KLR algebra corresponding to Murphy and ...
Chris Bowman's user avatar
  • 1,191
7 votes
0 answers
399 views

What is the endomorphism cooperad?

In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
Mark.Neuhaus's user avatar
  • 2,004
7 votes
0 answers
161 views

Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?

This is a follow up of this question. Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...
André Henriques's user avatar
7 votes
0 answers
312 views

Flat connection from gauged WZW model

$\newcommand{\g}{\mathfrak g}$ $\newcommand{\h}{\mathfrak h}$ In short my question is : Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ? Some ...
Adrien's user avatar
  • 8,234
7 votes
0 answers
329 views

An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. There already exists a generalization of Cauchy theorem using exponent, see [KSZ06]. We are interesting in an alternative ...
Sebastien Palcoux's user avatar
7 votes
0 answers
473 views

Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$. $H$ is called maximal if it has no left coideal $\...
Sebastien Palcoux's user avatar

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