Questions tagged [qa.quantum-algebra]

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

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Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$. Proposition: A finite group $G$ is perfect iff every $1$-dimensional complex representation of $G$ is trivial. proof: First if $...
Sebastien Palcoux's user avatar
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Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element: The square of $a$ equals Muger's "squared dimension" of $X$, an ...
Bruce Bartlett's user avatar
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Asymptotics of the quantum exponential

Let $\epsilon$ be an $N$th root of unity, and $q=\epsilon e^h$ where $h<0$. I am trying to give a derivation of the lead term of $$(z;q)_{\infty}=\prod_{n=1}^{\infty}(1-zq^n),$$ as $h\rightarrow 0^...
Charlie Frohman's user avatar
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On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials. Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...
Satoshi  Nawata's user avatar
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deformed Gauss Bonnet formula?

I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss ...
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Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked Definition: the Second-Hand Lion trace distance $D_k$ Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
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Coloured Jones polynomial at 4th root of unity and Arf invariant

Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
hopftype's user avatar
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Are there infinitely many simple integral fusion rings of rank $4$?

$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
Sebastien Palcoux's user avatar
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Is there a strongly noncommutative Grothendieck ring?

This sequel of Is there a strongly noncommutative fusion category? is motivated to know whether every fusion category is "equivalent" (in some sense) to one with a commutative Grothendieck ...
Sebastien Palcoux's user avatar
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Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
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Lusztig's root datum

In his "Introduction to quantum groups", Lusztig first defines a Cartan datum $(I,\cdot)$ and then a root datum of $(I,\cdot)$-type, which suggests (to me) that a root datum is not ...
Adam's user avatar
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Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
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Explicit construction of the Drinfeld double for quasi-bialgebras

Let $A$ be a quasi-bialgebra over a field $k$ and $A$-$\mathrm{mod}$ be the category of finite dimensional left $A$-modules. Since $A$-$\mathrm{mod}$ is a monoidal category, its Drinfel’d center $\...
yohei ohta's user avatar
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The fusion categories with a strict skeleton

We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post. A fusion category is skeletal if two isomorphic objects are always equal. Every ...
Sebastien Palcoux's user avatar
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On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is: We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...
Yingjin Bi's user avatar
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Are the finite quantum permutation groups, weakly group-theoretical?

Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
Sebastien Palcoux's user avatar
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Quantum dimension in the Drinfeld center

Let $\mathcal{C}$ be a spherical tensor category. It is known that the Drinfeld center of $\mathcal{C}$ is modular (and therefore also spherical), see for example, Corollary 8.20.14 in [1]. Recall the ...
Arthur's user avatar
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Star product on functions of a Poisson-Lie group

Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected). We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to ...
Rik Voorhaar's user avatar
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A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras

The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties: maximal, it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
Sebastien Palcoux's user avatar
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Are vertex operator algebras ever conspiratorial?

I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
Theo Johnson-Freyd's user avatar
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Scaling Yetter--Drinfeld Modules

A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
Nadia SUSY's user avatar
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Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...
Robert Rauch's user avatar
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Do global bases exist for quantum enveloping algebras at $q$ nonroot of unity?

Take $\Bbbk$ to be a field, $q \in \Bbbk$ a nonroot of unity, and $U = U_q(\mathfrak g)$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $U^-$ for its ...
Pablo Zadunaisky's user avatar
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Adjoint action of Lusztig's integral form preserves the De Concini-Kac integral form: reference?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra. I have seen in several papers the following fact stated: the adjoint action of Lusztig's integral form (...
Nicolas Dupré's user avatar
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Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
JC Arias's user avatar
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Characteristic classes of invariant star products

Let $\mathfrak{g}$ a Lie algebra, can one compute the $\mathfrak{g}$-invariant Deligne class of an invariant star product by using some kind of invariant local $\nu$-Euler derivation?
Chiara Esposito's user avatar
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Is the associated grouplike $\gamma=uS(u)^{-1}$ of a quasi-triangular Hopf algebra always the square of another grouplike?

Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let $...
Bipolar Minds's user avatar
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Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
Abo Kutis-Felan's user avatar
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Lusztig's definition of quantum groups

In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form $U^{\mathbb{Q}(q)}_q$ that is rather different from the usual approach (see [1,Ch.9.1] for ...
Bipolar Minds's user avatar
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Is the Nichols-Richmond theorem true for integral fusion rings?

The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper. It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here): Theorem: ...
Sebastien Palcoux's user avatar
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Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695). The particular question is about ...
Alexander Chervov's user avatar
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Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
Abo Kutis-Felan's user avatar
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Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...
Manuel Bärenz's user avatar
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What does "control of a deformation problem" mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
Jim Stasheff's user avatar
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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 $h(ab)=h_{(1)}(...
truebaran's user avatar
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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 ...
Jim Stasheff's user avatar
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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 d(...
Sebastien Palcoux's user avatar
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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 ...
Janos Erdmann's user avatar
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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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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 ...
Sebastien Palcoux's user avatar
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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 ...
Sebastien Palcoux's user avatar
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203 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 ...
kieffer's user avatar
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The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
Chuck Hague's user avatar
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Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
Adrien's user avatar
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Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
Adrien's user avatar
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Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping 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 ...
John Pardon's user avatar
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Is there a good reference for how ribbon structures change when one switches coproducts?

I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories. Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf ...
Ben Webster's user avatar
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Analogy between canonical basis of U(n_-) and Schur functors, each under restriction

.1. For any category $\mathcal C$, possibly enriched over schemes, define $Rep({\mathcal C})$ to be the functor category ${\mathcal C} \to {\bf Vec}$ with direct sum inherited from $\bf Vec$. (If $\...
Allen Knutson's user avatar
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Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?

In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension". From [DLN, Theorem II (iii)], where the ...
Sebastien Palcoux's user avatar
3 votes
0 answers
90 views

Isomorphic objects have the same dimension (pivotal categories)

I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e., $$ \mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
NoetherNerd's user avatar