Questions tagged [qa.quantum-algebra]
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
288 questions with no upvoted or accepted answers
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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 ...
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7
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331
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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 ...
7
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172
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When is Rep(U_q(g)) invariant under q -> -q and why?
Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
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7
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223
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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 $...
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7
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182
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Deformation of Noether's first theorem
Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
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7
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363
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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, $...
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528
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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 ...
- 8,866
7
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400
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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) \...
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460
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Quantum polynomial rings and singularities
Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
- 3,758
7
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213
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Decomposition of certain projectives for cyclotomic q-Schur algebras
In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard ...
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6
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349
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Quantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum ...
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128
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Unitary fusion category and subfactor
From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor.
By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
6
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442
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Conceptual proof of braid group actions on quantum groups
Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual.
The original paper ...
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186
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Why does the inverse Alexander polynomial appear in the MMR conjecture?
In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
- 2,533
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353
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Homotopy transfer of cyclic L-infinity algebras
Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
6
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Yangians as unique deformation
In Drinfeld's paper "Hopf algebras and the quantum Yang-Baxter equation" there is a statement (Theorem 2) that Yangian is a unique quantization of the corresponding Lie bialgebra.
My question is ...
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Relation between different versions of Bar-Natan homology
In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
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What quantum groups admit quantum topography space structure?
Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
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259
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Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?
A fusion ring is a finite dimensional $\mathbb{Z}$-module
$\mathbb{Z}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^...
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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 ...
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239
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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}, ..., d_{r}...
6
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377
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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 ...
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6
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512
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Is there a version of the 2d cobordism hypothesis for surfaces with non-empty incoming and outgoing boundary?
Question: Is there a condition on an object $x$ of an $(\infty,2)$-category $\mathcal C$ which is equivalent to $x = Z(pt_+)$ for a unique TFT $Z$ from the $(\infty,2)$-category of framed bordisms ...
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6
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369
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Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
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6
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578
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Jones Polynomial and Quantum Field Theory
I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...
- 61
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238
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Category of modules over a coPoisson-bialgebra
Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
$\pi(...
- 1,368
5
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128
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Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules
Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules).
All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...
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134
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Transferred $L_\infty$-structure from Hochschild dgLA
Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild ...
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207
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parameter of a quantum group
I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
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5
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183
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Strongly simple fusion categories: the known examples?
A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same ...
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123
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Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression
Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
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154
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Interpretation of superfactorial in terms of plane partitions
Recently I got interested in plane partitions and the following formula by MacMahon, which counts the number of plane partitions $\pi \in B(r,s,t)$ fitting in an $(r,s,t)$-box:
$$
\binom{r+s+t}{r,s,t}...
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85
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Drinfel'd polynomials for evaluation representations of $\mathbf{U}_q(\mathbf{L}\mathfrak{g})$?
We know for type $A$, there is an evaluation homomorphism from quantum affine algebra to quantum algebra,
$$\operatorname{ev}_a:\mathbf{U}_q(\mathbf{L}\mathfrak{g})\to \mathbf{U}_q(\mathfrak{g})$$
for ...
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255
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Is the category of topologically free $k[[h]]$-modules locally presentable?
$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...
- 8,524
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172
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Are the symmetric groups integrable as Hopf algebras?
Let $G$ be a group. For $g,h \in G$, let $[g,h]=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G' = \langle [g,h] \ | \ g,h \in G \rangle$ is called the commutator subgroup or derived subgroup. ...
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99
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Is there a coproduct on the Weyl algebra which gives the coproduct on $\mathcal{U}_q(\mathfrak{gl}_2)$?
In the paper Modular Double of Quantum Group, Fadeev gives a presentation of $\mathcal{U}_q(\mathfrak{gl}_2)$ in terms of a Weyl algebra $\mathcal{C}_q$ with generators $w_i, i \in \mathbb{Z}/4$ and ...
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345
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A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
- 309
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287
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Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$
I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial.
The results was ...
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219
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Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
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144
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Are the integral forms of quantized coordinate algebras always Noetherian?
Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
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158
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Can we see the symmetry of the quantum Schubert polynomial of a point
Let $X=G/B$ be a homogeneous space and consider the quantization map
$$
S_W\otimes\mathbb{C}[q]\to(S(\mathfrak{h})\otimes\mathbb{C}[q])/I_W^q\,,
$$
where
$S_W$ is the coinvariant algebra of the Weyl ...
- 1,066
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481
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Reference Request: Vertex Algebras
I am currently a graduate student in mathematics with an interest in vertex algebras. I am comfortable with the algebraic aspects and would like to learn more about the geometric aspects. The issue is ...
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113
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Non-semisimple representations of the braid group in a semisimple braided category
Suppose $\mathcal{C}$ is a semisimple braided tensor category (over $\mathbb{C}$, with finite dimensional hom spaces) and $X$ an object in $\mathcal{C}$.
Then for each n > 0 the braid group $B_n$ ...
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103
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Does this $SU(2)$ Chern-Simons Superconformal Index Example have Modular Properties?
Without any regards to the physics or the geometry used to generate this result, let's examine the formula of Gukov (see p. 32):
$$ \mathcal{I}_{SU(2)}(q,t) = \frac{1}{2}\sum_{m \in \mathbb{Z}}\int \...
- 22.8k
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198
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Analogue of Kontsevich's formality theorem for quantization of Courant algebroids
In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...
5
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114
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Differential form TQFT for Walker-Wang model?
In terms of the TQFTs in continuous differential form gauge fields, what would the Walker-Wang lattice model describe? Obviously, there is a $BF$ theory part:
$$\frac{N}{2 \pi}\int B dA$$
if it ...
- 755
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274
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Deformation quantization of Poisson bracket without star-product
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
- 654
5
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191
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Modular double of elliptic quantum group
By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...
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167
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Distinguishing the Duflo star product
$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$
For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of ...
- 8,524
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104
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Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws
Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital $*$-...
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