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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Representation of quantum groups
Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
- 121
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Indecomposable comodules
For a Hopf algebra $A$, we say that a comodule $V$ is indecomposable if it is not equivalent to a direct sum of irreducible comodules.
$\bullet$ What is an example of a finite dimensional ...
- 123
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111
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Rack cohomology as derived functor cohomology
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
- 1,066
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When is an affine Hecke algebra of type A, a quantum Lie algebra?
An affine Hecke algebra of type $A_{k-1}$ is an unital, universal, associative C-algebra generated by the elements $T_1, \dots, T_{k-1}$, $X_1, \dots , X_k$, $X_1^{-1}, \dots , X_k^{-1}$ subject to ...
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Are there non-semisimple complex "non-unital special Frobenius algebras"?
I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces)
$$\mu: V\...
- 3,001
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106
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Second Frobenius-Schur indicator and near-group categories G+|G|
A near-group category $G+m$ is a (spherical) fusion category whose simple objects are given by the element $g$ of the finite group $G$, plus one extra simple object $y$, with Grothendieck ring as ...
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Ordering in Cobordism Category
Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces.
Is it possible to treat the ...
- 341
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A reformulation of commutativity for intertwinning operators?
$\DeclareMathOperator{\Id}{\mathrm{Id}}\DeclareMathOperator{\Rep}{\operatorname{Rep}}$Let $V$ be a nice vertex algebra, and $M_1, M_2, M_3, M_4, M_5, M_6$ be modules over $V$. Assume that I have ...
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204
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Are there skein relations for the colored Alexander/ADO invariants?
The Alexander polynomial/Conway potential can be computed as the quantum invariant associated to (a certain quotient of) $\mathcal U_q(\mathfrak{sl}_2)$ for $q = i$. More generally this works for $q = ...
- 2,533
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Pants algebra $M_n$ as a dagger-special symmetric Frobenius algebra and $CP^*$
I'm looking at the paper Categorical Quantum Mechanics II: Classical-Quantum interaction by Coecke and Kissinger (arxiv link), and I'm having difficulty with one particular aspect.
Throughout the ...
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Reference request: classical polarization argument
I am reading the article ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS,
SOLOMON IDEMPOTENTS AND THE MAGNUS FORMULA of Frédéric Chapoton and Frédéric Patras. Many definitions and results used in this article ...
- 153
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285
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About the integral form of a quantum group
As far as I understood, in order to specialize a quantum group $U_q(\mathfrak{g})$, lets say over $\mathbb{Q}(q)$, to an element $\epsilon \in \mathbb{C}^\times$, it is necessary to find a $\mathbb{Z}[...
- 1,813
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Frobenius Monoids as Collapsed 2-Categories
Let $\mathbf{COB}_2$ denote the 2-category given by
$\bullet$ objects are finite sets of points
$\bullet$ 1-morphisms between these are 1d cobordisms
$\bullet$ 2-morphisms are 2d cobordisms with ...
- 433
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Reference request: Nichols algebras of a braided vector space with a diagonal braiding
Are there some references of the proof of the following result?
Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by
\...
- 6,201
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Deforming the category of representations of the Yangian of a simple Lie algebra?
Since I got a very good answer to my previous question,
217585,
I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...
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Weakly group theoretical fusion category and subsystems
Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...
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115
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Are the Standard Quantum Groups Coordinate Rings Noetherian?
Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?
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The planar algebra generated by the biprojections
Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two irreducible finite index subfactors.
Let $\mathcal{B}_i$ be the set of all the biprojections of $\mathcal{P}_{2+}(N_i \subset M_i)$.
Let $\...
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228
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Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
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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$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
- 1,271
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polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
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128
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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 \...
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94
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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 ...
- 307
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80
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Coproduct of Weak Bialgebras
Hi,
I have two questions concerning the coproduct of weak bialgebras.
First, I would like to know if there is a proof of the existence of the coproduct in the category of weak bialgebras?
Second, (...
- 465
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165
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Is there a two-variable E8 polynomial? (Conjectural or proven)
On MO I learnt about the two-variable E7 polynomial (status: conjectural).
What about a two-variable E8 polynomial? I have reasons to believe such a
thing exists too, but I do magic, not math, so my ...
- 4,829
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121
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Product knot invariants
Let $I_1(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_1$ and representation $\lambda_1$ having the S matrix $S_1$. Let $I_2(L)$ be a Reshetikhine-Turaev link ...
- 4,829
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Annulator of Tensor Power in a Quantum Group
There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it.
Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the ...
- 153
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105
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A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
- 105
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120
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Some version of non-commutative Wick formula
Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...
- 1,391
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An action on multiplicatively antisymmetric matrix
A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...
- 923
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Associativity of Quantum Double
Here is the statement about the associativity of the quantum double of bialgebras in Klimyk-Schmudgen "Quantum Groups ..." (Sec 8.2.1)
Can anyone help me derive the formula of on bottom of ...
- 2,390
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Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
- 1,144
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167
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Negative $q$-binomial series: reference request
There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$
$$
\sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots
(1+q^{r}x)^{-1}
$$
where the $q$...
- 1,143
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373
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Understanding a program for computing Khovanov homology
I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...
- 803
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355
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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 ...
- 254
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$h$-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 ...
- 1,776
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Schur's Di-Lemma: finite and Lie groups different?
For a finite group it's nothing special if two one-dimensional irreps pop up in a product, e.g. for $C_{3v}$ symmetry, $E\bigotimes{E}=A_1\bigoplus{A_2}\bigoplus{E}$ or in dimensions, $2*2=1+1+2$. ...
- 4,829
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Which is the configuration space of a finite dimensional Hilbert space ?
A quantum particle on the real line R has as configuration space this real line R, while its state space is the infinite dimensional complex Hilbert space of square integrable complex valued functions ...
