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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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 ...
- 3,637
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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 ...
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246
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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 ...
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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 ...
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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 ...
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4
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241
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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 $\...
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Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
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Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.
Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
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97
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What algebras generate polynomial count varieties as their representations spaces ? Is it preserved by the Koszul duality, Manin's endomorphisms?
Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to ...
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3
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Monoidal class vs gauge class vs Grothendieck class
In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are monoidal classes, gauge classes, and Grothendieck class. Could ...
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How to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac Fermions using $U(N \cdot k)_1 / SU(k)_N$?
It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A ...
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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}) = \...
- 143
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Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
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Is the Frobenius property invariant by Morita equivalence?
Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....
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Is there a non-pointed simple integral modular fusion category?
The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds:
Statement 1 (open): There is a ...
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Are the fusion categories weakly Frobenius?
A well-known open problem (generalizing Kaplansky 6th conjecture) asks whether every (spherical) fusion category $\mathcal{C}$ (over $\mathbb{C}$) is of Frobenius type, i.e. for every simple object $X$...
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Noncommutative group schemes corresponding to quantum groups
I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
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How to interpret compositional diagrams for quantum sets algebraically
$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ...
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Relative strength of Jones and colored Jones polynomials
this is my first post here.
I've been studying some Knot Theory and I came to a question concerning invariants.
We know that the Jones polynomial is related to the RT-invariant associated to the two-...
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Multiplication of two Pauli string
Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...
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Is there a finite depth irreducible subfactor of prime index and not group-subgroup?
Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
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134
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What are all the possible indices for the finite depth subfactors?
Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
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Is the category of Yetter-Drinfeld modules abelian?
Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
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752
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Where can I find Drinfeld's original papers on quantum groups?
Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
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143
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Understanding the intuition behind the $Q(z)$-tensor product
Let $z$ be a fixed non-zero complex number. Let $V$ be a vertex algebra, $W_1$, $W_2$, and $W_3$ be $V$-modules. Huang defines a $Q(z)$-intertwining map between these modules to be a linear map $F:W_1\...
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Hopf algebras structure and quantum affine algebras
I'm looking for some information about the Hopf algebras structure and the quantum groups.
In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
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Are the tangle functors based off Khovanov homology braided monoidal functors?
I was wondering if the tangle functors constructed in
"A functor-valued invariant of tangles"
https://arxiv.org/pdf/math/0103190.pdf
"An invariant of tangle cobordisms via subquotients of arc rings"
...
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Extended cyclotomic criterion for unitary categorification
According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object $a$ of a fusion category over $\mathbb{C}$ is a cyclotomic integer. Now, FPdim($a$) is the maximal ...
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Why is the RT invariant from $\mathcal Z(\mathcal C)$ the (norm) square of the one from $\mathcal C$?
The relationship between Turaev-Viro/state-sum invariants and Reshetikhin-Turaev/surgery invariants is roughly that
$$\tau_{TV, \mathcal C}(M) = |\tau_{RT, \mathcal C}(M)|^2.$$
Here $\mathcal C$ is a ...
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Is there a non-irreducible maximal subfactor other than two-sided TLJ?
A subfactor $N \subseteq M$ is called:
irreducible if $N' \cap M = \mathbb{C}$,
maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$.
The two-sided ...
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Quotient of the free Poisson algebra
Assume 𝑃 is the free Poisson algebra on the set of generators $𝑋=\{𝑥_{1},𝑥_{2},…,𝑥_{𝑛}\}$. It is well-known that 𝑃 is the polynomial algebra with infinitely many generators $y_{1}$, $y_{2}$, ......
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Is there a semisimple Hopf algebra Grothendieck equivalent to a strictly weak one?
By Corollary 2.22 in On fusion categories (by Pavel Etingof, Dmitri Nikshych and Viktor Ostrik) any fusion category is equivalent to the category of finite dimensional representations of a semisimple ...
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Quotient of quasi-isomorphic nonpositively graded cdga's
I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) concentrated in nonpositive degree, and $\mathfrak ...
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For any finite-dimensional Hopf C*-algebra, can one make the multiplication and co-multiplication cyclically symmetric simultanously?
For any finite-dimensional *-algebra, one can choose a basis such that the coefficients tensor of the anti-linear map $(a,b)\rightarrow (ab)^*$ becomes cyclically symmetric. (Any *-algebra is ...
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Generalisation of the quantum exterior algebra
One might generalise the classical exterior algebra as follows to the quantum exterior algebra:
$K<x_1,...x_n>/(x_i^2,x_i x_j + q_{i,j}x_j x_i)$ with nonzero field elements $q_{i,j}$ for $i<j$...
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Antipode action on quantum minors
Let ${\cal O}(SU_q(n))$ be the standard $q$-deformed coordinate algebra of $SU(n)$, with the canonical generators $x_{i,j}$. For $I = \{i_1,\ldots, i_r\}, J=\{j_1,\ldots,j_r\}\subseteq \{1, \ldots,n\}$...
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Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)
To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post).
A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together ...
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97
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Simple modules of quantum toroidal algebras
Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...
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Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
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A "nice" Orthogonal Basis for Translation Invariant Symmetric Polynomials
It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
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229
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The convolution on a semisimple finite quantum groupoid
Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{...
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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,
...
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105
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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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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 ...
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323
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Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types
Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-...
- 1,453
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269
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Hopf Algebra Pairings and Module-Comodule-Equivalences
Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...
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AS Cohen Macaulay algebras and dualizing complexes
Let $A$ be an $\mathbb N$-graded algebra such that $A_0 = k$ is a field. This are usually called graded connected algebras.
One can define a torsion functor with respect to the ideal $\mathfrak m = \...
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Does this inequality of negative relative entropy and quantum relative entropy hold?
Hello, everyone!
Question
I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices $...
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105
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Haar Functionals and Coquasi-triangular Structures
In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
- 1,453
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515
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What happens geometrically when you take associated-graded (or complete, ...) of a group ring at its augmentation ideal?
I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
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