Questions tagged [quantum-groups]
Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
192 questions with no upvoted or accepted answers
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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 ...
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22
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0
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811
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Combinatorics of Quantum Schubert Polynomials
Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
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18
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577
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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 ...
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15
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0
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1k
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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 ...
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14
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580
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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 ...
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13
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332
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Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
13
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0
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573
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Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
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12
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285
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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 (...
11
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0
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252
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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 ...
- 7,037
11
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259
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Status of the analog of the Haar measure on quantum groups
In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
- 561
10
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287
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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 ...
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10
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586
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canonical basis via Gelfand-Tsetlin basis
Do there exist explicit formulas for the action of Lusztig's canonical basis of
$U_q(\mathfrak n_+)$ in the Gelfand-Tsetlin basis of a Verma module for $sl(n)$?
- 7,037
9
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0
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144
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Ringel's interpretation of quantum groups as Hall algebras at $q=1$
Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
- 3,446
9
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381
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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 ...
- 1,144
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353
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Quantum Hamiltonian reduction and Quantum Airy structure
I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section ...
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9
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Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $
This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the $ ...
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8
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$U_q(\mathfrak{g})$ 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 $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
- 5,701
8
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192
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Two algebraic guises of Alternating Sign Matrices: any connection?
Alternating Sign Matrices (ASMs) have a famous history: they were discovered by Mills, Robbins, and Rumsey, who conjectured a product formula for their enumeration; this product formula was first ...
- 24.2k
8
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1
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A "concrete" example of a one-sided Hopf algebra
I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more ...
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8
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306
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Genus=2 theta functions, Arnold's relation, and KZ connection
Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
7
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164
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Nontrivial examples of locally compact quantum groups
What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
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151
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How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?
Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
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7
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385
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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 ...
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7
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116
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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 ...
- 3,001
7
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202
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What is the kernel of the action of the Iwahori-Hecke algebra?
The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy ...
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7
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183
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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 $Y(\mathfrak{g})$?
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140
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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(...
7
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161
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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 ...
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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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0
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606
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Is there an E8 symmetry in the zero-field Ising model?
In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules $\sigma^...
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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 $...
- 8,559
7
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171
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Embedding quantum affine algebras at different levels
For $l$ a positive integer, an affine Lie algebra
$\widehat{\mathfrak{g}}$ has a level $l$ embedding $\phi_l: \widehat{\mathfrak{g}} \longrightarrow \widehat{\mathfrak{g}}$ which takes $x\otimes t^k$ ...
- 213
7
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0
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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
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297
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Explicit Braid Group Reps from quantum SO(N) at roots of unity
This question is related to this one (and indeed the goals are similar).
Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ ...
- 1,639
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 ...
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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200
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What is a quantum analogue of the fact that the second fundamental group of every Lie group is trivial?
What is an appropriate version of the following fact in terms of Hopf algebras and quantum groups:
"For every connected Lie group $G$ the second fundamental group $\pi_2(G)$ is trivial?"
Is there ...
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338
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Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?
Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
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6
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118
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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 ...
- 99
6
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202
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What is the q-analogue of the Lefschetz decomposition?
The representation theory behind the Lefschetz decomposition in Kahler geometry was summarised very neatly by Victor Protsak in his answer to
29907
Let $W$ be a $2n$-dimensional symplectic vector ...
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5
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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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0
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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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Classification of connected finite affine type A crystals
In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
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5
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223
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Induction for quantum group
I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about ...
5
votes
0
answers
606
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Weak Hopf algebra structure on twisted group algebra
A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map
$\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
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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 ...
- 2,533
5
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0
answers
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 ...
- 671
5
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0
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218
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Lusztig's completion for universal enveloping algebra
In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
- 677
5
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0
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154
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Kazhdan-Lusztig positivity of monomials in the Hecke algebra of a Coxeter System
In 1990, Deodhar [3] showed that the non-negativity of Kazdan-Lusztig polynomials implies the expansion of any monomial of (dual) Kazhdan-Lusztig basis elements $C'_{s_{i_1}}\cdots C'_{s_{i_r}}=\sum_x ...
- 572