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.

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5
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0answers
114 views

Characterizing discrete quantum groups

Let $M$ be a von Neumann algebra, and let $\Delta$ be a unital normal $*$-homomorphism $M \rightarrow M \mathbin{\bar\otimes} M$ that satisfies the coassociativity condition $(\Delta \mathbin{\bar\...
3
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1answer
124 views

Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way. The axioms of Hopf algebras imply that $$ G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1. $$ ...
3
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0answers
59 views

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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1answer
107 views

Showing a product on a character space is continuous

Quoting from Timmermann's An invitation to quantum groups and duality: Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact quantum group. Then there exists a compact group $G$ ...
7
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1answer
257 views

What kind of object are the solutions of the Knizhnik-Zamolodchikov Equations

I am reading about the KZ equations in Kassel's Quantum groups. In definition XIX.3.1 (page 455) he defines the differential system $(KZ_n)$ as $$ dw = \frac{h}{2\pi\sqrt{-1}} \sum_{1 \leq i <j \...
3
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70 views

Quantum analogue of certain property of compact groups

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense ...
8
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1answer
216 views

DW, state sum models, and fully extended TQFTs

I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
4
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0answers
67 views

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
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0answers
57 views

Kontsevich integral on tangles and Fubini

I am reading about the Kontsevich integral, following this text : https://pdfs.semanticscholar.org/635b/c6370e8aba381724eaaa36abefba7f7a5bec.pdf At some point (page 10 to be exact) the author claims ...
4
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2answers
228 views

Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
3
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1answer
86 views

Irreducibility of product bicomodules

Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right, $H$-comodule respectively. The tensor product $$ V \otimes W $$ has an obvious $H$-$H$-bicomodule structure. If $V$ and $W$ are ...
2
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1answer
122 views

Infinitesimal categories and left duality

I have been reading Kassel's Quantum groups and there is something I can not understand. In Section 4 of chapter $XX$, he introduces the notion of a Infinitesimal symmetric category, that is a ...
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47 views

Proof that for any $u \in L_0$, there exists a $f(q) \in \mathbb{Q}[q]$ for which $f(q)u \in V^{A}$ in global bases theory

Setting We consider subrings of $\mathbb{Q}(q)$: $A_0$: localization at $(q)$ $A_{\infty}$: localization at $(q^{-1})$ $A$ = $\mathbb{Q}[q, q^{-1}]$ and free lattices of $\mathbb{Q}(q)$ $L_0$: ...
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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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120 views

Kashiwara's definition of normal crystal

Let $\mathfrak{g}$ be a symmetrisable Kac-Moody algebra, and $U_q(\mathfrak{g})$ its associated quantum group. Each integrable module of $U_q(\mathfrak{g})$ admits a crystal basis, as was first shown ...
3
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1answer
83 views

Reduced compact quantum group and left and right multiplication

Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product $$ \...
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135 views

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 ...
5
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2answers
293 views

Characters on Hopf algebras

For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
7
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2answers
166 views

Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
8
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3answers
341 views

Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
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111 views

Submodules of $V\otimes V^*$

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
2
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0answers
59 views

Embedding problems on quantum groups?

We work over the field of complex numbers. We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
2
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2answers
383 views

Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
3
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1answer
113 views

F-symbols for compact Lie groups

Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, ...
6
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301 views

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 $\otimes$-...
8
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2answers
392 views

Tannaka-Krein duality in Chari-Pressley's book

I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here. V.Chari and A.N.Pressley in their "Guide to Quantum ...
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1answer
99 views

Some quantities associated to finite dimensional Hopf algebras

let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H $ with $T_H=m\circ \Delta,\quad S_H=s$. Are there two finite dimensional ...
3
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1answer
296 views

A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

Does anyone have a proof for the following Lemma? Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
0
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1answer
193 views

Quantum (group) version of ${\mathbb Z}^n$?

As we know there are quantum analogue of tori called quantum tori generated by noncommuting operators $(A_1,\dots,A_n)$ with $A _iA_j=A_jA_ie^{2\pi i\alpha}$ where $\alpha$ is a irrational number as a ...
10
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3answers
632 views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
8
votes
1answer
322 views

Image of Comultiplication on Finite Quantum Groups/Hopf Algebras

Let $A=:F(G)$ be the algebra of functions on a finite quantum groups aka a finite dimensional C*-Hopf Algebra. Suppose that $F(G)$ is neither commutative nor cocommutative. In their 1966 paper Kac ...
4
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106 views

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 ...
4
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1answer
86 views

Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode

Let $(A,\Delta)=:F(G)$ be a finite dimensional $\mathrm{C}^*$-Hopf algebra, and so the algebra of functions on a quantum group $G$. Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$...
3
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1answer
158 views

Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles

In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?
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0answers
246 views

Is every finite quantum group a quantum symmetry group?

This post is basically a quantum extension of Is every finite group a group of “symmetries”? Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem ...
8
votes
2answers
308 views

q-difference equations and quantum mechanics

I have been trying to understand why the term quantum is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...). ...
4
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1answer
180 views

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 ...
8
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1answer
207 views

An inner product approach to Hopf algebras

We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$. Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^...
5
votes
1answer
163 views

Zero divisors in compact quantum groups

Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the polynomial Hopf algebra Pol$(\...
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0answers
95 views

almost magic unitary

A magic unitary is a unitary matrix $u=(p_{ij})_{ij}$ whose entries are all projections (in some Hilbert space) and in each row they sum to the identity and same holds for each column $(\sum_i p_{ij}=...
2
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0answers
89 views

Why are the quantum Fock spaces in FLOTW the same as Uglov's?

Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3] to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\...
5
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2answers
210 views

Indecomposable, non-simple, modules of quantum groups at roots of unity

Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
5
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0answers
124 views

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 ...
25
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1answer
1k views

Have the Quantum Group Theorists taught the Group Theorists Anything?

I will start with the general before moving to the specific. Consider for a moment the two (very) soft definitions. An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
4
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0answers
82 views

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 ...
14
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1answer
468 views

P-adic Volume Conjecture

Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...
10
votes
1answer
264 views

Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?

Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
9
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0answers
170 views

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 (...
2
votes
1answer
392 views

Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional equation

What is an example of a coalgebra $C$ which admit a linear operator $T$, different from scalar operators $T=\lambda Id$, which satisfy $$(T\otimes T)\circ \Delta=\Delta \circ T^2 $$ but $C$ is not ...
4
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0answers
95 views

Hopf algebras without coderivation

What is an example of a complex Hopf algebra $H$, different from $\mathbb{C}$, which does not admit a non zero coderivation? Is there a complete classification of all Hopf algebras with this property?

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