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.
565
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Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
6
votes
1
answer
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Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
3
votes
0
answers
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Fiber product of group rings
Let $K$ be a field. For a group $G$ we write $K[G]$ for the group ring of $G$.
Given group homomorphisms $F \to G, H \to G$
is the canonical map $$ K[F \times_G H] \to K[F] \times_{K[G]} K[H] $$ an ...
3
votes
0
answers
124
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Quantum orbit method at roots of unity
Chari and Pressley’s A guide to quantum groups, or the original work by Vaksman and Soibelman in 1989, explains that, similarly to the orbit method which relates quantised coadjoint orbits to unitary ...
4
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3
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Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional ...
2
votes
1
answer
571
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Quantum double vs Quantum group
I am trying to understand what is a difference between a quantum double and a quantum group? I thought these two were the same and now I have trouble figuring out how are they related?
3
votes
1
answer
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quantum affine $gl_2$
There are many sources of the relations and Hopf algebra structure of quantum affine $sl_2$ as a deformed enveloping algebra. However, for an application to integrable systems I need to look at ...
7
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2
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Low dimensional noncommutative non-cocommutative Hopf algebras
Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
9
votes
1
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The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
4
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0
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Why does Kashiwara define $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$?
When defining crystal bases, why do we typically view $U_q(\mathfrak{g})$ as an algebra over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$? In Kashiwara's original paper introducing crystal bases, he ...
6
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
1
vote
0
answers
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Superfluous axioms for ribbon Hopf algebra
In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying
(1) $v^2 ...
8
votes
1
answer
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Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig
"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
18
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2
answers
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Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
3
votes
1
answer
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What is the determinant of the R-matrix defining the colored Jones polynomial?
Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathcal U = \mathcal{U}_q(\mathfrak{sl}_2)$, and let $\mathbf R \in \mathcal{U} \widehat \otimes \mathcal{U}$ be the universal $R$-...
2
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1
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A comodule algebra map from a Hopf algebra to itself
Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
2
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0
answers
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Relation Hopf categories and categorified quantum groups
In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been ...
6
votes
1
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Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
5
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1
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Matrix coefficients of a compact quantum group
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz).
Definition: A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a_{i,j}) \in M_n(A)$ such that
$$\...
8
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1
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Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
5
votes
1
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Relating different constructions of the universal compact quantum group
Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...
3
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Galois descent of a Hopf algebra
In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent.
As I ...
4
votes
1
answer
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Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
6
votes
1
answer
326
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Invertible elements of the Hopf algebra quantum $SU(2)$
Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see
https://en.wikipedia.org/wiki/Compact_quantum_group
(Note that on the ...
0
votes
0
answers
100
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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
vote
1
answer
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Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra
It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
3
votes
0
answers
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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)$ ...
6
votes
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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 ...
7
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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 ...
5
votes
1
answer
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What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes ...
2
votes
1
answer
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Peter-Weyl theorem (compact quantum groups)
I'm reading the paper Notes on compact quantum groups. In this paper, the following theorem is proven:
Question: Why is the marked equality true?
0
votes
1
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Direct sum of representations of a compact quantum group
Let $(A, \Delta)$ be a compact quantum group and $\{(H_\alpha, v_\alpha)\}$ be a collection of representations of $A$. That is,
$$v_\alpha \in M(B_0(H_\alpha) \otimes A); \quad \quad(\text{id}\otimes \...
0
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1
answer
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Definition intertwiner of representations of compact quantum groups
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
2
votes
2
answers
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Kernel of intertwiner is invariant (compact quantum groups)
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
4
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1
answer
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Constructing intertwiners between representations of compact quantum groups
Consider the following paper by Van Daele en Maes Notes on compact quantum groups. For convenience of the reader, here is a picture of the relevant section:
(1) How is compact operator defined in ...
2
votes
1
answer
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Extending $*$-morphisms to the multiplier algebras
I'm reading the following fragment in the paper "Notes on compact quantum groups":
While I'm familiar with the multiplier algebra (constructed via double centralizers) and its universal ...
3
votes
2
answers
248
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Convolution of functionals on compact quantum group
Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product
$$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
4
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0
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Are the finite quantum permutation groups, weakly group-theoretical?
Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
3
votes
0
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562
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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\...
2
votes
0
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Frobenius reciprocities
An adjunction of the form $\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies ...
5
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Covariant splittings of Hopf algebra projections
What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
5
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0
answers
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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(...
6
votes
2
answers
433
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Confusion around the reflection equation algebra
I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
2
votes
1
answer
124
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Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
15
votes
1
answer
740
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Tips to organize a successful math workshop
I am a PhD student from India working on representations of quantum groups. I want to organize a workshop on Hopf Algebra and Quantum groups but there are only 2 or 3 specialists in India currently ...
12
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3
answers
810
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Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
12
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2
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Relations between quantum groups at roots of unity, modular representation theory, and physics
I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations ...
7
votes
2
answers
540
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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\...
4
votes
3
answers
333
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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^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
3
votes
0
answers
89
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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 ...