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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 ...

9
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119 views
+200

Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...
10
votes
1answer
391 views

Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
4
votes
2answers
194 views

How to compute the inverse of a quantum determinant?

Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries: \begin{alignat*}{2} & x_{ij} x_{il} = q x_{il} x_{ij}, && j < l, \\ & x_{ij} x_{kj} = q ...
11
votes
3answers
270 views

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
4
votes
0answers
79 views

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 ...
5
votes
0answers
123 views

Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
2
votes
0answers
33 views

Branching rule for degenerate cyclotomic Hecke algebras

Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...
4
votes
1answer
258 views

Is there another quantum deformation of sl(2)?

By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are: $$ [E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F, $$ some ...
4
votes
0answers
51 views

Do global bases exist for quantum enveloping algebras at $q$ nonroot of unity?

Take $\Bbbk$ to be a field, $q \in \Bbbk$ a nonroot of unity, and $U = U_q(\mathfrak g)$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $U^-$ for its ...
5
votes
0answers
167 views

Projecting GxG onto subspace with tied irreducible representations

Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$. I could alternatively ...
16
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0answers
172 views

Limiting representation theory of quantum groups at roots of unity and SL(2,C)

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
9
votes
1answer
128 views

Separability of compact quantum groups

In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
9
votes
2answers
213 views

Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
3
votes
0answers
52 views

How to understand extremal vector?

Extremal vectors are defined in Kashiwara's paper. The definition is as follows. Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-...
3
votes
0answers
39 views

Adjoint action of Lusztig's integral form preserves the De Concini-Kac integral form: reference?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra. I have seen in several papers the following fact stated: the adjoint action of Lusztig's integral form (...
13
votes
3answers
555 views

Hopf dual of the Hopf dual

Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
4
votes
1answer
175 views

The Ungraded Milnor-Moore Theorem

Let $k$ be a field of characteristic $0$. There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
2
votes
0answers
77 views

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
3
votes
1answer
197 views

When is this map of Hopf algebras Surjective?

I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero. Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...
5
votes
0answers
104 views

Are the integral forms of quantized coordinate algebras always Noetherian?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
8
votes
0answers
182 views

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 ...
3
votes
0answers
112 views

Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)

Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows. $$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
7
votes
2answers
283 views

Hopf Subalgebras of Quantized Algebras

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...
2
votes
0answers
36 views

How to compute the upper global basis (dual canonical basis) of an irreducible $U_q(\mathfrak{g})$-module?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U_q(\mathfrak{g})$ the corresponding quantum group. Are there some general method to compute the upper global basis (dual canonical basis) of an ...
1
vote
0answers
37 views

What is the intuition of lower global bases?

In the paper: Crystallizing the Q-analogue of Universal Enveloping Algebras, Kashiwara introduced the upper global bases. In the paper: https://projecteuclid.org/euclid.dmj/1077295931, Kashiwara ...
0
votes
0answers
90 views

Presentations of $\mathbb{C}_q[G]$

Let $G$ be a semisimple Lie group and $q \in \mathbb{C}^{\times}$, note a root of unity. I am trying to understand the presentations of the quantum group $\mathbb{C}_q[G]$. In the paper FACTORIZABLE ...
6
votes
1answer
247 views

Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology

Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products ...
2
votes
0answers
112 views

Module algebras and comodule algebras

Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-...
7
votes
1answer
116 views

Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
1
vote
1answer
77 views

Question about a computation of $F_{\beta_3}$ related to canonical basis of $U_q(sl_3)$.

I am reading the paper: ELEMENTARY CONSTRUCTION OF LUSZTIG’S CANONICAL BASIS and want to compute $F_{\beta_3}$ in Example 3 on page 3. Maybe there is some mistake in my computations but I could not ...
1
vote
1answer
59 views

$\tilde{e}_i$ action on $e^{(n)} u$

I am reading Kashiwara's paper GLOBAL CRYSTAL BASES OF QUANTUM GROUPS. On page 462, the action of $\tilde{e}_i$ on an integrable $U_q(g)$-module $M$ is defined as follows. For $u \in \ker e_i \cap M_{\...
4
votes
1answer
137 views

Schur's Lemma for Quantized Universal Enveloping Algebra

Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$...
0
votes
0answers
85 views

Reference request: dual canonical basis

Let $\mathfrak{g}$ be a simple Lie algebra and $V({\lambda})$ the $\mathfrak{g}$-module with highest weight $\lambda$. It is said that there is an embedding $V({\lambda}) \subset \mathbb{C}[U]$, where ...
2
votes
0answers
44 views

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 ...
6
votes
0answers
93 views

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 ...
3
votes
1answer
125 views

Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?

This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined ...
5
votes
2answers
161 views

Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
32
votes
2answers
4k views

What is quantum algebra?

This might be a very naive question. But what is quantum algebra, really? Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a ...
6
votes
1answer
270 views

The difference between $q$-deformations and $h$-deformations

What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras? In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...
2
votes
0answers
107 views

Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
4
votes
1answer
97 views

Question on a paper by U. Krähmer (“Dirac operators on quantum flag manifolds”)

I don't know if this is an adequate question for MO. But I cannot understand many aspects of the said paper https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf by ...
7
votes
0answers
108 views

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 $Y(\mathfrak{g})$?
6
votes
1answer
274 views

Inner automorphisms of Hopf algebras

Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?
3
votes
1answer
148 views

What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?

Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...
8
votes
1answer
224 views

Name for the action of a bialgebra on an algebra

Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that $$ (b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a). $$ ...
3
votes
1answer
171 views

Number of Isomorphism Classes of Corepresentations of A Compact Quantum Group

Given a compact quantum group $(G,\Delta)$, with dense Hopf algebra $H$, is it always true that, up to isomorphism, $H$ will have a countable number of irreducible comodules?
16
votes
2answers
788 views

Examples of representations of quantum groups

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
4
votes
0answers
106 views

Idea of Dirac operator on quantum groups

This is a somewhat unexact question. I would like to now more on the principle of the Dirac operator, especially for quantum groups. I have learned in some articles about the Dirac operator on the ...
11
votes
2answers
958 views

Is there any published article where $q$-mathematics is applied?

Excuse me for the concern, but I want to ask you a question. In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science ...
13
votes
2answers
558 views

Can one define quantized universal enveloping algebras in a basis-free way?

(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...