**3**

votes

**1**answer

72 views

### Comodules of Cosemisimple Hopf Algebras

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...

**0**

votes

**0**answers

50 views

### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...

**1**

vote

**0**answers

122 views

### Deformation of a Hopf algebra

A deformation of a Hopf algebra is defined as follows.
On page 171 of the book a guide to quantum groups, Remark 2, it is said that Any deformation of a Hopf algebra $A$ as a bialgebra is ...

**1**

vote

**1**answer

53 views

### Is the antipode anti-bracketed?

In the book Algebras of Functions on Quantum Groups: Part I, Remark 3.1.4, we have the following result.
Let $A$ be a Poisson Hopf algebra. That is, $A$ is both a Hopf algebra and a Poisson algebra ...

**7**

votes

**2**answers

374 views

### Computing in quantum groups

I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by ...

**5**

votes

**2**answers

94 views

### When are the braid relations in a quasitriangular Hopf algebra equivalent?

Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} ...

**6**

votes

**1**answer

101 views

### The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$

For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...

**2**

votes

**0**answers

28 views

### Constant solution of the CYBE

I am learing how to solve the system of equations
\begin{align}
r_{12}+r_{21}=t,\ [[r,r]]=0,
\end{align}
where $t$ is the Casimir element of $g\otimes g$ corresponding to a non-degenerate invariant ...

**3**

votes

**1**answer

117 views

### The Jacobi identity of a Lie algebra?

Let $g$ be a finite dimensional real Lie algebra and $(,)$ be a nondegenerate invariant symmetric bilinear form on $g$. Let $r\in g\bigotimes g$ be a skew-symmetric solution of the MCYBE. We may ...

**0**

votes

**1**answer

61 views

### Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...

**6**

votes

**0**answers

83 views

### Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as ...

**2**

votes

**0**answers

40 views

### How to value $\Omega$ in T-system for twisted quantum affine algebras?

Let us proceed to the unrestricted T-systems. Choose $h\in {\mathbb{C}\backslash 2\pi \sqrt{-1} \mathbb{Q}}$ arbitrarily.
The unrestricted T-system for $U_{q}(X_{N}^{(\mathfrak{k})})$ is the following ...

**5**

votes

**1**answer

120 views

### Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on ...

**4**

votes

**1**answer

203 views

### Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...

**2**

votes

**0**answers

75 views

### Quantum Groups and quantum spaces - From algebra to Analysis

My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...

**1**

vote

**0**answers

46 views

### An equation about the Lie bialgebra of a Poisson-Lie group

Let $w^{R}: G \rightarrow \mathfrak{g}\otimes \mathfrak{g}$ be the right translate of the Poisson bivector $w$ of $G$ to the identity, and let $\delta : \mathfrak{g}\rightarrow \mathfrak{g} \otimes ...

**6**

votes

**1**answer

117 views

### The quantum group SUq(n) as von Neumann algebra

i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...

**4**

votes

**1**answer

208 views

### The coxeter number condtion in the quantum Lusztig conjecture

This is a question about the second point in Geordie Williamson's answer in
What to do now that Lusztig's and James' conjectures have been shown to be false?
,
which says that the Lusztig ...

**3**

votes

**1**answer

92 views

### How to obtain an quantization of the algebra of functions of a given space?

My question is inspired by quantum spheres and quantum projective spaces (studied by Dijkhuizen & Noumi, Vaksman & Soibelman, etc., i.e. "A FAMILY OF QUANTUM PROJECTIVE SPACES AND RELATED ...

**3**

votes

**1**answer

92 views

### Are all the Lie bialgebra structure on $sl_n$ coboundary?

In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore ...

**1**

vote

**0**answers

55 views

### Low-dimensional classical r-matrices

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, ...

**2**

votes

**2**answers

165 views

### Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...

**2**

votes

**1**answer

146 views

### Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets ...

**6**

votes

**1**answer

100 views

### Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions?
To make my question more specific, have all solutions for dimension $2$ and $3$ been ...

**3**

votes

**1**answer

86 views

### Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...

**1**

vote

**0**answers

78 views

### Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...

**5**

votes

**2**answers

174 views

### How to make a premodular category a modular tensor category?

A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...

**2**

votes

**1**answer

118 views

### How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations.
In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...

**1**

vote

**1**answer

111 views

### Questions about Sklyanin bracket

For every classical r-matrix $r$, there is a Poisson bracket called Sklyanin bracket associated to $r$. It is defined in (3.3) of page 5 in (http://arxiv.org/pdf/1101.0015v2.pdf) as follows.
...

**0**

votes

**0**answers

66 views

### Relation between quantum affine algebras and W-algebras

In the paper, there is a concept $qq$-characters for W-algebras $W_{q_1,q_2}$. The usual $q$-character is defined for quantum affine algebra in the paper. A $q$-character can be obtained from a ...

**19**

votes

**4**answers

1k views

### Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...

**7**

votes

**1**answer

172 views

### Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A ...

**8**

votes

**1**answer

183 views

### $U_q(\mathfrak{sl}_2)$ representations of “quantum dimension” zero

I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...

**0**

votes

**0**answers

79 views

### What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$?

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$ in terms of $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$? Any help will be greatly appreciated!

**2**

votes

**0**answers

100 views

### Center of $U_q(sl_3)$ and $U_q(sl_4)$

In the book a guide to quantum groups, page 285, the center of $U_q(g)$ is described in Theorem 9.1.6. The center of $U_q(sl_2)$ is computed explicitly in Example 9.1.7. I tried to compute the center ...

**2**

votes

**1**answer

89 views

### Number of cluster variables

In the paper cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for ...

**5**

votes

**1**answer

114 views

### Compact Quantum Groups and FRT-Algebras

As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...

**2**

votes

**1**answer

130 views

### Representations of the algebra of odd quantum spheres

I read the article by Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf). Here they describe in section $4$ what the algebra of functions on the total space of a family of quantum ...

**1**

vote

**0**answers

74 views

### Irreducible representations of quantum affine algebras

The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...

**5**

votes

**1**answer

183 views

### Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...

**7**

votes

**2**answers

262 views

### The Irreducible Representations of the Sekine Quantum Groups

Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
...

**6**

votes

**0**answers

123 views

### 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 here.
We are interesting in an alternative ...

**7**

votes

**0**answers

196 views

### 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 ...

**3**

votes

**1**answer

137 views

### What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ ...

**1**

vote

**1**answer

91 views

### Why $A^*A =A$ implies that $A$ is a C$^*$ algebra (Proposition 5.2.8 of An Invitation to Quantum Groups and Duality by Thomas Timmermann)

I am reading "An Invitation to Quantum Groups and Duality
From Hopf Algebras to Multiplicative Unitaries and Beyond" by Thomas Timmermann.
In the proposition 5.2.8 (page 117) the author provide a ...

**0**

votes

**0**answers

37 views

### Deforming the category of representations of the Yangian of a simple Lie algebra?

Since I got a very good answer to my previous question,
217585,
I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...

**2**

votes

**1**answer

102 views

### Deforming the category of representations of a simple Lie algebra?

This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question 105221.
This answer explained that the quantised enveloping algebra, $U_q(\mathfrak{g})$, (defined by a ...

**6**

votes

**0**answers

141 views

### 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 ...

**21**

votes

**1**answer

413 views

### Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...

**1**

vote

**1**answer

61 views

### Radicals of co-quasitriangular map

Let $B=\mathcal{O}_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", ...