7
votes
2answers
647 views

What are Quantum Groups?

I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. ...
1
vote
0answers
48 views

A characterisation of the cyclic subfactors by the existence of a cyclic vector?

A cyclic subfactor is a subfactor admitting a distributive intermediate subfactors lattice. Let's start with the finite index irreducible depth 2 subfactors, i.e. the class of subfactors of the form ...
1
vote
1answer
68 views

Fusion rules and comultiplication for a Kac algebra

Let $\mathbb{A}$ be a Kac algebra (i.e. a Hopf C*-algebra) of finite dimension, and $\mathbb{A}^{*}$ its dual : Is there a formula revealing the fusion rules for the irreducible ...
2
votes
1answer
68 views

A question on a Hopf algebra

Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode. I am reading a proof of the fact $S(xy)=S(y)S(x)$. First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by ...
1
vote
1answer
84 views

Different definition of antipode for $SL_q(2)$?

In Majid's book (A Quantum Groups Primer) (pg11), the antipode for the Hopf algebra $SL_q(2)$ is defined as $Sd=a$, $Sa=d$, $Sb=-qb$, $Sc=-q^{-1}c$. However, in Kassel's book (Quantum Groups) (pg ...
6
votes
1answer
132 views

Quantum Groups: prove $U_1'\cong U[K]/(K^2-1)$ and $U\cong U_1'/(K-1)$

Regarding this theorem, which is in Kassel pg 126, I have two questions. I have typed in the relevant material for reference. 1) How does $$U_1'\cong U[K]/(K^2-1)$$ imply $$U\cong U_1'/(K-1)$$? ...
2
votes
1answer
74 views

Converse of Hopf Algebra Theorem

There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode. May I know if the converse is true? (i.e. if $S^2=id_H$, does it ...
6
votes
0answers
163 views

Basis of $SL_q(2)$

While trying to show that $SL_q(2)$ is noncocommutative, I needed to prove the following fact: Show that the set $\{a^ib^jc^k\}_{i,j,k\geq 0}\cup\{b^ic^jd^k\}_{i,j\geq 0,k>0}$ is a basis of ...
2
votes
1answer
46 views

Tensor Product Question in Kassel's Quantum Groups

In Kassel's book on Quantum Groups, I am stuck on the following computation: \begin{eqnarray*} [\Delta (E), \Delta (F)] &=& \Delta (E)\Delta (F)-\Delta (F)\Delta (E)\\ &=& (1\otimes E ...
1
vote
2answers
101 views

How to show that $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are noncommutative and noncocommutative

It is known that the two quantum groups $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are both noncommutative and noncocommutative. May I ask how do we show that? I have attempted the following: To prove ...
1
vote
1answer
58 views

$U_q$ Quantum group and the four variables: E, F, K, K^{-1}

In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations ...
2
votes
1answer
92 views

Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.

I read a textbook on Quantum Groups by Kassel, and did not understand the following proof of well-definedness of $\Delta$ (comult.) and $\epsilon$ (counit) on $GL_q(2)$ and $SL_q(2)$: (pg 84) The ...
1
vote
1answer
45 views

Kernel of a Comodule Map is a Sub-Comodule

Let $H$ be a Hopf algebra, $(V,\Delta_R)$ a right $H$-comodule map, and $f:V \to V$ a right $H$-comodule map. Since by definition we must have, for all $v \in V$, that $$ \Delta_R(f(v)) = \sum ...
0
votes
0answers
65 views

Monoidal Structures on the Category of Module-Comodules of a Hopf Algebra

Let $H$ be a Hopf algebra, and let ${\cal M}^H_H$ the category of right $H$-module-comodules, that is, the objects of ${\cal M}^H_H$ are right $H$-modules, and right $H$-comodules, such that, for $V$ ...
3
votes
1answer
63 views

proving that this coproduct and product get along well enough to make a bialgebra (Quantum Groups, Kassel)

I'm working through Kassel's Quantum Groups book and I'm stuck on what looks like a pretty simple problem (ch3, ex 2 in the book): Let $k$ be a field and $C= k[t]$ be a coalgebra with coproduct: $$ ...
8
votes
0answers
172 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
1
vote
0answers
212 views

A simple Hopf algebra problems

I have a little question when I read an article. Someone can give me any clue? Let $\mathrm{H}$ be a Hopf algebra and $\mathrm{B}$ be a braided bialgebra in Left-left-Yetter-Drinfled-module over ...