Tagged Questions
2
votes
1answer
43 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
25 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
118 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
41 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
145 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
41 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
80 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
44 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
84 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
0answers
31 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
$$
\Delta_R(f(v)) = f(v_{(0)}) ...
0
votes
0answers
53 views
Monoidal Structures on the Category of Module-Comodules of a Hopf Algebra
Let $H$ be a Hopf algebra, and let ${\cal M}^H_H$ the category of right $H$-module-comodules, that is, the objects of ${\cal M}^H_H$ are right $H$-modules, and right $H$-comodules, such that, for $V$ ...
3
votes
1answer
54 views
proving that this coproduct and product get along well enough to make a bialgebra (Quantum Groups, Kassel)
I'm working through Kassel's Quantum Groups book and I'm stuck on what looks like a pretty simple problem (ch3, ex 2 in the book):
Let $k$ be a field and $C= k[t]$ be a coalgebra with coproduct:
$$ ...
5
votes
0answers
133 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 ...
0
votes
0answers
178 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 ...
