Questions tagged [hopf-algebras]
For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.
525
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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 $...
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4
votes
1
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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 ...
- 19.8k
2
votes
2
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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 ...
- 19.8k
6
votes
2
answers
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$U_q$ 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
\begin{...
- 19.8k
5
votes
2
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645
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When do counital coalgebras have a basis of grouplike elements?
Question.
Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras (or Hopf ...
- 6,031
2
votes
1
answer
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proving that a action of hopf algebra k(G) on A implies a G-grading on A
Let $k(G)$ be the hopf algebra of functions on $G$ with values in $k$ with pointwise multiplication and a comultiplication given by $\Delta(f)(x,y) = f(xy)$ and let $A$ be a $k$-algebra.
I read (in ...
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2
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1
answer
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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 ...
- 19.8k
5
votes
1
answer
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Hopf algebra: Identity under convolution
In Hopf algebra texts, it is usually stated that $1=\eta\epsilon\in$Hom($H^C,H^A$) is the identity under convolution.
$\eta$ is the unit, $\epsilon$ is the counit.
My question is, is that a ...
- 19.8k
1
vote
1
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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
$$
\Delta_R(f(v)) = \sum f(v_{(...
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21
votes
0
answers
512
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Why is the Quasitriangular Hopf algebra called "Quasitriangular"?
The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia.
What is the reason behind the word "Quasitriangular"?
Is it because the R-matrix is a triangular matrix, ...
- 19.8k
4
votes
1
answer
160
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proving that this coproduct and product get along well enough to make a bialgebra (Quantum Groups, Kassel)
I'm working through Kassel's Quantum Groups book and I'm stuck on what looks like a pretty simple problem (ch3, ex 2 in the book):
Let $k$ be a field and $C= k[t]$ be a coalgebra with coproduct:
$$ \...
- 1,300
14
votes
0
answers
394
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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 ...
- 283
2
votes
0
answers
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The exterior product in the Hopf algebra
Here is a exercise in the Eiichi Abe's book Hopf Algebras, Ex2.4,Ch2, p.83.
If $X$ is a left coideal, show that $X\wedge Y$ is a right coideal and that $X\subset X\wedge Y$.
Maybe you can explain ...
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59
votes
4
answers
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How to think of the group ring as a Hopf algebra?
Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
- 1,800
3
votes
0
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328
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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 $\...
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