Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with hopf-algebras
Search options not deleted
user 11877
A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.
1
vote
1
answer
138
views
Analog between groups and Hopf algebras
"Subgroups" of a group correspond to "left coideal subalgebras" of a Hopf algebra. Why "subgroups" do not corresponds to "Hopf subalgebras" but "left coideal subalgebras"?
- 6,201
1
vote
0
answers
51
views
Is $\mathcal{O}_q(GL_n(k))$ a Nichols algebra?
Let $k$ be a field. The quantized coordinate ring of the group $GL_n(k)$ is defined in Section 3.1 in the paper. Is $\mathcal{O}_q(GL_n(k))$ a Nichols algebra? Thank you very much.
- 6,201
3
votes
1
answer
242
views
Are there p-adic or finite field analogue of Drinfeld-Jimbo's quantum groups
Drinfeld-Jimbo's quantum groups are associated algebras over the field of complex numbers. Are there some references about the analogue of Drinfeld-Jimbo's quantum groups over a p-adic field or a fini …
- 6,201
0
votes
How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding
We can also prove that Yetter-Drinfeld condition implies that $\Psi$ is a braiding as follows.
Let $u, v, w \in V $. Then
\begin{align}
& \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes w) \\
& = …
- 6,201
2
votes
1
answer
179
views
Is the cross product $A \rtimes H$ a bialgebra?
Let $H$ be a bialgebra and $A$ a $H$-module algebra. The cross product $A \rtimes H$ is defined as follows. As a vector space $A \rtimes H = A \otimes H$. The multiplication on $A \rtimes H$ is define …
- 6,201
1
vote
2
answers
250
views
Motivations of cross product
Let $H$ be a bialgebra and $B$ an $H$-module. The cross product $B \rtimes H$ of $B$ and $H$ is $B \otimes H$ as a vector space and the multiplication in $B \rtimes H$ is defined as follows: $(a \otim …
- 6,201
1
vote
0
answers
61
views
Is $T(V)$ a Yetter-Drinfeld module over $T(V^* \otimes V)$?
Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$. Is there some action $T(V^* \otimes V) \otimes T(V) \to T(V)$ and coaction $T(V) \to T(V^* \otimes V …
- 6,201
1
vote
1
answer
141
views
Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules
There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the article. Are there some references which derive these compatibili …
- 6,201
1
vote
1
answer
126
views
Action is determined by a braiding
Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by
\begin{align}
& \Psi: V \otimes W \to W \otimes V, \\
& \P …
- 6,201
4
votes
1
answer
865
views
How to show that the graded dual of the universal enveloping algebra of a free Lie algebra o...
In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X.
It is said that the graded dual of the universal envel …
- 6,201
4
votes
0
answers
301
views
Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie al...
Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$.
Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the …
- 6,201
1
vote
1
answer
199
views
Why in the definition of a Nichols algebra we require that $V$ is a Yetter-Drinfeld module?
In the article, a Nichols algebra is defined as follows. Let ${\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}}$. There exists a largest ideal ${\displaystyle {\mathfrak {I}}\subset TV} $ with the follo …
- 6,201
9
votes
2
answers
1k
views
Algebra in a category
I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \oti …
- 6,201
8
votes
1
answer
418
views
Compatibility conditions for Yetter-Drinfeld modules
In the paper, page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is
$$
h_{(1)} v_{(-1)} \otimes h_{(2)}.v_{(0)} = (h_{(1)}.v)_{(-1)}h_{(2)} \otimes (h_{(1)}. …
- 6,201
3
votes
2
answers
243
views
How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding
Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if
\begin{align}
( v \triangleleft h_{(2)} )_{(0)} \otimes …
- 6,201