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 85967
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 $G$.
2
votes
Accepted
Cobraided and coquasitriangular Hopf algebras
The definitions of:
cobraided (according to the terminology of Kassel's book; see (5.1)-(5.2)-(5.3) p. 184-185 or equivalently (5.4)-(5.5)-(5.6)-(5.7), p.185) and
coquasitriangular (according to …
- 7,746
6
votes
Cartier-Kostant-Milnor-Moore theorem
The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).
…
- 7,746
6
votes
Cocommutativity, comultiplication and coalgebra maps
Given a (coassociative and counital) coalgebra $(C,\Delta,\varepsilon)$, over a field $k$, we can form the tensor product coalgebra $(C\otimes C,\Delta_{C\otimes C},\varepsilon_{C\otimes C})$ through: …
- 7,746
4
votes
1
answer
664
views
Primitive elements in group hopf algebras over fields of non-zero characteristic
An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:
"If $H$ is a $ …
- 7,746
1
vote
Comodule Morita equivalence for Hopf algebras
There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops t …
- 7,746
2
votes
Accepted
What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?
No, in general the claim is not true:
To see why, consider a field $k$ of characteristic $p$ and take the polynomial hopf algebra $k[x]$ (in a single variable). Then $x$ is primitive and so is $x^p$ …
- 7,746
6
votes
Accepted
When is this map of Hopf algebras Surjective?
Some thoughts, regarding question (a):
In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classifica …
- 7,746
6
votes
Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional eq...
About your first question:
Since you are asking for an example, take any group hopf algebra $k\mathbb{G}$, pick some subset $S\subset \mathbb{G}$ and denote $kS$ the linear subspace of $k\mathbb{G}$ …
- 7,746
3
votes
Hopf algebra kernels vs. algebra kernels
-too long for a comment-
I am a little confused about the way terminology is used in the OP.
Maybe i'm missing the point; in case i do not, the closest result i know of -quite general and does not ref …
- 7,746
8
votes
Accepted
Inner automorphisms of Hopf algebras
I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad_l h)\triangleright k=\sum h_1 kS(h_2)$ of a hopf algebra $H$ on itself.
(It is known tha …
- 7,746
3
votes
Accepted
Bialgebra maps and Hopf algebra maps
Yes it is.
It is actually a standard result, for any hopf algebra, that under the circumstances you are describing any bialgebra map $\phi$ commutes with the antipode, i.e. we have: $$S_{H'}\circ \phi …
- 7,746
4
votes
2
answers
651
views
Cocommutativity, comultiplication and coalgebra maps
Given a coalgebra $(C,\Delta,\varepsilon)$, over a field, the following is a well-known property:
the comultiplication $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is cocommutat …
- 7,746
10
votes
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
This is a question on an active area of research, with lots of work on it (for the general case of algebraically closed fields of char zero). It is historically and conceptually closely connected to K …
- 7,746
7
votes
1
answer
643
views
Classification of quasitriangular Hopf algebras
The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)commutat …
- 7,746
1
vote
Commutative and Cocommutative Quantum Groups
If the definition of a finite quantum group, you use, is a pair $(A,\Phi)$ of a finite dimensional $C^*$-algebra $A$, with a comultiplication $\Phi$, such that $(A,\Phi)$ is a Hopf $*$-algebra, then t …
- 7,746