All Questions
Tagged with hopf-algebras linear-algebra
18
questions
4
votes
0
answers
56
views
A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra
Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
- 1,350
0
votes
0
answers
29
views
Associated graded space as a (bi)algebra
A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$...
- 738
0
votes
0
answers
47
views
Change of scalars for comodules as adjunctions?
Let $k$ be a commutative ring and $f: C \to C'$ be a homomorphism of $k$-coalgebras (for simplicity, we can suppose that it is surjective so that $f(C) = C'$). There is a functor:
$$f_*: {}^lC-comod \...
- 493
2
votes
1
answer
142
views
Grouplike elements is a group in a Hopf algebra
I have been looking at this question and two questions arose for me:
(1) A simpler one: Why is $(g \otimes g)(h \otimes h)=(g h \otimes g h)$? My guess:
$$
(g \otimes g)(h \otimes h)
= g^2(1 \...
- 1,139
1
vote
0
answers
36
views
cofinite subspaces in weak* topology
I am learning hopf algebras by D.E Radford, questions about cofiniteness occur to me but I can't find out the answers.
My question is
Let $U$ be a vector space over a field $k$. Is there any subspace ...
- 103
1
vote
1
answer
138
views
Unimodularity: How are these notions related?
1. Definitions
We call a Hopf algebra $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right integrals $I_r(H)$.
We call a square integer matrix $M$ unimodular if $det(...
- 3,068
1
vote
1
answer
86
views
Taft-Hopf Algebra has dimension $N^2$?
Definition of the Taft-Hopf Algebra
Let $k$ be a field. Let be $N$ a positive integer such that there exists a primitive $N$-th root of unitiy $\zeta$ over $k$.
Denote by $(H, \mu, 1_H)$ the unital, ...
- 3,068
5
votes
0
answers
124
views
Coideals in the grouplike colagebra are spanned by differences
Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ ...
- 1,279
3
votes
1
answer
419
views
Universal enveloping algebra vs algebra of continuous functions
I'm a physicist studying quantum groups, but this question is about the usual classical Lie groups (though it is related to the language that is used to describe the deformation to quantum groups, ...
- 555
12
votes
1
answer
800
views
Example of $V^* \otimes V^*$ not isomorphic to $(V \otimes V)^*$
There is always an injection between $V^* \otimes V^*$ and $(V \otimes V)^*$ given by
$$
f(v^* \otimes w^*)(x \otimes y)=v^*(x)w^*(y),
$$
where $x,y \in V$. I've been given to understand that in ...
- 9,254
1
vote
0
answers
38
views
Monoidal Morita theory
Is it possible for two Hopf algebras $H$ and $H'$ such that $H$ has skew-primitives(i.e. elements $x$ such that $\Delta(x)=g\otimes x+x\otimes h$ with $g,h$ grouplike elements) but $H'$ does not have ...
- 12.6k
5
votes
0
answers
156
views
On group graded algebras and Brauer groups
I was reading the paper "Algebras graded by groups" by Knus. I want to test and further my understanding of the paper by asking several questions.
Since the paper is not readily available I ...
- 12.6k
2
votes
0
answers
84
views
On Brauer groups
The Brauer group of a braided monoidal category $\mathcal{C}$ is defined in general in this paper. Essentially it's defined as the equivalence classes of Azumaya algebras in $\mathcal{C}$ (see the ...
- 12.6k
1
vote
0
answers
55
views
Associators of quasi-Hopf algebras
I've read in a paper that given a basic finite-dimensional radically graded quasi-Hopf algebra $H$, its associator lives in degree $0$. I don't understand why this is true. The statement can be found ...
- 12.6k
3
votes
1
answer
410
views
Bialgebra and Hopf algebra
I'm reading ''HOPF ALGEBRAS'' by S. Dascalescu, C. Nastasescu and S. Raianu. At the beginning of the four chapter the following is explained:
''Proposition 4.1.6. Let H be a finite dimensional ...
- 97