All Questions
Tagged with hopf-algebras modules
32
questions
1
vote
0
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29
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$\mathbb{C}G$-modules and $\mathbb{C}^{G}$-comodules
I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this ...
- 659
0
votes
0
answers
24
views
About the regular representation of weak hopf algebra
In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
- 11
0
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0
answers
47
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Action on associated graded algebra inducing action on filtered algebra
Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
- 479
1
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1
answer
639
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Can we define tensor product of modules over an algebra?
Let $A$ be an algebra over a commutative ring $k$ and $M$ and $N$ be modules over $A.$ Is there any natural way to define tensor product of $M$ and $N$ over the algebra $A\ $?
My idea is that since $...
- 2,912
5
votes
1
answer
146
views
Group algebra for quaternion group
I'm trying to understand Hopf Galois Theory, and I decided to try studying some example of a non commutative ring extension. The papers I've studied tell me that, for a strongly $G$-graded algebra $A$ ...
- 3,564
4
votes
0
answers
87
views
$H$-comodule structure of $A\otimes_K A$
I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider ...
- 3,564
1
vote
0
answers
134
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Prove $A$ is $K[G]$-comodule algebra
I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and there's something I'm trying to prove for what I may need some help.
More precisely, is this statement from ...
- 3,564
1
vote
1
answer
68
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Question about Hopf algebra and actions on endomorphisms rings
Let $H$ be a Hopf algebra and $M, N$ two left $H$-modules, then we can define a left $H$-module structure on $\mathrm{Hom}(M,N)$ given by $(h * f)(m) = h_1 f(S(h_2) m)$, where $S$ is the antipode. My ...
- 1,684
2
votes
0
answers
56
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Isomorphism between two Hopf algebras
Let $k$ be a field, over which we consider algebras and coalgebras. A $k$-coalgebra is a comonoid object in $k$-modules, and a $k$-algebra is a monoid object in $k$-modules. A $k$-bialgebra is ...
user900250
2
votes
0
answers
45
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Bialgebras with rigid representation categories
Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its category of modules.
Suppose now that we fix a rigid structure on $B\text{-mod}$ such that the dual of ...
- 2,075
5
votes
1
answer
211
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Relation: Module structure on the dual and braiding?
1. Context
Let $H$ be a Hopf algebra over a field $\mathbb k$. Let $(V, p)$ be a finite dimensional (left) $H$-module.
We want to endow its dual vector space $V^*$ with the structure of a (left) $H$-...
- 3,068
8
votes
1
answer
337
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Example of a cocommutative, non-unimodular Hopf algebra?
1. Definitions: Unimodularity and cocommutativity
Let $H$ be a Hopf algebra over a field $\mathbb k$.
We call $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right ...
- 3,068
17
votes
1
answer
869
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Integrals of a Hopf algebra: Why that name?
1. Context: The notion of an integral
Let $H$ be a Hopf algebra over a field $\mathbb k$. We call its $\mathbb k$-linear subspace
$$
I_l(H)= \{x \in H; h \cdot x=\epsilon(h)x \quad for \>all\>h\...
- 3,068
2
votes
1
answer
106
views
Unique 2-dimensional irreducible representation
Let $K$ be a field whose characteristic is different from $2$ containing a primitive fourth root of unity. $H$ is defined as follows: It is generated by $x,y,z$ that satisfy the relations $x^2=y^2=1, ...
- 1,327
1
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0
answers
100
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Definition: Invariants of Hopf modules
In my lecture notes it says:
Let H be a $k$-Hopf algebra. Let M be a left H-module. The invariants of H on M are defined as the $k$-vector subspace
$M^{H}$:= {m $\in$ M | h.m=$\epsilon$(h)m for ...
- 3,068