All Questions
Tagged with duality gr.group-theory
7 questions
4
votes
0
answers
113
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
2
votes
0
answers
124
views
Heap torsors and dual objects
Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...
0
votes
0
answers
124
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When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
- 1,215
1
vote
0
answers
127
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Proof/reference for a variant of Pontryagin duality
Let $X,X'$ be locally compact abelian groups with a non-degenerate quadratic form
$\left<\bullet ,\bullet \right>\colon X\times X' \to \mu_{l}$,
where $l$ is a prime, and $\mu_l$ the group of $...
- 3,362
18
votes
2
answers
731
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What categorical property of monoidal categories picks out the ones with duals?
Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
- 54.6k
1
vote
1
answer
908
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A bijective correspondence induced by Fourier Transform
Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection ...
- 2,452
5
votes
1
answer
2k
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Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
- 285