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Identification of Maharam extension

All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
4 votes
1 answer
232 views

Definition of free group action on a scheme

This may be an elementary question, but I didn't know what it means for a group action on a scheme to be free from the textbooks I read. I guess that at least for schemes locally of finite ...
• 435
3 votes
0 answers
105 views

How much a general a theory of matrices equivalence under group actions we have?

Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$. My question is: Do we have some theory about the ...
• 125
1 vote
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72 views

Liouville-Arnold and fibration relative to a convex polytope

Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...
1 vote
0 answers
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What is the quotient of the unit sphere by the bilateral shift on infinite-dimensional separable real Hilbert space?

Let H denote the real Hilbert space 𝓁2(ℤ) with its usual inner product. If {en | n ∈ ℤ} denotes its standard orthonormal basis, define the unitary mapping W : H → H via W(en) = en+1, extended ...
• 2,618
0 votes
0 answers
27 views

Why $d\mu (q)\delta (k,q)$ is $G$-invariant?

Let $G$ be a Lie group acting transitively on a smooth manifold $M$ endowed with a quasi-invariant measure $\mu$ (then there exists Radon-Nikodym derivative $\rho_f$ for every $f\in G$). For $k\in M$,...
• 287
5 votes
0 answers
172 views

• 1,427
1 vote
0 answers
102 views

Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
• 881
1 vote
0 answers
64 views

Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$. Let us consider a Schwartz space $\mathcal{S}$ defined as \mathcal{S}:= \Bigl\{ \...
• 3,113
6 votes
1 answer
163 views

References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
2 votes
0 answers
44 views

• 6,457
3 votes
0 answers
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Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...
• 383
1 vote
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Smooth action on cotangent space of the plane

Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by $$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$ acts via ...
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0 votes
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Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on

For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
• 101
7 votes
1 answer
172 views

Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle. Considering the moduli space of connections $\mathscr{B}$...
4 votes
1 answer
154 views

Examples of discrete quantum group actions on commutative von Neumann algebras

Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...
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1 vote
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• 159
2 votes
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53 views

Different invariants of group actions from isomorphic subgroups

Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...
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