Abelian Subgroup

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Joseph W. Iverson - One of the best experts on this subject based on the ideXlab platform.

  • The Zak transform and representations induced from characters of an Abelian Subgroup
    arXiv: Functional Analysis, 2019
    Co-Authors: Joseph W. Iverson
    Abstract:

    We consider a variant of the Zak transform for a finite group $G$ with respect to a fixed Abelian Subgroup $H$, and demonstrate a relationship with representations of $G$ induced from characters of $H$. We also show how the Zak transform can be used to study right translations by $H$ in $L^2(G)$, and give some examples of applications for equiangular tight frames.

  • The Zak transform and representations induced from characters of an Abelian Subgroup
    2019 13th International conference on Sampling Theory and Applications (SampTA), 2019
    Co-Authors: Joseph W. Iverson
    Abstract:

    We consider a variant of the Zak transform for a finite group G with respect to a fixed Abelian Subgroup H, and demonstrate a relationship with representations of G induced from characters of H. We also show how the Zak transform can be used to study right translations by H in L2(G), and give some examples of applications for equiangular tight frames.The purpose of this note is to demonstrate some connections between the Zak transform and the theory of induced representations. We also show some applications for right shift-invariant spaces, and for equiangular tight frames that occur as orbits of induced representations. For the sake of clarity, we restrict our attention to the setting of finite groups, where we can safely ignore convergence issues. However, the results in Sections I and II should also hold (with suitable modification) on locally compact groups.

  • Subspaces of L2(G) invariant under translations by an Abelian Subgroup, preprint
    2016
    Co-Authors: Joseph W. Iverson
    Abstract:

    Abstract. For a second countable locally compact group G and a closed Abelian Subgroup H, we give a range function classification of closed subspaces in L2(G) invariant under left translation by H. For a family A ⊆ L2(G), this classification ties with a set of conditions under which the translations of A by H form a continuous frame or a Riesz sequence. When G is Abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on a new group-theoretic tool: for a closed Subgroup Γ ⊆ G, we produce a measure on the space Γ\G of right cosets that gives a measure space isomorphism G ∼ = Γ × Γ\G. Outside of the group setting, we consider a more general problem: for a measure space X and a Hilbert space H, we investigate conditions under which a family of functions in L2(X;H) multiplies with a basis-like system in L2(X) to produce a continuous frame or a Riesz sequence in L2(X;H). Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernández et al. in [22]. 1

  • subspaces of l2 g invariant under translation by an Abelian Subgroup
    Journal of Functional Analysis, 2015
    Co-Authors: Joseph W. Iverson
    Abstract:

    Abstract For a second countable locally compact group G and a closed Abelian Subgroup H , we give a range function classification of closed subspaces in L 2 ( G ) invariant under left translation by H . For a family A ⊆ L 2 ( G ) , this classification ties with a set of conditions under which the translations of A by H form a continuous frame or a Riesz sequence. When G is Abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on a new group-theoretic tool: for a closed Subgroup Γ ⊆ G , we produce a measure on the space Γ \ G of right cosets that gives a measure space isomorphism G ≅ Γ × Γ \ G . Outside of the group setting, we consider a more general problem: for a measure space X and a Hilbert space H , we investigate conditions under which a family of functions in L 2 ( X ; H ) multiplies with a basis-like system in L 2 ( X ) to produce a continuous frame or a Riesz sequence in L 2 ( X ; H ) . Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernandez et al. in [25] .

Ignasi Mundet I Riera - One of the best experts on this subject based on the ideXlab platform.

  • finite group actions on 4 manifolds with nonzero euler characteristic
    Mathematische Zeitschrift, 2016
    Co-Authors: Ignasi Mundet I Riera
    Abstract:

    We prove that if X is a compact, oriented, connected 4-dimensional smooth manifold, possibly with boundary, satisfying \(\chi (X)\ne 0\), then there exists a natural number C such that any finite group G acting smoothly and effectively on X has an Abelian Subgroup A generated by two elements which satisfies \([G:A]\le C\) and \(\chi (X^A)=\chi (X)\). Furthermore, if \(\chi (X)<0\) then A is cyclic. This answers positively, for any such X, a question of Etienne Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.

  • finite group actions on homology spheres and manifolds with nonzero euler characteristic
    arXiv: Differential Geometry, 2014
    Co-Authors: Ignasi Mundet I Riera
    Abstract:

    Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite Subgroup $G$ of $Diff(X)$ has an Abelian Subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.

  • finite group actions on 4 manifolds with nonzero euler characteristic
    arXiv: Differential Geometry, 2013
    Co-Authors: Ignasi Mundet I Riera
    Abstract:

    We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and effectively on $X$ has an Abelian Subgroup $A$ satisfying $[G:A]\leq C$, $\chi(X^A)=\chi(X)$, and $A$ can be generated by at most $2$ elements. Furthermore, if $\chi(X)<0$ then $A$ is cyclic. This proves, for any such $X$, a conjecture of Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.

  • finite group actions on manifolds without odd cohomology
    arXiv: Differential Geometry, 2013
    Co-Authors: Ignasi Mundet I Riera
    Abstract:

    Let $X$ be a compact smooth manifold, possibly with boundary. Denote by $X_1,\dots,X_r$ the connected components of $X$. Assume that the integral cohomology of $X$ is torsion free and supported in even degrees. We prove that there exists a constant $C$ such that any finite group $G$ acting smoothly and effectively on $X$ has an Abelian Subgroup $A$ of index at most $C$, which can be generated by at most $\sum_i[\dim X_i/2]$ elements, and which satisfies $\chi(X_i^A)=\chi(X_i)$ for every $i$. This proves, for all such manifolds $X$, a conjecture of Etienne Ghys. An essential ingredient of the proof is a result on finite groups by Alexandre Turull and the author which uses the classification of finite simple groups.

Pawel Kasprzak - One of the best experts on this subject based on the ideXlab platform.

  • rieffel deformation of homogeneous spaces
    Journal of Functional Analysis, 2011
    Co-Authors: Pawel Kasprzak
    Abstract:

    Let G1⊂G be a closed Subgroup of a locally compact group G and let X=G/G1 be the quotient space of left cosets. Let X=(C0(X),ΔX) be the corresponding G-C∗-algebra where G=(C0(G),Δ). Suppose that Γ is a closed Abelian Subgroup of G1 and let Ψ be a 2-cocycle on the dual group Γˆ. Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C∗-algebra XΨ – the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the Subgroup G1 obtaining the closed quantum Subgroup G1Ψ⊂GΨ, which in turn, by the results of S. Vaes, leads to the GΨ-C∗-algebra GΨ/G1Ψ. In this paper we show that GΨ/G1Ψ≅XΨ. We also consider the case where Γ⊂G is not a Subgroup of G1, for which we cannot construct the Subgroup G1Ψ. Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C∗-algebras.

  • rieffel deformation of homogeneous spaces
    arXiv: Operator Algebras, 2010
    Co-Authors: Pawel Kasprzak
    Abstract:

    Let H be a closed Subgroup of a locally compact group G and let X=G/H be the quotient space of left cosets. Let C*X be the corresponding G-C*-algebra of continuous functions on X, vanishing at infinity. Suppose that L is a closed Abelian Subgroup of H and let f be a 2-cocycle on the dual group of L. Let G(f) be the Rieffel deformation of G. Using these data we may construct G(f)-C*-algebra C*X(f) - the Rieffel deformation of C*X. On the other hand we may perform the Rieffel deformation of the Subgroup H obtaining the closed quantum Subgroup H(f) of G(f) which in turn, by the results of Vaes, leads to the G(f)-C*-algebra G(f)/H(f). In this paper we show that G(f)/H(f) and C*X(f) are isomorphic G(f)-C*-algebras. We also consider the case where L is a Subgroup of G but not of H, for which we cannot construct the Subgroup H(f). Then C*X(f) cannot be identified with a quantum quotient. What may be shown is that it is a G(f)-simple object in the category of G(f)-C*-algebras.

Goansu Kim - One of the best experts on this subject based on the ideXlab platform.

Wei Zhou - One of the best experts on this subject based on the ideXlab platform.

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