Inductive Algebras for Finite Heisenberg Groups

Prasad, Amritanshu; Vemuri, M. K.

doi:10.1080/00927870902828520

Cited by 4 publications

(8 citation statements)

References 6 publications

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“…The perfect realizations of the canonical representation of H correspond precisely to systems of imprimitivity, as is explained in [16]. The maximal systems of imprimitivity for the canonical representation are expected to correspond to the maximal isotropic subgroups of L. This has been shown when L is a finite dimensional real vector space in [16] and when L is a finite abelian group in [12].…”

Section: Motivation: Heisenberg Groupsmentioning

confidence: 89%

Locally compact abelian groups with symplectic self-duality

Prasad¹,

Shapiro²,

Vemuri³

2010

Advances in Mathematics

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Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).Comment: 23 page

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Section: Motivation: Heisenberg Groupsmentioning

confidence: 89%

Locally compact abelian groups with symplectic self-duality

Prasad¹,

Shapiro²,

Vemuri³

2010

Advances in Mathematics

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“…By a famous theorem by Stone, von Neumann and Mackey, the Heisenberg group H 3 (M 6 , C) has a unique irreducible representation (see, e.g., [53,54]). The vector space of this representation is the Hilbert space of the seven-dimensional topological quantum field theory, or equivalently, the "partition vector space."…”

Section: The Defect Groupmentioning

confidence: 99%

“…The vector space of this representation is the Hilbert space of the seven-dimensional topological quantum field theory, or equivalently, the "partition vector space." This vector space can be built out of a maximal isotropic subgroup L of H 3 (M 6 , C) [14,40,53]. First, there is a unique ray in the vector space, which we represent by the vector Z L 0 that is invariant under the action of L. Denoting the coset…”

Section: The Defect Groupmentioning

confidence: 99%

On the Defect Group of a 6D SCFT

Zotto¹,

Heckman²,

Park³

et al. 2016

Lett Math Phys

125

227

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We use the F-theory realization of 6D superconformal field theories (SCFTs) to study the corresponding spectrum of stringlike, i.e. surface defects. On the tensor branch, all of the stringlike excitations pick up a finite tension, and there is a corresponding lattice of string charges, as well as a dual lattice of charges for the surface defects. The defect group is data intrinsic to the SCFT and measures the surface defect charges which are not screened by dynamical strings. When non-trivial, it indicates that the associated theory has a partition vector rather than a partition function. We compute the defect group for all known 6D SCFTs, and find that it is just the abelianization of the discrete subgroup of U(2) which appears in the classification of 6D SCFTs realized in F-theory. We also explain how the defect group specifies defining data in the compactification of a (1,0) SCFT.Comment: 24 page

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“…This gives rise to a canonical map: In case G carries a locally compact group topology and Z(G) ≅ T, the target group in (1) is nothing else but the Pontryagin dual of G Z(G) (see Remark 4.2 for more detail). In order to describe the properties of such a group G related to its irreducible unitary representations, Mumford [19] imposed additionally the condition that M is a topological isomorphism (these groups were used also in [20,21]). Clearly, every Mackey -Weil group has this property.…”

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confidence: 99%

“…We only keep the condition on the Mumford map to be a topological isomorphism and we call these groups Mumford groups (Definition 5.1). Clearly, the locally compact Mumford groups G with center topologically isomorphic to T are precisely the above mentioned groups isolated in [19,20,21]. We give criteria for a generalized Heisenberg group to be a Mumford group, and we give criteria for a Mumford group to be a generalized Heisenberg group.…”

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confidence: 99%

Generalize Heisenberg Groups and Self-Duality

Bonatto¹,

Dikranjan²

2016

Preprint

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This paper compares two generalizations of Heisenberg groups and studies their connection to one of the major open problems in the field of locally compact abelian groups, namely the description of the selfdual locally compact abelian groups ([12], [13]). The first generalization is presented by the so called generalized Heisenberg groups H(ω), defined in analogy with the classical Heisenberg group, and the second one is inspired by the construction proposed by Mumford in [19] and named after him as Mumford groups. These two families can be defined also in the framework of topological groups. We investigate the relationship between locally compact Mumford groups, locally compact generalized Heisenberg groups with center isomorphic to T, and symplectic self-dualities.

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Inductive Algebras for Finite Heisenberg Groups

Abstract: Abstract. A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.

Cited by 4 publications

References 6 publications

Locally compact abelian groups with symplectic self-duality

Locally compact abelian groups with symplectic self-duality

On the Defect Group of a 6D SCFT

Generalize Heisenberg Groups and Self-Duality

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