A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups

Shtern, A. I.

doi:10.1070/im2008v072n01abeh002397

Cited by 26 publications

(25 citation statements)

References 48 publications

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“…Theorem III (see [9] We recall that any commutative connected locally compact group is infinitely divisible (see [9], Corollary 1), as is any compact connected group (see [6],Theorem 9.35) and the successive derived groups of a connected topological group are connected, as are the quotient groups by it; therefore any connected soluble locally compact group satisfies all the conditions of Theorem I. Hence we have the following result.…”

Section: Lemmamentioning

confidence: 91%

“…Somewhat later, van der Waerden [2] proved that any (not a priori continuous) finite-dimensional representation of a (not necessarily compact) Lie group is automatically continuous. This work all but replaces Cartan's result in the memory of the mathematical community, but in many sources ([3]- [8]), including the author's publication [9], Cartan's result is ascribed to van der Waerden. I am grateful to Dieter Remus [10], who drew my attention to the publications [11] and [12] in which the historical truth is established.…”

Section: Introductionmentioning

confidence: 99%

“…We recall some definitions and results from [9] used in the proofs of this paper; all the proofs not given here can be found in [9]. Definition 3.…”

Section: Preliminary Informationmentioning

confidence: 99%

“…In the author's paper [9] we obtained a number of results on the automatic continuity of locally bounded finite-dimensional linear representations of connected Lie groups, which proved to be productive in investigating a number of questions about the structure of embeddings, of more general homomorphisms and representations of connected Lie groups and connected locally compact groups. Using these, in [24] we obtained a number of results on the structure of homomorphisms taking connected locally compact groups into compact groups.…”

Section: Introductionmentioning

confidence: 99%

“…Using these, in [24] we obtained a number of results on the structure of homomorphisms taking connected locally compact groups into compact groups. In this paper we consider applying the results in [9] to study the question of the existence of a faithful locally bounded (not necessarily continuous) finite-dimensional linear representation of a given locally compact group and to give a description of the Hochschild kernel urk(G) of a given locally compact group G (or of the intersection of the kernels of continuous finite-dimensional linear representations of a topological group G).…”

Section: Introductionmentioning

confidence: 99%

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Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations

Shtern

2011

Trans. Moscow Math. Soc.

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Abstract. We obtain a number of consequences of the theorem on the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the derived subgroup of the group, as well as an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of connected soluble locally compact groups. In particular, we give a description of connected Lie groups admitting a (not necessarily continuous) faithful locally bounded finitedimensional representation; as it turns out, such groups are linear. Furthermore, we give a description of the intersection of the kernels of continuous finite-dimensional representations of a given connected locally compact group, obtain a generalization of Hochschild's theorem on the kernel of the universal representation in terms of locally bounded (not necessarily continuous) finite-dimensional linear representations, and find the intersection of the kernels of such representations for a connected reductive Lie group.

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Section: Lemmamentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

“…We recall some definitions and results from [9] used in the proofs of this paper; all the proofs not given here can be found in [9]. Definition 3.…”

Section: Preliminary Informationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations

Shtern

2011

Trans. Moscow Math. Soc.

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An Analogue of the Coleman–Mandula Theorem for Quantum Field Theory in Curved Spacetimes

Fewster

2017

Commun. Math. Phys.

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Abstract:The Coleman-Mandula (CM) theorem states that the Poincaré and internal symmetries of a Minkowski spacetime quantum field theory cannot combine nontrivially in an extended symmetry group. We establish an analogous result for quantum field theory in curved spacetimes, assuming local covariance, the timeslice property, a local dynamical form of Lorentz invariance, and additivity. Unlike the CM theorem, our result is valid in dimensions n ≥ 2 and for free or interacting theories. It is formulated for theories defined on a category of all globally hyperbolic spacetimes equipped with a global coframe, on which the restricted Lorentz group acts, and makes use of a general analysis of symmetries induced by the action of a group G on the category of spacetimes. Such symmetries are shown to be canonically associated with a cohomology class in the second degree nonabelian cohomology of G with coefficients in the global gauge group of the theory. Our main result proves that the cohomology class is trivial if G is the universal cover S of the restricted Lorentz group. Among other consequences, it follows that the extended symmetry group is a direct product of the global gauge group and S, all fields transform in multiplets of S, fields of different spin do not mix under the extended group, and the occurrence of noninteger spin is controlled by the centre of the global gauge group. The general analysis is also applied to rigid scale covariance.

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A Difference Property for Functions with Bounded Second Differences on Amenable Topological Groups

Shtern

2016

J Math Sci

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A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups

Cited by 26 publications

References 48 publications

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations

An Analogue of the Coleman–Mandula Theorem for Quantum Field Theory in Curved Spacetimes

A Difference Property for Functions with Bounded Second Differences on Amenable Topological Groups

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