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Van der Waerden's continuity theorem for the commutator subgroups of connected Lie groups and Mishchenko's conjecture

A. I. Shtern
장전수학회 (Jangjeon Mathematical Society)|2006.10|Advanced Studies in Contemporary Mathematics|제13권 제2호|pp. 143-158

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Abstract

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. In this paper, with the help of an earlier result of the author claiming that the van der Waerden theorem holds for any connected semisimple Lie group, it is proved that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup of the group; moreover, it turns out that a connected Lie group satis¯es the assertion of the van der Waerden theorem (i.e., all locally bounded finite-dimensional representations of the group are continuous) if and only if the group is perfect (i.e., coincides with the commutator subgroup). Thus, for perfect connected linear Lie groups, the structure of (totally) bounded sets de¯nes the topology, and any boundedness-preserving group isomorphism of a perfect connected linear Lie group onto another perfect connected linear Lie group is automatically continuous. To study this phenomenon, the notion of discontinuity group of a locally bounded finite-dimensional representation of a topological group is introduced and studied. The notion of local boundedness of a representation is naturally related to the notion of point oscillation (at the identity element of the group) introduced by the author in 2002. According to a conjecture expressed by A. S. Mishchenko, the finite-dimensional representations of Lie groups can take only three possible values for the (reasonably defined) point oscillation, namely, 0, 2, and 1. We prove the validity of the conjecture. As a corollary, we prove that the class of connected Lie groups for which the point oscillation of a finite-dimensional representation can take only two values, 0 and 1, is the very class of perfect connected Lie groups. Related open problems are indicated.

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관련 논문AI추천

Analog of van der Waerden's continuity theorem and the validity of Mishchenko's conjecture for relatively compact homomorphisms of arbitrary locally compact groupsA. I. Shtern장전수학회 · Advanced Studies in Contemporary Mathematics2007 · 제14권 제1호Continuity conditions for homomorphisms of connected locally compact groups into Lie groupsA. I. Shtern장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2025 · 제28권 제2호A criterion for the continuity of the restriction to the commutator subgroup of a locally bounded finite-dimensional representation of a connected Lie group with respect to the ordinary group topologyA. I. Shtern장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2019 · 제22권 제1호Locally bounded automorphisms of connected Lie groupsA. I. Shtern장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2025 · 제28권 제3호Continuity criteria for locally bounded homomorphisms of certain connected Lie groupsA. I. SHTERN장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2023 · 제26권 제4호Bounded structure and continuity for homomorphisms of perfect connected locally compact groupsA. I. Shtern장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2012 · 제15권 제3호

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