Quantum-mechanical calculations in the algebraic group theory
type: Item
instance of: scholarly article
title: Quantum-mechanical calculations in the algebraic group theory
zbMATH Open document ID: 0713.20041
DOI: 10.1007/BF02096657
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: [This review concerns also the following two items Zbl 0713.20042 and Zbl 0713.20043.] Methods of quantum mechanics can be used not only to derive some formulas in algebraic group theory, but also to give to these formulas a physical interpretation. Given a simple Lie algebra \({\mathfrak g}\) and a finite field k, a Chevalley group is a finite group generated by \(\exp (tX_{\alpha})\), \(t\in k\), \(X_{\alpha}\in {\mathfrak g}\) (in a Chevalley basis). Here the orders of all Chevalley groups of Lie type over finite fields are shown to be simply related to the partition function Z of n harmonic oscillators over states of a particular symmetry type. There is also an interesting conformal limit when \(n\to \infty\).
Mathematics Subject Classification ID: 20G45
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B81
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 4173254
zbMATH Keywords: quantum mechanics
zbMATH Keywords: simple Lie algebra
zbMATH Keywords: Chevalley basis
zbMATH Keywords: Chevalley groups of Lie type over finite fields
zbMATH Keywords: partition function
zbMATH Keywords: harmonic oscillators
zbMATH Keywords: states
zbMATH Keywords: conformal limit
author: M. Olshanetsky
reviewed by: A. O. Barut
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02096657
OpenAlex ID: W1986381246
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description: scientific article; zbMATH DE number 4173254