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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some model theory of compact Lie groups
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by Ali Nesin and Anand Pillay PDF
Trans. Amer. Math. Soc. 326 (1991), 453-463 Request permission

Abstract:

We consider questions of first order definability in a compact Lie group $G$. Our main result is that if such $G$ is simple (and centerless) then the Lie group structure of $G$ is first order definable from the abstract group structure. Along the way we also show (i) if $G$ is non-Abelian and connected then a copy of the field $\mathbb {R}$ is interpretable. in $(G, \cdot )$, and (ii) any "$1$-dimensional" field interpretable in $(\mathbb {R}, +, \cdot )$ is definably (i.e., semialgebraically) isomorphic to the ground field $\mathbb {R}$.
References
    F. Bachman, Geometric structures, Nauka, Moscow, 1969.
  • Armand Borel, Groupes linéaires algébriques, Ann. of Math. (2) 64 (1956), 20–82 (French). MR 93006, DOI 10.2307/1969949
  • Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
  • Nicolas Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1982 (French). Chapitre 9. Groupes de Lie réels compacts. [Chapter 9. Compact real Lie groups]. MR 682756
  • C. Chevalley, Theory of Lie groups, Princeton Univ. Press, Princeton, N. J., 1946.
  • Ali Nesin, Nonsolvable groups of Morley rank $3$, J. Algebra 124 (1989), no. 1, 199–218. MR 1005703, DOI 10.1016/0021-8693(89)90159-2
  • Anand Pillay, On groups and fields definable in $o$-minimal structures, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255. MR 961362, DOI 10.1016/0022-4049(88)90125-9
  • An application of model theory to real and $p$-adic algebraic groups, preprint, 1988.
  • Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
  • L. S. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 0201557
  • Masahiro Shiota, Nash manifolds, Lecture Notes in Mathematics, vol. 1269, Springer-Verlag, Berlin, 1987. MR 904479, DOI 10.1007/BFb0078571
  • B. I. Zil′ber, Some model theory of simple algebraic groups over algebraically closed fields, Colloq. Math. 48 (1984), no. 2, 173–180. MR 758524, DOI 10.4064/cm-48-2-173-180
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 453-463
  • MSC: Primary 03C40; Secondary 03C60, 22E15
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1002922-9
  • MathSciNet review: 1002922