On the linearity of torsion-free nilpotent groups of finite Morley rank

Altinel, Tuna; Wilson, John

doi:10.1090/S0002-9939-08-09695-0

On the linearity of torsion-free nilpotent groups of finite Morley rank
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by Tuna Altinel and John S. Wilson

Proc. Amer. Math. Soc. 137 (2009), 1813-1821

DOI: https://doi.org/10.1090/S0002-9939-08-09695-0

Published electronically: December 15, 2008

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Abstract:

It is proved that every torsion-free nilpotent group of finite Morley rank is isomorphic to a matrix group over a field of characteristic zero.

References

I. D. Ado, The representation of Lie algebras by matrices, Uspehi Matem. Nauk (N.S.) 2 (1947), no. 6(22), 159–173 (Russian). MR 0027753
Andreas Baudisch, A new uncountably categorical group, Trans. Amer. Math. Soc. 348 (1996), no. 10, 3889–3940. MR 1351488, DOI 10.1090/S0002-9947-96-01623-6
Alexandre Borovik and Ali Nesin, Groups of finite Morley rank, Oxford Logic Guides, vol. 26, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1321141
Gregory L. Cherlin and Joachim Reineke, Categoricity and stability of commutative rings, Ann. Math. Logic 9 (1976), no. 4, 367–399. MR 480007, DOI 10.1016/0003-4843(76)90017-6
I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 16094, DOI 10.1090/S0002-9947-1946-0016094-3
O. Frécon. Linearity of solvable groups of finite Morley rank. Preprint, 2007.
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
Angus Macintyre, On $\omega _{1}$-categorical theories of fields, Fund. Math. 71 (1971), no. 1, 1–25. (errata insert). MR 290954, DOI 10.4064/fm-71-1-1-25
Ali Nesin, Nonassociative rings of finite Morley rank, The model theory of groups (Notre Dame, IN, 1985–1987) Notre Dame Math. Lectures, vol. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 117–137. MR 985343
I. N. Stewart, An algebraic treatment of Mal′cev’s theorems concerning nilpotent Lie groups and their Lie algebras, Compositio Math. 22 (1970), 289–312. MR 288158
Robert B. Warfield Jr., Nilpotent groups, Lecture Notes in Mathematics, Vol. 513, Springer-Verlag, Berlin-New York, 1976. MR 0409661, DOI 10.1007/BFb0080152
B. I. Zil′ber, Rings whose theory is $\aleph _{1}$-categorical, Algebra i Logika 13 (1974), 168–187, 235. MR 0366650