On the module structure of free Lie algebras

Bryant, R.; Stöhr, Ralph

doi:10.1090/S0002-9947-99-02369-7

On the module structure of free Lie algebras
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by R. M. Bryant and Ralph Stöhr

Trans. Amer. Math. Soc. 352 (2000), 901-934

DOI: https://doi.org/10.1090/S0002-9947-99-02369-7

Published electronically: October 6, 1999

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Abstract:

We study the free Lie algebra $L$ over a field of non-zero characteristic $p$ as a module for the cyclic group of order $p$ acting on $L$ by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of $L$ as a direct sum of indecomposable modules.

References

Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
R. M. Bryant, ‘Cyclic groups acting on free Lie algebras’, in P. H. Kropholler, G. A. Niblo and R. Stöhr (editors) ‘Geometry and Cohomology in Group Theory’, London Mathematical Society Lecture Note Series, 252, Cambridge University Press, Cambridge, 1998, pp. 39–44.
R. M. Bryant and R. Stöhr, Fixed points of automorphisms of free Lie algebras, Arch. Math. (Basel) 67 (1996), no. 4, 281–289. MR 1407330, DOI 10.1007/BF01197591
Stephen Donkin and Karin Erdmann, Tilting modules, symmetric functions, and the module structure of the free Lie algebra, J. Algebra 203 (1998), no. 1, 69–90. MR 1620709, DOI 10.1006/jabr.1997.7327
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
L. G. Kovács and R. Stöhr, ‘Lie powers of the natural module for $GL(2)$’, J. Algebra, to appear.
G. P. Kukin, Subalgebras of free Lie $p$-algebras, Algebra i Logika 11 (1972), 535–550, 614 (Russian). MR 0318251
Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
M. W. Short, A conjecture about free Lie algebras, Comm. Algebra 23 (1995), no. 8, 3051–3057. MR 1332165, DOI 10.1080/00927879508825387
Ralph Stöhr, On torsion in free central extensions of some torsion-free groups, J. Pure Appl. Algebra 46 (1987), no. 2-3, 249–289. MR 897018, DOI 10.1016/0022-4049(87)90096-X
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
G. E. Wall, On the Lie ring of a group of prime exponent, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 667–690. MR 0357630
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3