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On the module structure of a group action on a Lie algebra

Part of: Representation theory of groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Athanassios I. Papistas
Athanassios I. Papistas
Affiliation:
Faculty of Sciences Department of Mathematics Aristotle University of ThessalonikiGR 541 24, ThessalonikiGreece e-mail: apapist@math.auth.gr
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Abstract

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Let G be a finite group, K a field, and V a finite-dimensional K G-module. Write L(V) for the free Lie algebra on V; similarly, let M ( V) be the free metabelian Lie algebra. The action of G extends naturally to these algebras, so they become KG-modules, which are direct sums of finite-dimensional submodules. This paper explores whether indecomposable direct summands of such a KG-module (for some specific choices of G, K and V) must fall into finitely many isomorphism classes. Of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic) and dim V > 1.

The first two results show that the answer is positive for M(V) when K is finite and dim V = 2, but negative when G is the Klein four-group, the characteristic of K is 2, and V is the unique 3-dimnsional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of charateristic 2 with more than 2 elements, V is any faithfull module of dimension 2, and B is the unique 3-dimensional quotient of D; the answer is positive for L(V) if and only if it is positive for each of L(B), L(D), and L(V ⊗ V).

MSC classification

Secondary: 17B01: Identities, free Lie (super)algebras 17B30: Solvable, nilpotent (super)algebras 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 237 - 248
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alperin, J. and Kovács, L. G., ‘Periodicity of Weyl modules for SL(2, q)’, J. Algebra 74 (1982), 52–54.CrossRefGoogle Scholar
[2]Alperin, J. L., Local representation theory, Cambridge Studies in Advanced Mathematics 11 (Cambridge University Press, Cambridge, 1993).Google Scholar
[3]Bakhturin, Yu. A., Identical relations in Lie algebras (Nauka, Moscow, 1985), in Russian; English translation: (VNU Science Press, Utrecht, 1987).Google Scholar
[4]Bryant, R. M., ‘Free Lie algebras and formal power series’, J. Algebra 253 (2002), 167–188.CrossRefGoogle Scholar
[5]Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Lie powers of modules for groups of prime order’, Proc. London Math. Soc. (3) 84 (2002), 343–374.CrossRefGoogle Scholar
[6]Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Lie powers of modules for GL(2,p)’, J. Algebra 260 (2003), 617–630.CrossRefGoogle Scholar
[7]Conlon, S. B., ‘Certain representation algebras’, J. Aust. Math. Soc. 5 (1965), 83–99.CrossRefGoogle Scholar
[8]Fulton, W. and Harris, J., Representation theory: a first course (Springer, New York, 1991).Google Scholar
[9]Hannebauer, T. and Stöhr, R., ‘Homology of groups with coefficients in free metabelian Lie powers and exterior powers of relation modules and applications to group theory’, in: Proceedings of the Second International Group Theory Conference (Bressanone, 1989), Rend. Circ. Mat. Palermo (2) Suppl. 23 (1990), 77–113.Google Scholar
[10]Huppert, B. and Blackburn, N., Finite groups II (Springer, Berlin, 1982).CrossRefGoogle Scholar
[11]Karagueuzian, D. and Symonds, P., ‘The module structure of a group action on a polynomial ring’, J. Algebra 218 (1999), 672–692.CrossRefGoogle Scholar
[12]Stöhr, R., ‘Restricted Lazard elimination and modular Lie powers’, J. Aust. Math. Soc. 71 (2001), 259–277.CrossRefGoogle Scholar