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On the Growth of Subalgebras in Lie p-Algebras

✍ Scribed by David Riley; Vladimir Tasić

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
117 KB
Volume
237
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let L be a finitely generated Lie p-algebra over a finite field F. Then the number, a n L , of p-subalgebras of finite codimension n in L is finite. We say that L has PSG (polynomial p-subalgebras growth) if the growth of a n L is bounded above by some polynomial in F n . We show that if L has PSG then the lower central series of L stabilises after a finite number of steps. On the other hand, if L is nilpotent then L has PSG. We deduce the following group-theoretic result. Let G be a group and let G p denote a pro-p completion of G. Then the associated Lie p-algebra p G of G has PSG if and only if G p is a p-adic analytic Lie group.

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