On Characterizing Nilpotent Lie algebras by their Multipliers

Abstract

Authors have turned their attentions to special classes of nilpotent Lie algebras such as two-step nilpotent and filiform Lie algebras, in particular filiform Lie algebras are classified up to dimension eleven [8]. These techniques have not worked well in higher dimensions. For a nilpotent Lie algebra L, of dimension n, we consider central extensions 0->M->C->L->0 where M is contained or equal to cˆ2 and Z(C), where cˆ2 is the derived algebra of C and Z(C) is the center of C. Let M(L) be the M of largest dimension and call it the multiplier of L due to it's analogy with the Schur multiplier. The maximum dimension that M can obtain is 1/2n(n-1) and this is met if and only if L is abelian. Let t(L) =1/2n(n-1) - dimM(L). Then t(L) =1 if and only if L=H(1), where H(n) is the Heisenberg algebra of dimension 2n + 1.

A recent technique to classify nilpotent Lie algebra is to use the dimension of the multiplier of L. In particular, to find those algebras whose multipliers have dimension close to the maximum, we call this invariant t(L). Algebras with t(L) less than or equal to 8 have been classified [10]. It's the purpose of this work to use this technique on filiform Lie algebras along with three main tools namely: Propositions 0, 3, and theorem 4. All algebras in this work will be taken over any field whereas in previous works, they have been taken over the field of real and complex numbers.