**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXI, 1(2002)

p. 51

On $k$-abelian $p$-filiform Lie algebras I

O. R. Campoamor Stursberg

**Abstract**.
We classify the $\left( n-5\right) $-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. We show that this property is strongly related with the structure of the Lie algebra of derivations; explicitly we show that if a $\left( n-5\right) $-filiform Lie algebra is characteristically nilpotent, then it must be $2$-abelian. We also give applications to the construction of solvable rigid laws whose nilradical is $k$-abelian with mixed characteristic sequence, as well as applications to the theory of nilalgebras of parabolic subalgebras of the exceptional simple Lie algebra $E_ 6 $.

**AMS subject classification**:
17B30 17B56

**Keywords**:

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