Local and 2-local anti-derivations on null-filiform and filiform Leibniz algebras and their solvable extensions
In this paper, we study anti-derivations, local anti-derivations and 2-local anti-
derivations on several finite-dimensional complex Leibniz algebras. More precisely, we consider the
null-filiform Leibniz algebra NF n , the naturally graded non-Lie filiform Leibniz algebras F 1
n
and F 2
n ,
and their associated solvable extensions. Here, an anti-derivation is understood as a (−1)-derivation
satisfying
D([x,y]) = −[D(x),y] − [x,D(y)].
We first determine the spaces of anti-derivations for all algebras under consideration. We then ob-
tain complete descriptions of their local anti-derivations. In particular, every local anti-derivation is
an anti-derivation precisely for the algebras R, R(F 1
n ) and L 2 , whereas NF n , F
1
n , F
2
n
and L 1 admit
local anti-derivations which are not anti-derivations. Furthermore, we prove that every 2-local anti-
derivation on NF n and on each of the four solvable extensions is an anti-derivation. For F 1
n
and F 2
n ,
we construct families of nonlinear 2-local anti-derivations with values in the center. The members
of these families are anti-derivations exactly when the corresponding defining maps are linear.
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