Capability of Nilpotent Lie Algebras of Small Dimension

Given a nilpotent Lie algebra L of dimension ≤ 6 on an arbitrary field of characteristic ≠ 2, we show a direct method to detect whether L is capable or not via computations on the size of its nonabelian exterior square L ∧ L. For dimensions higher than 6, we show a result of general nature, based on the evidences of the low dimensional case, but also on the evidences of large families of nilpotent Lie algebras, namely the generalized Heisenberg algebras. Indeed, we detect the capability of L ∧ L via the size of the Schur multiplier M(L/Z∧(L)) of L/Z∧(L), where Z∧(L) denotes the exterior center of L.