Capability of nilpotent Lie algebras of small dimension

Abstract

Given a nilpotent Lie algebra $L$ of dimension $\le 6$ on an arbitrary field of characteristic $\ne 2$, we show a direct method which allows us to detect the capability of $L$ via computations on the size of its nonabelian exterior square $L\wedge L$. For dimensions higher than $6$, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of $L\wedge L$ via the size of the Schur multiplier $M(L/Z^{\wedge }(L))$ of $L/Z^{\wedge }(L)$, where $Z^{\wedge }(L)$ denotes the exterior center of $L$.

Cited by in corpus (1)

• On characterizing nilpotent Lie algebra by their multiplier, $ s(L)=6, 7