Finite-dimensional nilpotent Lie algebras of class two and alternating bilinear maps

Abstract

Let F be a field of characteristic different from two. Suppose that N2 is the category of finite-dimensional nilpotent Lie algebras of class two over the field F and that ALT is the category of alternating bilinear maps of F-vector spaces. We establish a relation between the category N2 and the category ALT. Then we show that the problem of determining the capability of these Lie algebras reduces to determining the epicenter of the corresponding objects in ALT. As an application of this technique, we describe the structure of Lie algebras corresponding to alternating bilinear maps of rank one (that is, to alternating bilinear forms). Also, we describe the epicenter of decomposable nondegenerate alternating bilinear maps of rank two.