Finite-dimensional nilpotent Lie algebras of class two and alternating bilinear maps (Q2049387)

Overlapping the abstract: ``Let \(\mathbb{F}\) be a field of characteristic different from two. Suppose that \(\mathcal{N}_2\) is the category of finite-dimensional nilpotent Lie algebras of class two over the field \(\mathbb{F}\) and that \(\mathcal{ALT}\) is the category of alternating bilinear maps of \(\mathbb{F}\)-vector spaces. We establish a relation between the category \(\mathcal{N}_2\) and the category \(\mathcal{ALT}\). Then we show that the problem of determining the capability of these Lie algebras reduces to determining the epicenter of the corresponding objects in \(\mathcal{ALT}\). 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.'' I should also mention that there is a wide use of Heisenberg algebras, nonabelian tensor products and Schur multipliers in the main proofs of this elegant contribution.