Metric Lie algebras and quadratic extensions
A metric Lie algebra is a finite-dimensional real Lie algebra equipped with an invariant nondegenerate symmetric bilinear form. An example of a metric Lie algebra is the Lie algebra of the transvection group of a pseudo-Riemannian symmetric space. The paper contains a systematic study of the structure of metric Lie algebras. A metric Lie algebra \({\mathfrak g}\) is indecomposable if it does not contain a proper ideal on which the bilinear form is nondegenerate, and in this case \({\mathfrak g}\) is either simple or does not contain a simple ideal. Let \({\mathfrak l}\) be a Lie algebra and \((\rho,{\mathfrak a})\) be an orthogonal \({\mathfrak l}\)-module. A quadratic extension of \({\mathfrak l}\) by \({\mathfrak a}\) is a quadruple \(({\mathfrak g},{\mathfrak i},i,p)\), where \({\mathfrak g}\) is a metric Lie algebra, \({\mathfrak i}\) is an isotropic ideal in \({\mathfrak g}\), and \(i : {\mathfrak a} \to {\mathfrak g}/{\mathfrak i}\) and \(p : {\mathfrak g}/{\mathfrak i} \to {\mathfrak l}\) are Lie algebra homomorphisms such that \(0 \to {\mathfrak a} \to {\mathfrak g}/{\mathfrak i} \to {\mathfrak l} \to 0\) is an exact sequence, \(i(\rho(L)A) = [\tilde{L},i(A)] \in i({\mathfrak a})\) holds for all \(A \in {\mathfrak a}\) and \(\tilde{L} \in {\mathfrak g}/{\mathfrak i}\) with \(\rho(\tilde{L}) = L\), \(i({\mathfrak a}) = {\mathfrak i}^\perp/{\mathfrak i}\) and \(i : {\mathfrak a} \to {\mathfrak i}^\perp/{\mathfrak i}\) is an isometry. Such a quadratic extension does not contain a simple ideal. The authors show that any metric Lie algebra \({\mathfrak g}\) without simple ideals has the structure of a balanced quadratic extension. Here, balanced means that \({\mathfrak i}\) coincides with a canonical isotropic ideal \({\mathfrak i}({\mathfrak g})\) in \({\mathfrak g}\). For a balanced quadratic extension the representation \(\rho\) of \({\mathfrak l}\) on \({\mathfrak a}\) is semisimple.\break The balanced quadratic extensions can be identified with elements in certain quadratic cohomology sets \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})\). The investigation of \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})\) is the main technical tool in the paper. The authors show that there is a one-to-one correspondence between the set of balanced quadratic extensions and the set \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})_0\) of certain admissible cohomology classes in \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})\). This leads to the characterization of the set of isomorphism classes of nonsimple indecomposable metric Lie algebras as the union \(\amalg_{{\mathfrak l},{\mathfrak a}} {\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})_0/G_{{\mathfrak l},{\mathfrak a}}\) over a set of representatives of isomorphism classes of pairs \(({\mathfrak l},{\mathfrak a})\) consisting of real finite-dimensional Lie algebras \({\mathfrak l}\) and a semisimple orthogonal \({\mathfrak l}\)-module \({\mathfrak a}\), where \(G_{{\mathfrak l},{\mathfrak a}}\) is the automorphism group of the pair \(({\mathfrak l},{\mathfrak a})\) and the orbit space \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})_0/G_{{\mathfrak l},{\mathfrak a}}\) is with respect to the induced action of \(G_{{\mathfrak l},{\mathfrak a}}\) on \({\mathcal H}^2_Q({\mathfrak l},{\mathfrak a})_0\).\break This characterization of nonsimple indecomposable metric Lie algebras by certain equivalence classes of cohomology sets indicates a classification scheme for such Lie algebras. The authors use this scheme to obtain an explicit classification of all nonsimple indecomposable metric Lie algebras of index \(3\).
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