This article concerns the equivalence of different characterisations of 2-step nilpotent quadratic Lie algebras of finite dimension: double extensions, \(T^*\)-extensions and \(n\)-quadratic families of matrices.
Double extensions, \(T^*\)-extensions and \(n\)-quadratic families of matrices are used as tools to construct quadratic Lie algebras. The quadratic Lie algebras arising from these 3 methods are formulated in terms of very different algebraic ingredients and, in consequence, it is hard to determine when two of these methods will produce isometric algebras.
For 2-step nilpotent quadratic Lie algebras, the authors lay out a detailed a description of the ingredients involved in each construction. This enables them to establish a direct relationship between the three methods and to give an explicit equivalence between the underlying algebras and the algebraic operators required for each construction. Every quadratic Lie algebra decomposes as a direct sum of proper ideals: a quadratic abelian Lie algebra and a quadratic reduced Lie algebra. Hence the authors restrict their attention to quadratic reduced Lie algebras. Moreover, the class of reduced 2-step nilpotent Lie algebras is equivalent to those Lie algebras whose derived ideal equals the centre.
One dimensional double extensions
Given a nilpotent quadratic Lie algebra \(\mathfrak{g}\), in order to construct double extensions \(\overline{\mathfrak{g}}\) by one dimensional Lie algebras, one first takes the extension of \(\mathfrak{g}\) by a central element followed by an extension using a derivation.
It is known that any quadratic Lie algebra can be constructed as a sequence of double extensions by one dimensional Lie algebras, which the authors refer to as a chain. Indeed, Kac shows that any solvable quadratic Lie algebra \((\mathfrak{g},q)\) is the one dimensional double extension \(\mathfrak{g}=\overline{\mathfrak{h}}\) of a quadratic Lie algebra \((\mathfrak{h},q')\). Conversely, he shows that a solvable quadratic Lie algebra extends to a Lie algebra of the same type. [V. G. Kac, Infinite dimensional Lie algebras. 3rd ed. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)] Medina and Revoy prove that Kac’s results hold for arbitrary quadratic Lie algebras. [A. Medina and P. Revoy, Ann. Sci. Éc. Norm. Supér. (4) 18, 553–561 (1985; Zbl 0592.17006)].
For a chain of one dimensional double extensions, the authors introduce the non-null and the two step properties, NNP and 2SP, respectively. It is shown that these two properties are preserved when taking any further one dimensional double extensions and, chains with such properties produce 2-step nilpotent Lie algebras. Departing from these properties, the authors are able to give sufficient conditions that guarantee that the final element of the chain is a 2-step nilpotent quadratic Lie algebra, whose quadratic form and Lie bracket take a very explicit form. They also determine, in terms of these properties, necessary and sufficient conditions for the final algebra to be reduced.
\(T^*\)-extensions
\(T^*\)-extensions produce Lie algebras which are built from a Lie algebra and some cohomological data. More precisely, if \(\mathfrak{g}\) is a Lie algebra and a 2-cocycle \(\omega\colon \mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}^*\), one defines the \(T^*\)-extension of \(\mathfrak{g}\) by equipping \(T^*_\omega(\mathfrak{g})=\mathfrak{g}\oplus\mathfrak{g}^*\) with a Lie bracket consisting of a sum of the Lie bracket of \(\mathfrak{g}\), \(\omega\) and the commutator of the coadjoint representation. A cyclic \(\omega\), gives a quadratic \(T^*_\omega(\mathfrak{g})\). [M. Bordemann, Acta Math. Univ. Comen., New Ser. 66, No. 2, 151–201 (1997; Zbl 1014.17003)].
Duong characterises quadratic 2-step reduced nilpotent Lie algebras in terms of \(T^*\)-extensions: any such algebra is isometrically isomorphic to the \(T^*\)-extension of an abelian Lie algebra with a non-degenerate cyclic 2-cocycle \(\omega\). [Minh Thanh Duong, Vietnam J. Math. 41, No. 2, 135–148 (2013; Zbl 1323.17013)] An alternate proof of this fact is given by the authors.
\(n\)-quadratic families of matrices
Quadratic Lie algebras can be equivalently described in terms of a family of \(n\) quadratic matrices \(\{M_1,\ldots,M_n\}\). In essence, these \(n\times n\) matrices are defined so that the \(j\)-th column of \(M_i\) is a Lie bracket of basis vectors \([e_i,e_j]\). An additional requirement of the skew-symmetry of each \(M_i\) ensures the existence of a quadratic form. One can then consider an \(n\times \left(\frac{n-1}{2}\right)\) matrix \(\mathcal{F}\) whose columns are the structure constants. The authors prove that 2-step reduced nilpotent Lie algebras are equivalent to those families where \(\mathcal{F}\) has maximum rank (non-degenerate).
From this identification, Benito, de-la-Concepción, Roldán-López and Sesma, derive an algorithm to find quadratic Lie algebras of fixed dimension and generators. [P. Benito et al., J. Symb. Comput. 94, 70–89 (2019; Zbl 1423.68613)]. This would require one to solve the linear equations prescribed by the definition of \(n\)-quadratic matrices subject to the constraint of the non-degeneracy of the family. Unknown to the authors, solving the non-degeneracy of the family is a particular case of the well known matrix completion problem in computer science. This can be an NP-hard problem, depending on the parameters resulting from the definition of the \(n\)-quadratic matrices.
Main theorem
The statement of the main theorem is as follows.
Theorem. Let \(n\geq3\) and fix a vector space \(B=\langle b_1,\ldots b_n\rangle\) and a bilinear form \(\omega\colon B\times B\to B^*\). Set \(\omega(b_i,b_j)(b_k) = c_{ijk}\) and \(\mathfrak{g} = B\oplus B^*\). For every \(b,b'\in B\) and \(\beta,\beta'\in B^*\), define the product \( [b+\beta , b'+ \beta'] = \omega(b,b') \) and the bilinear form \[ q'(b+\beta, b'+ \beta') = \beta(b') + \beta'(b). \] Let \(\{(\mathfrak{g}_k,q_k)\}_{k=0}^n,\) denote the chain obtained by extending, at step \(i\), the algebra \((\mathfrak{g}_0,q_0) = (\{0\},0)\) by the central element \(b_i\) and then by the derivation \(d_{i-1}(b^*_j) = 0\), \(d_{i-1}(b_j) = \sum_{k=1}^{i-1}c_{ijk}b_k^*\). Let \(\{ M_1,\ldots,M_n\}\) be the family of matrices with entries \((M_i)_{(k,j)} = c_{ijk}\). The following statements are equivalent

1.

\((\mathfrak{g}, q')\) is a 2-step reduced quadratic Lie algebra,

2.

\(\omega\) is a non-zero cyclic 2-cocycle and \((\mathfrak{g}, q')=(T_\omega B,q')\),

3.

the chain \(\{(\mathfrak{g}_k,q_k)\}\) satisfies NNP and 2SP and \((\mathfrak{g}_n,q_n) = (\mathfrak{g},q')\)

4.

\(\{ M_1,\ldots,M_n\}\) is a non-degenerate \(n\)-quadratic family of matrices whose associated algebra is \((\mathfrak{g},q')\).

Trivectors
The authors use the equivalence theorem to provide an alternate characterisation of 2-step nilpotent reduced quadratic Lie algebras: they show a one to one correspondence between 2-step nilpotent reduced quadratic Lie algebras and 3-alternating forms or trivectors.
The classification of trivectors is then used to provide a complete list of 2-step nilpotent reduced quadratic Lie algebras of dimension up to 17.