On an algorithm for finding derivations of Lie algebras (Q2889743)

The paper introduces a matrix-based algorithm to compute the derivation algebra of any given Lie algebra \(\mathfrak g\) over the real number field \(\mathbb R\), starting from its dimension and its structure constants \(c_{ij}^k\) with respect to a fixed basis \(\mathcal B\). According to the authors, the advantage of the algorithm proposed in this article lies in the fact that no restrictions must be imposed on the Lie algebra for computing its derivation algebra.NEWLINENEWLINEThe authors use the matrix representation \((d_{ij})^T\) of any given linear transformation \(D\) on \(\mathfrak g\) with respect to the basis \(\mathcal B\) in order to translate the condition of being a derivation: NEWLINE\[NEWLINED([X,Y])=[D(X),Y]+[X,D(Y)], \quad \mathrm{for} \;X,Y\in\mathfrak gNEWLINE\]NEWLINE into a system of linear equations where the unknowns are precisely the values of \(d_{ij}\) and the system coefficients are the structure constants \(c_{ij}^k\). Concretely, the system to be solved has \(n^2\) unknowns for the \(n^3\) linear equations, with the following expression: NEWLINE\[NEWLINE\sum_{k=1}^n c_{ij}^k d_{kp}=\sum_{k=1}^n(d_{ik}c_{kj}^p + d_{jk}c_{ik}^p), \quad \mathrm{for} \;1\leq i,j,p\leq n.NEWLINE\]NEWLINE By using the reflexivity and skew-symmetry of the Lie bracket, the authors reduce the number of equations up to \(n^2(n-1)/2\), and hence the number of computations to be carried out by the algorithm. However, the complexity order is the same in both cases.NEWLINENEWLINEAs example of application, the derivation algebra is computed for the orthogonal Lie algebra and the Heisenberg algebra of dimension 3, as well as for a particular Lie algebra of dimension 4. This is done to show how the algorithm runs easier when the non-zero brackets in the law of the Lie algebra \(\mathfrak g\) correspond to the generators of \(\mathfrak g\). The third example shows that the algorithm also runs when some non-zero bracket is a linear combination of the generators of \(\mathfrak g\).NEWLINENEWLINEFinally, the authors conclude the article with some possible applications of the algorithm to problems related to control systems on Lie groups in virtue of some results proved in [\textit{V. Ayala} and \textit{J. Tirao}, Proc. Symp. Pure Math. 64, 47--64 (1999; Zbl 0916.93015)]. Concretely, the following three problems are commented: First, the existence of an efficient algorithm to compute derivations of Lie algebras makes determining trajectories easier in linear control systems for connected Lie groups. Second, derivations provide a condition to decide when a (transitive) linear control system on a connected Lie group is locally controllable. Third, spectrum conditions on derivations could be also used to obtain globally null controllable systems on connected, simply connected Lie groups.