Computer construction of split Cartan subalgebras
The author gives in this paper a polynomial time algorithm that finds a split Cartan subalgebra of a finite Chevalley Lie algebra. A Chevalley Lie algebra is the Lie algebra of an untwisted group of Lie type. Every Chevalley Lie algebra contains a split Cartan subalgebra and has a Chevalley basis that consists of common eigenvectors for such a subalgebra. The author supposes that a Chevalley Lie algebra is specified as a vector space together with a list of multiplicative structure constants that define the products of pairs of vectors from an arbitrary basis. The main problem posed in the paper is the following: \textit{Given a set of multiplicative structure constants of a finite Chevalley Lie algebra, compute a basis for a split Cartan subalgebra.} No Chevalley basis is assumed to be known in this problem (in the cases considered, the construction of a split Cartan subalgebra is equivalent to the construction of a Chevalley basis). The author gives a polynomial time Las Vegas algorithm that solves those instances of the former problem where the characteristic is odd and the Lie algebra does not have type \(A_{2}\) or \(G_{2}\) in characteristic 3. The algorithm employs random operations, in such a way that, even if the random choices are unfortunate, the algorithm merely fails to produce output rather than produce an incorrect output. It succeeds with a probability of at least 1/2 and its running time is bounding by some polynomials in its input size. The algorithm selects any available Cartan subalgebra and repeatedly simplifies the Cartan subalgebra until it is split. The simplification process is recursive, and relies on the fact that initial Cartan subalgebras can be found reasonably quickly. Sections 2, 3, 4 and 5 provide the theoretical justification for the algorithm. Once the algorithm is established in Sections 6 and 7, it is applied to the Chevalley Lie algebra of \(E_{8} (3)\) in the example presented in Section 8. In its recursive calls, the example also exhibits applications of the algorithm to the Chevalley Lie algebras \(E_{6}(3)\), \(D_{5}(3)\), \(A_{3}(3)\) and \(A_{1}(3)\). In Section 9, the author gives an example to show that, with a small modification, the algorithm can also be applied to certain Chevalley Lie algebras in characteristic 2.
scientific article; zbMATH DE number 5141725