Bardy, Nicole Biregular subalgebras of a Kac-Moody-Borcherds algebra. (Sous-algèbres birégulières d’une algèbre de Kac-Moody-Borcherds.) (French) Zbl 1007.17018 Nagoya Math. J. 156, 1-83 (1999). Summary: Let \({\mathfrak g}\) be a Kac-Moody-Borcherds algebra on a field \(\mathbb{K}\) associated to a symmetrizable matrix and with Cartan subalgebra \({\mathfrak h}\). Let \({\mathfrak L}\) be an ad \({\mathfrak h}\)-invariant subalgebra such that the restriction to \({\mathfrak L}\) of the standard bilinear form is nondegenerate. We show that the root system \(\Psi\) of \(({\mathfrak L},{\mathfrak h})\) is a subsystem according to [N. Bardy, Mém. Soc. Math. Fr., Nouv. Sér. 65, 188 (1996)] of \(\Delta( {\mathfrak g},{\mathfrak h})\). Moreover, if a subsystem \(\Omega\) satisfies some conditions (i.e. \(\Omega\) is “réduit et presque-clos”) of \(\Psi\), we construct inside of \({\mathfrak L}\) a Kac-Moody-Borcherds algebra with root system \(\Omega\).Let \(k\) be a subfield of \(\mathbb{K}\). We prove similar results in the case of an action of a finite group of \(k\)-semi-automorphisms. In particular, we obtain a generalization to the Kac-Moody case of a result by Borel and Tits. Let \({\mathfrak g}\) be an almost-\(k\)-split form of a Kac-Moody algebra. We construct a Kac-Moody \(k\)-algebra with root system similar to the system of \({\mathfrak g}\) (save for some multiples of certain roots). Cited in 2 Documents MSC: 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B65 Infinite-dimensional Lie (super)algebras Keywords:almost \(k\)-split form; Kac-Moody-Borcherds algebra; Kac-Moody algebra; root system × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Tsukuba J. Math 11 pp 77– (1987) · Zbl 0632.17012 · doi:10.21099/tkbjm/1496160503 [2] Infinite dimensional Lie algebras, troisième édition (1990) [3] Linear algebraic groups (1975) [4] Amer. math. Soc. Transl., Ser 6 pp 111– [5] DOI: 10.1006/jabr.1995.1004 · Zbl 0823.17034 · doi:10.1006/jabr.1995.1004 [6] DOI: 10.3792/pjaa.67.117 · Zbl 0752.17025 · doi:10.3792/pjaa.67.117 [7] J. algebra 115 pp 501– (1989) [8] Groupes et algèbres de Lie · Zbl 0464.22001 [9] Mémoires de la SMF 65 (1996) [10] Séminaire Bourbaki 119 pp 01– (1955) [11] Publ. Math. I.H.E.S 27 (1965) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.