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Inverses of Cartan matrices of Lie algebras and Lie superalgebras
Stockholm University, Faculty of Science, Department of Mathematics. New York University Abu Dhabi, United Arab Emirates.
Number of Authors: 22019 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 583, p. 195-256Article in journal (Refereed) Published
Abstract [en]

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the Os only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic). We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).

Place, publisher, year, edition, pages
2019. Vol. 583, p. 195-256
Keywords [en]
Cartan matrix, Difference equation, Lie algebra, Lie superalgebra
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-175920DOI: 10.1016/j.laa.2019.08.026ISI: 000491616900011OAI: oai:DiVA.org:su-175920DiVA, id: diva2:1371531
Available from: 2019-11-20 Created: 2019-11-20 Last updated: 2019-11-20Bibliographically approved

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