paper

Inverses of Cartan matrices of Lie algebras and Lie superalgebras

arXiv:1905.12408 · doi:10.1016/j.laa.2019.08.026

Abstract

The inverses of indecomposable Cartan matrices are computed for finite-dimension\-al 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. We discovered three new phenomena concerning 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 0s only occur on the main diagonals of the inverses; (c) the determinants of pairwise non-equivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic). Here, we explain all three. The inverse has a 0 at the th place of its main diagonal precisely if the deletion of the th node of the Dynkin diagram leaves an affine algebra, and it has no 0 anywhere else; so (a) singles out the almost affine Lie (super)algebras for which the deletion of any node leaves a finite-dimensional one. We interpret most of the results of Wei Yangjiang and Zou Yi Ming in Linear Alg. Appl., 521 (2017) 283--298 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.

arXiv comment52 pages. We reproduce definition of root spaces from arXiv:0710.5149 and arXiv:0906.1860. Version 3 answers several questions of the published version

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