Tsukuba Journal of Mathematics

Contragredient Lie algebras and Lie algebras associated with a standard pentad

Nagatoshi Sasano

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Abstract

From a given standard pentad, we can construct a finite or infinite-dimensional graded Lie algebra. In this paper, we will define standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. Using pentads of Cartan type, we can obtain arbitrary contragredient Lie algebras with an invertible symmetrizable Cartan matrix. Moreover, we can use pentads of Cartan type in order to find the structure of a Lie algebra. When a given standard pentad consists of a finite-dimensional reductive Lie algebra, its finite-dimensional completely reducible representation and a symmetric bilinear form, we can find the structure of its corresponding Lie algebra under some assumptions.

Article information

Source
Tsukuba J. Math., Volume 42, Number 1 (2018), 1-51.

Dates
Received: 27 December 2016
Revised: 4 January 2018
First available in Project Euclid: 7 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1541559647

Digital Object Identifier
doi:10.21099/tkbjm/1541559647

Mathematical Reviews number (MathSciNet)
MR3873530

Zentralblatt MATH identifier
07055223

Subjects
Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Secondary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 17B70: Graded Lie (super)algebras

Keywords
contragredient Lie algebras Kac-Moody Lie algebras Cartan matrices standard pentads

Citation

Sasano, Nagatoshi. Contragredient Lie algebras and Lie algebras associated with a standard pentad. Tsukuba J. Math. 42 (2018), no. 1, 1--51. doi:10.21099/tkbjm/1541559647. https://projecteuclid.org/euclid.tkbjm/1541559647


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