Classification of simple Lie algebras on a lattice. Issue 3 (14th September 2012)

• Record Type: Journal Article
• Authors: Iohara, Kenji; Mathieu, Olivier
• Is Part Of: Proceedings of the London Mathematical Society. Volume 106:Issue 3(2012:Mar.)
• Page Start: 508
• Page End: 564
• Publication Date: 2012-09-14
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pds042
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract : Let Λ=ℤ n for some n ⩾1. The aim of the paper is to classify all simple Λ‐graded Lie algebras ℒ=⊕λ∈Λ ℒλ such that dim ℒλ =1 for all λ. The classification involves two affine Lie algebras, namely A 1 (1) and A 2 (2), and a family ( W π ), parametrized by a dense open set of the space of all embeddings π : Λ→ℂ 2 . The family ( W l ) of generalized Witt algebras, indexed by all embeddings l :Λ→ℂ, appears as a subfamily. In general, the algebras W π are described as Lie algebras of symbols of twisted pseudo‐differential operators.