Lattice automorphisms of semi-simple Lie algebras

Author:

Barnes D. W.

Abstract

Let L be a Lie algebra over the field F. A lattice automorphism of ℒ is an automorphism φ: ℒ(L) → ℒ(L) of the lattice ℒ(L) of all subalgebras of L. We seek to describe the lattice automorphisms in terms of maps σ: LL of the underlying algebra. A semi-automorphism σ of L is an automorphism of the algebraic system consisting of the pair (F, L), that is, a pair of maps σ: FF, σ: LL preserving the operations. Thus σ: FF is an automorphism of F and (x + y)σ = xσ + yσ, (xy)σ = xσyσ (λx)σ = λσxσ for all x, y ε L and λ ε F. Clearly, any semi-automorphism of L induces a lattice automorphism. To study a given lattice automorphism φ, we select a semi-automorphism a such that φσ-1 fixes certain subalgebras, and so we reduce the problem to the investigation of lattice automorphisms leaving these subalgebras fixed.

Publisher

Cambridge University Press (CUP)

Reference3 articles.

1. On normaliser preserving lattice isomorphisms between nilpotent groups

2. Lattic isomorphisms of lie algebras

3. [3] Jacobson N. , Lie Algebras (Interscience Tracts No. 10, 1962).

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3. Lattice isomorphisms of Lie algebras and algebraic groups;Journal of Algebra;1987-04

4. Lattice automorphisms of Lie algebras;Archiv der Mathematik;1986-01

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