Lie algebras of skew-adjoint endomorphisms of a bilinear form #
When a module carries a bilinear form, the Lie algebra of endomorphisms of the module contains a distinguished Lie subalgebra: the skew-adjoint endomorphisms. Such subalgebras are important because they provide a simple, explicit construction of the so-called classical Lie algebras.
This file defines the Lie subalgebra of skew-adjoint endomorphims cut out by a bilinear form on a module and proves some basic related results. It also provides the corresponding definitions and results for the Lie algebra of square matrices.
Main definitions #
lie algebra, skew-adjoint, bilinear form
M, equipped with a bilinear form, the skew-adjoint endomorphisms form a
Lie subalgebra of the Lie algebra of endomorphisms.
An equivalence of modules with bilinear forms gives equivalence of Lie algebras of skew-adjoint endomorphisms.
The Lie subalgebra of skew-adjoint square matrices corresponding to a square matrix
An invertible matrix
P gives a Lie algebra equivalence between those endomorphisms that are
skew-adjoint with respect to a square matrix
J and those with respect to
An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an equivalence of Lie algebras of skew-adjoint matrices.