ABSTRACT
The goal of this section is to introduce the topological Lie algebras and to illustrate this notion by a number of examples. To begin with, we explain what a Lie algebra is.
DEFINITION 1.1 A Lie algebra over K ∈ {R,C} is a vector space g over K equipped with a bilinear mapping
g× g→ g, (x, y) 7→ [x, y],
such that for all x, y, z ∈ g we have [x, y] = −[y, x] (anti-symmetry)
and [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 (Jacobi identity).
