A Lie algebra is defined as a non-associative algebraic structure that describes a Lie group locally. The operation defined by the Lie bracket, denoted as [a, b], is fundamentally different from the associative group multiplication. Specifically, for a Lie group G, the corresponding Lie algebra represents the tangent space at the identity element of G, which consists of left invariant vector fields. This distinction clarifies the relationship between Lie algebras and Lie groups.
Mathematicians, physicists, and students of advanced algebra who are interested in the interplay between algebraic structures and geometric representations in the context of Lie groups and Lie algebras.
The way it "describes a Lie group locally" is somewhat involved. In any case, the product operation of the Lie group and the bracket operation of the corresponding Lie algebra are by no means the same. Specifically, for a Lie group G, the corresponding Lie algebra is the tangent space of G at the identity, which 'is' the space of left invariant vector fields. The Lie bracket is the usual commutator of two vector fields. This is an entirely different operation than the group multiplication of G.Lapidus said:a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally?