Understanding the Relationship Between Lie Algebras and Lie Groups

[Lapidus]

How come that a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally? I wonder because groups are, again by definition, asscioative.

thanks

 

[Dickfore]

Let

$$a \otimes b \equiv [a, b] = a b - b a$$

Then:

$$a \otimes (b \otimes c) = [a,[b,c]] = [a, b c] - [a, c b] = a b c - b c a - a c b + c b a$$

But:

$$(a \otimes b) \otimes c = [[a, b], c] = [a b, c] - [b a, c] = a b c - c a b - b a c + c b a$$

These two expressions are not the same. Thus, the operation $\otimes$ is not associative. But, the Lie algebra is precisely defined through such an operation.

 

[Landau]

a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally?

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.

For more information, see wikipedia.