Lie groups: Bianchi classification

This lecture is part of an online graduate course on Lie groups.
We give a sketch of the Bianchi classification of the Lie algebras and groups of dimension at most 3. We mention that this is related to the Thurston geometries of 3-manifolds.
For the other lectures in the course see • Lie groups

[@rosieshen8431]

7:04 should be "x commutes with M"

[@migarsormrapophis2755]

yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

[@atomiccompiler9495]

wise words

[@vladkovalchuk8299]

How much does this "correspondence"(or at least an overlap) between Lie groups and geometries on manifolds (compact) of a fixed dimension n, extend to general n > 3? For the Lie groups, I figure it has to do with the structure of left-invariant bilinear forms, which is an interesting space.

[@leostein128]

Yes, we can construct the Killing 'form' on G as that left-invariant symmetric bilinear. This can turn G into a (homogeneous) metric space.

[@richarde.borcherds7998]

In dimensions 4 and 5 the possible geometries have been classified by Filipkiewicz and Geng arxiv.org/pdf/1605.07545.pdf and there seems to be a fair amount of overlap with the Lie groups in these dimensions. However 4-manifolds do not always have a geometric decomposition (see Hillman arxiv.org/abs/math/0212142)