In low dimensions, there are isomorphisms among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, owing to low dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have
- Spin(1) = O(1)
- Spin(2) = U(1) = SO(2) which acts on by double phase rotation
- Spin(3) = Sp(1) = SU(2), corresponding to
- Spin(4) = Sp(1) × Sp(1), corresponding to
- Spin(5) = Sp(2), corresponding to
- Spin(6) = SU(4), corresponding to
There are certain vestiges of these isomorphisms left over for n = 7, 8 (see Spin(8) for more details). For higher n , these isomorphisms disappear entirely.
Read more about this topic: Spin Group
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