SL(2,C) × SL(2,C) - Knowlpedia

The product Lie group $\mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C)$ has two copies of $\operatorname{SL}(2,\mathbb C)$ as factors, with componentwise multiplication.

Dimensions and structure

This group has real dimension $12$ , complex dimension $6$ . The two factors commute, and the manifold dimension is the sum of their dimensions. It is simply connected.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

Direct verification

Products carry componentwise multiplication and inversion and the product manifold structure. Dimensions add, and the two projections and the inclusions of each factor are smooth homomorphisms.