## Inversion of double-covering map $$SPIN(N) \to SO (N, {\mathbb R})$$ for $$N \leq 6$$.(English)Zbl 1376.22012

For any integer $$n$$, the spin group $$\text{Spin}(n)$$ is the double cover of the special orthogonal group $$\text{SO}(n,\mathbb{R})$$ such that there is a short exact sequence of Lie groups $1 \rightarrow \mathbb{Z}/2 \mathbb{Z} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n,\mathbb{R}) \rightarrow 1.$ The cases $$n=3$$ and $$4$$, where $$\text{Spin}(n)=\text{SU}(2)$$ and $$\text{SU}(2) \times \text{SU}(2)$$ respectively, are classical and date back to Hamilton, who described both the covering maps and the co-images using the quaternion algebra $$\mathbb{H}$$.
In the paper under discussion, the authors consider the cases $$n=5$$ and 6, where $$\text{Spin}(n)=\text{Sp}(4)$$ and $$\text{SU}(4)$$ respectively. They provide explicit formulas for the covering maps, and algorithms for finding the co-images.

### MSC:

 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 20G20 Linear algebraic groups over the reals, the complexes, the quaternions

### Keywords:

Lie algebras; linear algebraic groups
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