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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
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