Degenerate spin groups as semi-direct products
Let \(\langle\; , \; \rangle\) be a symmetric bilinear form on \(\mathbb{R}^n = \mathbb{R}^{p+q+r}\) with rank \(p+q\), co-rank \(r\) and a linear basis \(\{e_1,\dots,e_p, e_{p+1},\dots,e_{p+q},f_1,\dots,f_r\}\) of \(\mathbb{R}^{p+q+r}\) such that \(\langle e_i , e_j\rangle = \delta_{ij}\) (\(1\leq i,j \leq p\)), \(\langle e_{p+i,p+j}\rangle = -\delta_{ij}\) (\(1\leq i,j\leq q\)), \(\langle f_i, f_j\rangle = 0\) (\(1\leq i,j \leq r\)). Let us consider the Clifford algebras \(\mathcal{C}\ell_{p,q,r} = \mathcal{C}\ell(\mathbb{R}^{p+q+r}, \langle\; , \; \rangle)\) and \(\mathcal{C}\ell_{p,q} = \mathcal{C}\ell(\mathbb{R}^{p+q},\langle\; , \; \rangle)\), and the spin group \(\text{Spin}(p,q) = \{v_1\cdots v_m\mid v_i = \sum_{j=1}^{p+q} a_{ij} e_j, \, \langle v_i,v_i\rangle = \pm 1,\;1 \leq i \leq m\}\). Let us also consider \(S_{p,q,r} = \{ s \gamma_1 \dots \gamma_{p+q}(1+G)\mid s \in \text{Spin}(p,q),\;\gamma_i = 1 + e_i \sum_{l=1}^r c_{il} f_l , \; G \in \Lambda(f)\}\), with \(\Lambda(f) = \text{span}\{f_{k_1}\dots f_{k_j}\mid 1\leq k_1 < k_2 < \dots k_j\leq r\}\), and \(\Delta = \{ 1 + G\mid G \in \Lambda(f)\}\). The authors show that \(S_{p,q,r}\) is a group under Clifford multiplication and that \(\Delta\) is a normal subgroup of \(S_{p,q,r}\). Then, defining the degenerate spin group \(\text{Spin}(p,q,r) = S_{p,q,r}/\Delta\), they also show that \(\text{Spin}(p,q,r)\) is given by the semi-direct product of \(\text{Spin}(p,q)\) with the additive group of \((p+q)\times r\) matrices \(\text{Mat}(p+q,r)\).
scientific article; zbMATH DE number 5818546