Boson representations of symplectic algebras (Q1088791)

The boson operators \(a_{\alpha i}\), \(a^+_{\alpha i}\) \((\alpha =1,2\); \(i=1,...,k)\) in their bilinear products generate, as is well known, the group Sp(4k, \({\mathbb{R}}):\{a^+_{\alpha i}a^+_{\beta j}\); \(a_{\alpha i}a_{\beta j}\), \(a^+_{\alpha i}a_{\beta j}\); \(\alpha,\beta =1,2\); \(i,j=1,...,k\}\). In this paper the subgroup chains \[ Sp(4k, {\mathbb{R}})\quad \to \quad U(2k)\quad \to \quad SU(2)\times SU(u)\quad \to \quad SU(k) \] \[ Sp(4k, {\mathbb{R}})\quad \to \quad U(k,k)\quad \to \quad U(k)\times U(k)\quad \to \quad SU(k) \] are considered, and the examples \(k=1\), \(k=2\), and \(k=3\) are analysed in detail.