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Siegel-Satake cross and associated Clifford algebras. (English) Zbl 1506.11059

Authors’ abstract: In the present self-contained paper, we want, first, to construct a fundamental diagram, called (S.C), in homage to Carl Siegel and I. Satake that connects the following groups: \(\mathrm{SU}(m, m)\), \(\mathrm{SO}^\ast(2m)\), \(\mathrm{Sp}(2m, \mathbb{R})\), \(\mathrm{Sp}(4m, \mathbb{R})\), \(\mathrm{SO}^\ast(4m)\). Then, we define and study three Clifford algebras related to that diagram. First, we consider the morphism from \(\mathrm{Sp}(2m, \mathbb{R})\) into \(\mathrm{SU}(m, m)\), shown in the construction of the diagram (S.C.). Then, we define a Clifford algebra \(Cl^{m, m}\), naturally associated with the group \(\mathrm{U}(m, m)\). Let \((E, b)\) be an \(m\)-dimensional skew-hermitian space over \(\mathbb{H}\). For any \(x, y\in E\), write \(b(x, y) = h(x,y) + ja(x, y)\). It is well known that \(h\) is a skew-hermitian complex form on \(\mathbb{E}_{2m}\), the complex \(2m\)-dimensional vector space underlying \(E\), and \(a\) is a symmetric bilinear complex form on \(\mathbb{E}_{2m}\). We proved previously in [in: Clifford algebras and their applications in mathematical physics, Proc. Workshop, Canterbury/U.K. 1985, NATO ASI Ser., Ser. C 183, 79–91 (1986; Zbl 0609.53006)] that the special unitary group \(\mathrm{SU}(E, b)\) of a skew-hermitian \(\mathbb{H}\)-right vector space \((E, b)\), \(m\)-dimensional over \(\mathbb{H}\), can be identified with the group \(\mathrm{SO}^\ast(2m)\) defined by E. Cartan. We define a real Clifford algebra, namely \(Cl_{\mathbb{R}}^\ast(2m)\), whose complexified algebra is \(C_{2m}^+(\mathbb{E}_{2m}, a)\), the even complex Clifford algebra associated with \(a\). Both algebras are associated with the geometry of the skew-hermitian \(\mathbb{H}\)-space \((E, b)\). Let \(V = (\mathbb{R}^{2m}, \mathrm{Sp}(2m,\mathbb{R}))\) be the standard model of a real symplectic space. We present some connections between the geometry of \(V\) and the algebras \(Cl^{m,m}\), \(C_{2m}^+(\mathbb{E}_{2m}, a)\), \(Cl_{\mathbb{R}}^*(2m)\). The last section wants to give a sketch of the prospects offered by these algebras for the study of the real conformal symplectic geometry. An appendix gives some indispensable recalls and some complements.

MSC:

11E88 Quadratic spaces; Clifford algebras
15A63 Quadratic and bilinear forms, inner products
15A66 Clifford algebras, spinors
53D05 Symplectic manifolds (general theory)
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Citations:

Zbl 0609.53006
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References:

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