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CR-quadrics with a symmetry property. (English) Zbl 1202.32032
Summary: For non-degenerate CR-quadrics \({Q \subset \mathbb{C}^{n}}\) it is well known that the real Lie algebra \({\mathfrak{g} = \mathfrak{hol}(Q)}\) of all infinitesimal CR-automorphisms has a canonical grading \({\mathfrak{g} = \mathfrak{g}^{-2} \oplus\mathfrak{g}^{-1} \oplus\mathfrak{g}^{0} \oplus\mathfrak{g}^{1} \oplus\mathfrak{g}^{2}}\). While the first three spaces in this grading, responsible for the affine automorphisms of \(Q\), are always easy to describe this is not the case for the last two. In general, it is even difficult to determine the dimensions of \({\mathfrak{g}^{1}}\) and \({\mathfrak{g}^{2}}\). Here we consider a class of quadrics with a certain symmetry property for which \(\mathfrak{g}^{1}\), \(\mathfrak{g}^{2}\) can be determined explicitly. The task then is to verify that there exist enough interesting examples. By generalizing the Šilov boundaries of irreducible bounded symmetric domains of non-tube type we get a collection of basic examples. Further examples are obtained by ‘tensoring’ any quadric having the symmetry property with an arbitrary commutative (associative) unital \(^\star\)-algebra \(A\) (of finite dimension). For certain quadrics this also works if \(A\) is not necessarily commutative.

MSC:
32V20 Analysis on CR manifolds
17B66 Lie algebras of vector fields and related (super) algebras
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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