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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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