CR-quadrics with a symmetry property

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 Silov 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 *-algebra A (of finite dimension). For certain quadrics this also works if A is not necessarily commutative.