On Automorphisms of Direct Products of CR Manifolds

A criterion is given for the Lie algebra of infinitesimal holomorphic automorphisms of the direct product of two germs of real analytic generic CR manifolds to be equal to the direct sum of the algebras of the factors. In particular, this equality holds if the algebra of the direct product is finite-dimensional (for example, in the case of the product of holomorphically nondegenerate germs of finite Bloom–Graham type). The notion of the sum of Bloom–Graham types is introduced. It is shown that the type of a direct product is equal to the sum of the types of factors.