Higgs potentials as “inheritance” from higher space-time dimensions I. Dimensional reduction and scalar fields

The dimensional reduction of multidimensional “free” gauge theories with additional space-time symmetry is considered. In the framework of this approach, a general method for finding explicitly the scalar fields in the reduced theory is developed. These fields are described by a certain intertwining operator, for the construction of which a root lattice technique for complex classical Lie algebras which generalizes the Dynkin diagram technique is developed.