The equivariant embeddings of $n$-dimensional space into the Hilbert space

The present paper is devoted to the study of equivariant embeddings of $n$-dimensional space into the Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle of this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of additive group of real numbers and construct an embedding of $n$-dimensional space supplied with a metric $d(x,y)=|x-y|^\alpha$ into the Hilbert space.