Linear representations of the Lie algebra of the diffeomorphism group in \(\mathbb{R}^d\)

Abstract

A family of representations of the Lie algebra of the diffeomorphism group in \(\mathbb{R}^d\) is studied. A method for constructing representations of this family is proposed. Equations for matrices describing the action of the Lie algebra on the representation space are obtained. It is shown that the developed formalism is suitable for describing representations under which fields of linear homogeneous geometric objects are transformed. The formalism is shown to allow describing representations for which the representation space vectors cannot be expressed in terms of fields of linear homogeneous geometric objects. An example of such a representation is studied.