On Linear and Non-Linear Representations of the Generalized Poincaré Groups in the Class of Lie Vector Fields

We study representations of the generalized Poincaré group and its extensions in the class of Lie vector fields acting in a space of n + m independent and one dependent variables. We prove that an arbitrary representation of the group P(n, m) with max {n, m} ≥ 3 is equivalent to the standard one, while the conformal group C (n, m) has non-trivial nonlinear representations. Besides that, we investigate in detail representations of the Poincaré group P (2, 2), extended Poincaré groups ̃P (1, 2), ̃P (2, 2), and conformal groups C (1, 2), C (2, 2) and obtain their linear and nonlinear representations.