Some nonunitary, indecomposable representations of the Euclidean algebra \mathfrak {e}(3)

The Euclidean group E(3) is the noncompact, semidirect product group . It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra is the complexification of the Lie algebra of E(3). We construct three distinct families of finite-dimensional, nonunitary representations of and show that each representation is indecomposable. The representations of the first family are explicitly realized as subspaces of the polynomial ring with the action of given by differential operators. The other families are constructed via duals and tensor products of the representations within the first family. We describe subrepresentations, quotients and duals of these indecomposable representations.

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