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INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 83 / Issue 3 / June 2011
- Published online by Cambridge University Press:
- 01 April 2011, pp. 439-449
- Print publication:
- June 2011
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- Article
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The Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra
and show that the irreducible representations V (m,0,0) and V (0,0,m) of are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.
