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The Euclidean algebra in rank 2 classical Lie algebras. (English) Zbl 1328.17010
The Euclidean algebra \(e(2)\) is the Lie algebra of orientation-preserving isometries of two-dimensional Euclidean space. The authors classify, up to inner automorphisms, the embeddings of \(e(2)\) into the rank two semisimple Lie algebras; \(sl(2,\mathbb C)\oplus sl(2,\mathbb C), sl(3, \mathbb C)\) and \(sp(4,\mathbb C)\). In each of the first two cases, there are exactly two such embeddings. In the final case there are two single embeddings and also a family of embeddings. Then the authors study how representations of these three algebras restrict to \(e(2)\). In the first two algebras, irreducible representations remain indecomposable when restricted to \(e(2)\). In the third algebra, such restrictions may or may not decompose when restricted.

17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
Full Text: DOI
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