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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.

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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##### References:
 [1] Douglas, A.; Premat, A., A class of nonunitary representations of the Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$$\mathfrak{e}(2)$$\end{document}, Commun. Algebra, 35, 5, 1433-1448, (2007) · Zbl 1232.17012 [2] Douglas, A.; de Guise, H.; Repka, J., The Poincaré algebra in rank 3 simple Lie algebras, J. Math. Phys., 54, 023508, (2013) [3] Douglas, A.; Kahrobaei, D.; Repka, J., Classification of embeddings of abelian extensions of D_{n} into E_{n + 1}, J. Pure Appl. Algebra, 217, 1942-1954, (2013) · Zbl 1329.17009 [4] Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S., 30, 72, 349-462, (1952); Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S., 30, 72, 349-462, (1952); · Zbl 0077.03404 [5] Hall, B. C., Lie Groups, Lie Algebras, and Representations, (2003), Springer-Verlag: Springer-Verlag, New York · Zbl 1026.22001 [6] de Graaf, W. A., Constructing semisimple subalgebras of semisimple Lie algebras, J. Algebra, 325, 416-430, (2011) · Zbl 1255.17007 [7] Littelmann, P., Cones, crystals, and patterns, Transform. Groups, 55, 3, 564-566, (1998) · Zbl 0908.17010 [8] Minchenko, A. N., The semisimple subalgebras of exceptional Lie algebras, Trans. Moscow Math. Soc., 67, 225-259, (2006) · Zbl 1152.17003
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