We classify embeddings of the Poincaré algebra into the rank 3 simple Lie algebras. Up to inner automorphism, we show that there are exactly two embeddings of into , which are, however, related by an outer automorphism of . Next, we show that there is a unique embedding of into , up to inner automorphism, but no embeddings of into . All embeddings are explicitly described. As an application, we show that each irreducible highest weight module of (not necessarily finite-dimensional) remains indecomposable when restricted to , with respect to any embedding of into .
Received 27 August 2012Accepted 17 January 2013Published online 15 February 2013
The work of A.D. is partially supported by a research grant from the Professional Staff Congress/City University of New York (PSC/CUNY). The work of H.deG. and J.R. is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
The authors would like to thank Andrey Minchenko and Willem de Graaf for valuable correspondence regarding semisimple subalgebras in simple Lie algebras. Finally, the authors would like to thank the anonymous reviewer for beneficial comments.
Article outline: I. INTRODUCTION II. THE POINCARÉ GROUP AND ALGEBRA IN PHYSICS III. THE SEMISIMPLE LIE ALGEBRA AND ITS REPRESENTATIONS IV. THE RANK 3 SIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS V. ADDITIONAL DEFINITIONS AND NOTATION VI. CLASSIFICATION OF EMBEDDINGS OF INTO THE RANK 3 SIMPLE LIE ALGEBRAS VII. CLASSIFICATION OF EMBEDDINGS OF INTO VIII. CLASSIFICATION OF EMBEDDINGS OF INTO IX. NO EMBEDDINGS OF INTO X. CONCLUSIONS
5.W. Fulton and J. Harris, Representation Theory (Springer-Verlag, New York, 1991).
6.B. C. Hall, Lie Groups, Lie Algebras, and Representation Theory (Springer-Verlag, New York, 2003).
7.J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer-Verlag, New York, 1972).
8.J. Kowalski-Glikman, “Introduction to doubly special relativity,” Planck Scale Effects in Astrophysics and Cosmology, Lecture Notes in Physics Vol. 669, edited by J. Kowalski-Glikman and G. Amelino-Camelia (Springer, Berlin/Heidelberg, 2005).