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In algebra, a hypoalgebra is a generalization of a subalgebra of a Lie algebra introduced by Patera, Sharp & Slansky (1980). The relation between an algebra and a hypoalgebra is called a subjoining (Patera & Sharp 1980), which generalizes the notion of an inclusion of subalgebras. There is also a notion of restriction of a representation of a Lie algebra to a subjoined hypoalgebra, with branching rules similar to those for restriction to subalgebras except that some of the multiplicities in the branching rule may be negative. W. G. McKay, J. Patera, and D. W. Rand (1990) calculated many of these branching rules for hypoalgebras.

References

McKay, W. G.; Patera, J.; Rand, D. W. (1990), Tables of representations of simple Lie algebras. Vol. I, Montreal, QC: Université de Montréal Centre de Recherches Mathématiques, ISBN 978-2-921120-06-7, MR 1109497
Patera, J.; Sharp, R. T. (1980), "Generating functions for plethysms of finite and continuous groups", Journal of Physics A: Mathematical and General, 13 (2): 397–416, doi:10.1088/0305-4470/13/2/008, ISSN 0305-4470, MR 0558637
Patera, J.; Sharp, R. T.; Slansky, R. (1980), "On a new relation between semisimple Lie algebras", Journal of Mathematical Physics, 21 (9): 2335–2341, doi:10.1063/1.524689, ISSN 0022-2488, MR 0585584

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