Hypoalgebra

In algebra, a hypoalgebra is a generalization of a subalgebra of a Lie algebra introduced by (Patera Sharp). The relation between an algebra and a hypoalgebra is called a subjoining (Patera Sharp), 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 
• 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, http://stacks.iop.org/0305-4470/13/397 
• 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