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λ-Mappings Between Representation Rings of Lie Algebras

Published online by Cambridge University Press:  20 November 2018

R. V. Moody  and
A. Pianzola
R. V. Moody
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
A. Pianzola
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.

The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagram

Inside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 5 , 01 October 1983 , pp. 898 - 960
Copyright
Copyright © Canadian Mathematical Society 1983

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