On Lie induction and the exceptional series

Grabowski, Jan (2005) On Lie induction and the exceptional series. Journal of Algebra and Its Applications, 4 (6). pp. 707-737. ISSN 0219-4988

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Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

Item Type:
Journal Article
Journal or Publication Title:
Journal of Algebra and Its Applications
Additional Information:
Preprint of an article submitted for consideration in Journal of Algebra and Its Applications © 2005 copyright World Scientific Publishing Company http://www.worldscinet.com/jaa/
Uncontrolled Keywords:
?? lie bialgebra; simple lie algebra; exceptional series; inductionalgebra and number theoryapplied mathematics ??
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Deposited On:
09 Mar 2012 02:46
Last Modified:
18 Dec 2023 01:14