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Maximal nilpotent subalgebras II: A correspondence theorem within solvable associative algebras. With 242 exercises

Textbook 2017 193 Pages

Mathematics - Algebra

Summary

Within series II we extend the theory of maximal nilpotent substructures to solvable associative algebras, especially for their group of units and their associated Lie algebra.
We construct all maximal nilpotent Lie subalgebras and characterize them by simple and double centralizer properties. They possess distinctive attractor and repeller characteristics. Their number of isomorphic classes is finite and can be bounded by Bell numbers. Cartan subalgebras and the Lie nilradical are extremal among all maximal nilpotent Lie subalgebras.
The maximal nilpotent Lie subalgebras are connected to the maximal nilpotent subgroups. This correspondence is bijective via forming the group of units and creating the linear span. Cartan subalgebras and Carter subgroups as well as the Lie nilradical and the Fitting subgroup are linked by this correspondence. All partners possess the same class of nilpotency based on a theorem of Xiankun Du.
By using this correspondence we transfer all results to maximal nilpotent subgroups of the group of units. Carter subgroups and the Fitting subgroup turn out to be extremal among all maximal nilpotent subgroups.
All four extremal substructures are proven to be Fischer subgroups, Fischer subalgebras, nilpotent injectors and projectors.
Numerous examples (like group algebras and Solomon (Tits-) algebras) illustrate the results to the reader. Within the numerous exercises these results can be applied by the reader to get a deeper insight in this theory.

Details

Pages
193
Type of Edition
Erstausgabe
Year
2017
ISBN (eBook)
9783960676966
ISBN (Book)
9783960671961
File size
12.8 MB
Language
English
Catalog Number
v380467
Grade
Tags
Cartan subalgebra Carter subgroup Lie algebra Associative algebra Maximal nilpotent Group algebra Exercise Lie nilpotency Nilradical Fitting subgroup Soluble

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Title: Maximal nilpotent subalgebras II: A correspondence theorem within solvable associative algebras. With 242 exercises