Maximal Nilpotent Subalgebras II: A Correspondence Theorem Within Solvable Associative Algebras. With 242 ExercisesAnchor Academic Publishing, 1 de nov. 2017 - 196 pàgines 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. |
Continguts
Introduction | 7 |
Standard examples symbols and notations | 13 |
Radical algebras and the theorem of Xiankun Du | 21 |
Solvability | 37 |
Carter subgroups of the group of units of solvable associative | 59 |
6 | 85 |
7 | 125 |
8 | 137 |
Fischer subgroups nilpotent projectors and injectors | 163 |
Outlook on series III | 173 |
Bibliography | 179 |
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Maximal nilpotent subalgebras II: A correspondence theorem within solvable ... Sven Bodo Wirsing Previsualització limitada - 2017 |
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abelian analyze ascending central chain associative algebras associative subalgebra associative unitary solvable attraction section bijection Borel subalgebras Cartan subalgebras Carter subgroups CE(T char(K class of nilpotency commutative complement of rad(A corollary deduce derive Determine division algebra double-centralizing exactly Excercise field of characteristic field possessing finite field finite-dimensional associative unitary Fischer subgroup Fitting subgroup following statements fully separable gebras group algebra group of units Hence ideal idempotents identity isomorphic classes least three elements Lie nilpotent subalgebras Lie nilradical main theorem maximal Lie nilpotent maximal nilpotent Lie maximal nilpotent subgroups nilpotent Lie subalgebras normal subgroup p-group possessing a separable possessing at least Proof proposition prove rad(B rad(M radical algebra radical complement resp separable factor algebra sequence solvable associative algebras solvable classes solvable group solvable K-algebra possessing statements are valid subalge subgroup of E(A subgroup of G theorem of Xiankun unital subalgebras unitary solvable K-algebra VSEP(M