Maximal nilpotent subalgebras I: Nilradicals and Cartan subalgebras in associative algebras. With 428 exercises

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Chapter 5 Cartan subalgebras in Lie algebras associated to associative algebras:
In the previous chapter we have deeply investigated the nilradical of Lie algebras associated to associative algebras and described its structure as an associative subalgebra. The nilradical is one special member of the set all maximal Lie nilpotent substructures. Other important ones are the Cartan subalgebras which are in focus within this chapter. They are defined as nilpotent and self-normalizing subalgebras of a Lie algebra. In our context Cartan subalgebras do exist. We will show that they are associative subalgebras and analyze the associative structure more deeply. Some of the results of this chapter are based on an article by Salvatore Siciliano [59], others are enhancements of his theory which we will widen to other classes of algebras (like basic algebras or algebras with separable factor algebra by its nilradical). Finally, we will analyze our standard examples, the group algebras, the Solomon-Tits algebras, the Solomon algebras in characteristic zero and the algebras of upper and lower triangular matrices over an arbitrary field, concerning Cartan subalgebras.
5.1 Cartan subalgebras are associative solvable sub-algebras:
In this section we focus on the associative structure of Cartan subalgebras in Lie algebras associated to associative algebras. We will prove that these Lie subalgebras are indeed associative subalgebras and, in addition, that they are solvable. For a perfect field we prove further that every fully separable element is central. Therefor exactly one radical complement exists which is central. The main result for nilradicals in chapter 4 play an important role for proving this result.
5.1.1 Associative structure of Cartan subalgebras:
Salvatore Siciliano proved in [59], theorem 1 in particulary that for a finitedimensional associative unitary K-algebra every Cartan subalgebra of its associated Lie algebra is indeed an associative

Sven Bodo Wirsing was born in Neumunster in 1975. After graduating from high school at KKS in Itzehoe (with a focus on mathematics and physics), he studied mathematics with a minor in business administration (especially logistics) at CAU university in Kiel. He did his doctorate in 2005 on group and algebra theory. During his years of study in Kiel he gained experience in the analysis of interdisciplinary processes, which are reflected in different disciplines of algebra, such as group theory, representation theory, theory of Lie and associative algebras. From this experience, he also studied and analyzed the subject matter of the present work.
Since the end of his doctorate, Dr. Wirsing has been working as a senior IT consultant for logistics processes at a renowned IT consulting firm, where he is responsible for logistics optimization and maintenance. Since 2012, he has published several books on algebras.