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

1. Auflage 2018
ISBN: 978-3-96067-696-6
Verlag: Diplomica Verlag
Format: PDF
Kopierschutz: 0 - No protection

E-Book, Englisch, 193 Seiten

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.

Autoren/Hrsg.

Wirsing, Sven Bodo

Leseproben

Text Sample:

Chapter 2 Radical algebras and the theorem of Xiankun Du:

Within this chapter we focus on radical algebras. For these algebras a deep nilpotent connection between the associated Lie algebra and the circle or adjoint group proven by Xiankun Du is presented. This chapter is designed based on the manuscript of Hartmut Laue in [40]. Within this manuscript results of the diploma thesis of Karsten Scholz are used (see [58]). Proofs are available in chapter 4 in [40] and not stated here.
Based on Du’s theorem the nilpotency classes of the associated Lie algebra and the adjoint circle group resp. the group of units are identical. Thus, the determination of the nilpotency class of the circle group can be handled by calculations purely within the associated Lie algebra and vice versa. In some applications it is much more easier to do the calculation within the Lie algebra as within the circle group. For this transfer principle some applications are presented within the exercises. In addition, one theorem about the p-power structure of the circle group in characteristics p is proven by using this transfer principle to the Lie algebra.
Radical algebras and their analysis concerning nilpotency of the associated Lie algebra and the circle group will play an important role later on in this work: the correspondence theorem between maximal nilpotent Lie subalgebras and subgroups. […].
2.4 Open-ended questions and exercises:

Open-ended questions 1 (i) Does a pendant of the theorem of Du exist for radical algebras concerning solvability?
(ii) Is it true that the factor groups along the descending central chain of the quasi regular group of a radical algebra are elementary-p-abelian if the base field is of characteristic p > 0 (except for the derived subgroup itself)?
(iii) Determine the order of the elementary-p-abelian factor groups along the ascending central chain of the quasi-regular group of a radical algebra if the base field is of characteristic p > 0 (except for the center itself)!
(iv) What is the answer for the previous questions for nilradicals of group algebras? This question is partly answered by the dissertation of M. Theede (see [69]).
(v) Determine those nilpotent algebras for which the sets of members of the series of upper and lower Lie central chains are identical.
(vi) Determine those nilpotent algebras for which the lower Lie central chain and the associative powers are identical.
(vii) Determine those nilpotent algebras for which the sets of members of the series of upper Lie central chain and associative powers are identical.
Excercise 2 For the algebras of strict upper and lower triangular matrices over a field analyze the connections between the upper and lower Lie central chain as well as the associative powers (Tip: see [38])!
Excercise 3 Prove proposition 1 in details!
Excercise 4 Prove proposition 2! […].
3.3 The theorem of Sophus Lie and Borel subalgebras:

Examples of solvable associative algebras will be presented is a separate section within this chapter. In every associative algebra there are solvable subalgebras (e.g. the semidirect sum of a torus and of the nilradical). One special solvable associative subalgebra is the so-called solvable radical: the sum of all solvable ideals is solvable. Thus, within finite-dimensional associative algebras there is always a biggest solvable ideal { the solvable radical. Within the theory of Lie algebra so-called Borel subalgebras play an important role: they are maximal solvable Lie subalgebras. They contain the solvable radical and they are closely connected to the Cartan subalgebras.1 In this section we will prove that Borel subalgebras of an associated Lie algebra based on an associative finite-dimensional algebra are associative subalgebras, if the base field is of characteristic zero.
Theorem 12 (Sophus Lie) Let K be an algebraically closed field of characteristic zero and L a finite-dimensional solvable K-Lie algebra. If V is an irreducible L-module, then V is one-dimensional.
In particular, L is isomorphic to a Lie subalgebra of the Lie algebra of upper triangular matrices over K.
The theorem of Sophus Lie implies that under its assumption the Lie algebra of upper triangular matrices is the mother of all solvable Lie algebras. If we focus on the associative span of a Lie algebra contained in the algebra of upper triangular matrices, then it is straightforward to prove that it is solvable as Lie algebra (and as associative algebra). This phenomenon is true in a wider context as proven by Hartmut Laue in [42]:

Corollary 1 (H. Laue, theorem of the associative span) Let A be a finite-dimensional associative algebra based on a field K of characteristic 0 and L be a solvable Lie subalgebra of A°. […].
Chapter 6 Maximal nilpotency in Lie algebras associated to solvable associative algebras:

The aim of this chapter is to determine all maximal nilpotent Lie subalgebras in Lie algebras associated to solvable associative algebras with respect to the following questions and topics:

- Is it true that a maximal Lie nilpotent subalgebra is an associative unital subalgebra?
- What is the the inner associative structure of maximal Lie nilpotent subalgebras?
- Is it possible to determine maximal Lie nilpotent subalgebras in a constructive way?
- Is it possible to characterize maximal Lie nilpotent subalgebras by a special property?
- In what way are Cartan subalgebras and the nilradical special among all maximal Lie nilpotent subalgebras?
- Is it possible to bound the number of pairwise non-isomorphic maximal Lie nilpotent subalgebras?
We will answer all of these questions within this chapter. One importantinstrument for our analysis are single, double and manifold centralizers of subalgebras.

Über Autor(innen)

Sven Bodo Wirsing was born in 1975 in Neumunster. 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.