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

Wirsing, Sven Bodo

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

Summary

During the author’s doctorate time at the Christian-Albrechts-Universitat to Kiel, Salvatore Siciliano gave a stimulating talk in the upper seminar algebra theory about Cartan subalgebras in Lie algebra associates to associative algebra. This talk was the incentive for the author to analyze maximal nilpotent substructures of the Lie algebra associated to associative algebras. In the present work Siciliano's theory about Cartan subalgebras is worked off and expanded to different special associative algebra classes. In addition, a second maximal nilpotent substructure is analyzed: the nilradical. Within this analysis the main focus is to describe these substructure with the associative structure of the underlying algebra. This is successfully realized in this work. 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.

Excerpt

4.6.1

Subalgebras . . . . . . . . . . . . . . . . . . . . . . . .

4.6.2

Right and left ideals . . . . . . . . . . . . . . . . . . .

4.6.3

Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.4

Factor algebras . . . . . . . . . . . . . . . . . . . . . .

4.6.5

The opposite algebra . . . . . . . . . . . . . . . . . . .

4.6.6

Matrix algebras . . . . . . . . . . . . . . . . . . . . . .

4.6.7

Adjunction of a unit . . . . . . . . . . . . . . . . . . .

4.6.8

Tensor products

. . . . . . . . . . . . . . . . . . . . .

4.7

The group algebra once again . . . . . . . . . . . . . . . . . .

4.8

Open-ended questions and exercises

. . . . . . . . . . . . . .

Cartan subalgebras in Lie algebras associated to associative

algebras

5.1

Cartan subalgebras are associative solvable subalgebras

. . .

5.1.1

Associative structure of Cartan subalgebras . . . . . .

5.1.2

Open-ended questions and exercises

. . . . . . . . . .

5.2

Maximal tori and Cartan subalgebras

. . . . . . . . . . . . .

5.2.1

Maximal tori . . . . . . . . . . . . . . . . . . . . . . .

5.2.2

Maximal tori of group algebras based on dihedral and

quaternion groups . . . . . . . . . . . . . . . . . . . .

5.2.3

Cartan subalgebras . . . . . . . . . . . . . . . . . . . .

5.2.4

Cartan subalgebras of group algebras based on dihe-

dral and quaternion groups . . . . . . . . . . . . . . .

5.2.5

Maximal tori and Cartan subalgebras of Solomon al-

gebras, Solomon-Tits algebras and triangular matrices

5.2.6

Open-ended questions and exercises

. . . . . . . . . .

5.3

Division algebras . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

Central division algebras . . . . . . . . . . . . . . . . .

5.3.2

Non-central division algebras . . . . . . . . . . . . . .

5.3.3

Open-ended questions and exercises

. . . . . . . . . .

5.4

Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . .

5.4.1

The case characteristic = 2 . . . . . . . . . . . . . . .

5.4.2

Examples for the associative conjugacy

. . . . . . . .

5.4.3

Remark on the Lie conjugacy . . . . . . . . . . . . . .

5.4.4

The case characteristic 2 . . . . . . . . . . . . . . . . .

5.4.5

Open-ended questions and exercises

. . . . . . . . . .

5.5

Solvable algebras . . . . . . . . . . . . . . . . . . . . . . . . .

5.5.1

Unitary solvable algebras . . . . . . . . . . . . . . . .

5.5.2

The quasi regular group and non-unitary solvable as-

sociative algebras . . . . . . . . . . . . . . . . . . . . .

5.5.3

Solvable group algebras . . . . . . . . . . . . . . . . .

5.5.4

Solvable group algebras of dihedral groups . . . . . . .

5.5.5

Solvable group algebras of quaternion groups . . . . . 100

5.5.6

Splitting solvable algebras . . . . . . . . . . . . . . . . 101

5.5.7

Open-ended questions and exercises

. . . . . . . . . . 105

5.6

Lie nilpotent associative algebras . . . . . . . . . . . . . . . . 107

5.6.1

Lie nilpotency

. . . . . . . . . . . . . . . . . . . . . . 107

5.6.2

Nilpotent group of units . . . . . . . . . . . . . . . . . 110

5.6.3

Group algebras and Lie nilpotency . . . . . . . . . . . 112

5.6.4

Exercises

. . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7

Simple, semisimple and separable algebras . . . . . . . . . . . 116

5.7.1

Central-simple algebras . . . . . . . . . . . . . . . . . 116

5.7.2

Simple algebras . . . . . . . . . . . . . . . . . . . . . . 119

5.7.3

Semisimple and separable algebras . . . . . . . . . . . 119

5.7.4

Open-ended questions and exercises

. . . . . . . . . . 122

5.8

Basic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.8.1

Characterizations of reduced algebras

. . . . . . . . . 125

5.8.2

Maximal solvable subalgebras . . . . . . . . . . . . . . 126

5.8.3

Cartan subalgebras . . . . . . . . . . . . . . . . . . . . 127

5.8.4

Maximal nilpotent subalgebras . . . . . . . . . . . . . 128

5.8.5

Reduced group algebras: the semisimple case . . . . . 129

5.8.6

Reduced group algebras: the modular case

. . . . . . 133

5.8.7

An example of Benjamin Steinberg . . . . . . . . . . . 134

5.8.8

Open-ended questions and exercises

. . . . . . . . . . 137

5.9

Associative algebras with separable factor algebra by its nil-

radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.9.1

A description by radical complements . . . . . . . . . 143

5.9.2

A strategy for the determination of Cartan subalgebras 144

5.9.3

Solvable algebras revised . . . . . . . . . . . . . . . . . 144

5.9.4

Group algebras for dihedral groups . . . . . . . . . . . 144

5.9.5

Open-ended questions and exercises

. . . . . . . . . . 147

5.10 Natural determination of Cartan subalgebras . . . . . . . . . 148

5.10.1 Sub- and factor structures . . . . . . . . . . . . . . . . 148

5.10.2 Direct products . . . . . . . . . . . . . . . . . . . . . . 150

5.10.3 Adjunction of a unit . . . . . . . . . . . . . . . . . . . 150

5.10.4 Cyclic algebras . . . . . . . . . . . . . . . . . . . . . . 151

5.10.5 Tensor products and extension of the base field . . . . 151

5.10.6 Matrix algebras . . . . . . . . . . . . . . . . . . . . . . 151

5.10.7 Group algebras . . . . . . . . . . . . . . . . . . . . . . 152

5.10.8 Open-ended questions and exercises

. . . . . . . . . . 154

Summation formulas for the dimension of maximal tori in

group algebras

157

6.1

The semisimple case . . . . . . . . . . . . . . . . . . . . . . . 157

6.2

Upper and lower bounds . . . . . . . . . . . . . . . . . . . . . 158

6.3

Special classes of groups . . . . . . . . . . . . . . . . . . . . . 160

6.4

The modular case . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5

Open-ended questions and exercises

. . . . . . . . . . . . . . 178

Invariants

189

7.1

Dimension formula for maximal tori

. . . . . . . . . . . . . . 189

7.2

Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . 191

7.3

Open-ended questions and exercises

. . . . . . . . . . . . . . 206

Outlook on series II

213

A Derived algebras

215

A.1 Definition and initial properties . . . . . . . . . . . . . . . . . 215

A.2 Structural properties . . . . . . . . . . . . . . . . . . . . . . . 217

A.3 Lie nilpotency . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

A.4 Isomorphism of derived algebras

. . . . . . . . . . . . . . . . 218

A.5 Open-ended questions and exercises

. . . . . . . . . . . . . . 223

List of figures

227

Bibliography

227

Index

234

Introduction

Maximal nilpotent are Cartan-subalgebras

as well as the nilradical.

Both will be studied in the magnificient sal

of associated Lie algebras.

(Sven Wirsing, December 2015)

Within the theory of Lie algebras Cartan subalgebras play an important role

for the classification of semisimple Lie algebras as well as within the theory

of symmetric spaces.

During my time of studying at the Christian-Albrechts-Universit¨

at of Kiel

Salvatore Siciliano presented his researches in the Oberseminar Algebren-

theorie about Cartan subalgebras in Lie algebras associated to associative

algebras. His presentation was the starting point for me to study maximal

nilpotent substructures in associated Lie algebras of associative algebras. In

this work we will present his theory of Cartan subalgebras and enhance it

to some special associative algebras (e.g. basic algebras, division algebras,

algebras with separable factor algebra by its nilradical). In addition, a sec-

ond maximal nilpotent substructure is analyzed, its the so-called nilradical

of a Lie algebra.

The first chapter introduces some special associative and Lie algebras, monoids

and groups. They will be important to visualize and illustrate the general

theorems proven within this work. Some applications are also transferred to

the exercises at the end of each section or chapter. There are some exercises

included enhancing the theory presented so far such that the reader gets a

deeper insight. In addition, at the beginning of each exercise series some

open-ended topics are included which can be used by the reader and also

by the author to do additional researches within this theory. The author

has included some (manually created) graphics mostly so called Hasse di-

agrams to visualize the results of each section or chapter.

Within chapter 2 basic results about finite subgroups of fields and divi-

sion algebras are summarized. Some will be proven in details, others will be

just presented without a proof. They will play a role later on in the next

chapters of this work such that their understanding leads to a better insight

of the latter results. In addition, the author includes some proofs of these

basic results because of personal interest on the proofs itself. The summary

will include the proof that finite subgroups of fields are cyclic, the theorem

of Wedderburn about finite division algebras as well as results of Herstein

and Amitsur about the classification of finite subgroups of division algebras.

Likewise structured is chapter 3. This chapter focusses on the normal and

subnormal subgroup structure of division algebras. We will prove the the-

orem of Cartan-Brauer-Hua about normal subgroups of division algebras

and the theorem of Scott about solvable group of units of division alge-

bras. Finally, the theorem of Stuth about subnormal subgroups is presented

(without proving it) enhancing the theorem of Scott.

For an associative algebra the associated Lie algebra can be derived in a

natural way. In chapter 4 we analyze the nilradical the greatest nilpotent

ideal of that Lie algebra and focus our analysis on its associative structure.

For this, the center and the nilradical of the associative algebra are of im-

portance: the nilradical is the sum of these two associative substructures. In

particular, its an associative subalgebra. For this theorem we assume that

the factor algebra by the nilradical of the associative algebra is separable

and thus we can use the theorem of Wedderburn-Malcev within the proof.

The analysis begins by determining the nilradical in the case of a solvable

associative algebra.

For this, results of the so-called generalized Jordan

decomposition are used. We demonstrate the theorem on special solvable

group algebras based on dihedral and quaternion groups, on the Solomon

algebras in characteristic zero, on the Solomon-Tits algebras and on the al-

gebras of upper and lower triangular matrices over an arbitrary field.

In a second step some results of Herstein about simple rings and their as-

sociated Lie ring are transferred to simple and semisimple algebras: we will

prove that the nilradical is identical to the greatest solvable Lie ideal the

so-called solvable radical. Both structures are identical to the center of the

associative algebra. For the semisimple case it is proven that the Lie nil-

radical of direct products is the direct product of the Lie nilradicals of the

corresponding components: there are no diagonals possible.

Both results for solvable and semisimple associative algebras are used to

determine the nilradical for arbitrary associative algebras.

The chapter is finalized to apply and enhance the theorem for algebra con-

structions like the tensor product, the adjunction of a unit and matrix al-

gebras over algebras. The idea is to determine the Lie nilradical by the

components of the algebra constructions, like by the factors for the tensor

product. We will give proofs or counterexamples for these constructions

with respect to this question.

In the previous chapter we have deeply analyzed the Lie nilradical of an

associative algebra with respect to its associative structure. The Lie nilrad-

ical is a maximal nilpotent substructure, and the Cartan subalgebras are

maximal nilpotent, too. They are in focus of the next chapter. They are

defined as being nilpotent and self-normalizing Lie subalgebras. The aim

of this chapter is the same as for Lie nilradical: their determination and

the description of their associative structure. Some results of this chapter

are based on an article of Salvatore Siciliano [59], others are enhancements

of his theory to other classes of associative algebras like division algebras,

simple, semisimple and separable associative algebras, reduced associative

algebras or associative algebras with separable factor algebra by their nil-

radical. Standard examples are investigated in details, in particular group

algebras, lower and upper triangular matrices and Solomon-(Tits) algebras

for illustrating the developed theory.

The main result of this chapter is the 1:1-connection between maximal

tori (maximal commutative separable subalgebras) und Cartan subalgebras.

Centralizing maximal tori is a bijection between these structures. The in-

verse calculates for every Cartan subalgebras a maximal torus by creating

the set of fully separable elements of the Cartan subalgebra.

In some cases both sets maximal tori and Cartan subalgebras are iden-

tical, like for separable associative algebras. Central divisions algebras are

separable, too, and we prove a theorem of Salvatore Siciliano (in a different

way) that maximal tori and Cartan subalgebras are exactly the maximal

separable subfields. We enhance the theorem by proving that these are ex-

actly the separable maximal subfields which is also an alternative proof of a

theorem of Emmy Noether. In particular, it is proven that all maximal tori

= Cartan subalgebras have the same dimension and are isomorphic as Lie

algebras. This theorem is transferred to non-central division algebras.

Solvable associative algebras have the property that maximal tori are exactly

the radical complements if the factor algebra by its associative nilradical is

separable. This result proven by Thorsten Bauer in his dissertation [4] and

by Salvatore Siciliano in [59] is proven by a different approach and revised

later on in the second to last section of this chapter. As a consequence of our

main theorem about Cartan subalgebras and the theorem of Wedderburn-

Malcev all maximal tori and Cartan subalgebras are conjugated, and the

Cartan-subalgebras are exactly the centralizers of the radical complements.

For basic algebras we transfer the determination of Cartan subalgebras to

Cartan subalgebras of maximal solvable substructures. These maximal ones

are describable as direct sums of maximal tori and the associative nilradical.

The centralizers of the maximal tori of the underlying algebra are identical

to the centralizer within these maximal solvable subalgebras. Afterwards we

focus on reduced group algebras. In the modular case the terms basic and

solvable are equivalent. For semisimple group algebras the situation is more

complex: the group is hamiltonian and the equation a

+ b

+ 1 = 0 has

no solution in special field extension based on roots of unities. Finally, we

determine the dimension of the Cartan subalgebras for these group algebras

based on the results of chapter 6.

In the second to last section we analyze how the determination of Cartan

subalgebras can be done based on separable radical complements. The max-

imal tori of the radical complement and of the whole algebra are identically.

For separable radical complements maximal tori and Cartan subalgebras are

identically, too. The centralizers of them are exactly the Cartan subalgebra

of the underlying algebra. Based on this result a strategy is developed for

determining Cartan subalgebras. For solvable algebras this strategy is used

and the determination of Cartan subalgebras is revised in a more transpar-

ent way. We apply this strategy also on group algebras of dihedral groups.

The chapter is finalized to apply and enhance the theorem for Cartan sub-

algebras for algebra constructions like the tensor product, the adjunction

of a unit and matrix algebras over algebras. The idea is to determine the

Lie nilradical by the components of the algebra constructions, like by the

factors for the tensor product. We will give proofs or counterexamples for

these constructions with respect to this question.

The next chapter is dedicated to the dimension of maximal tori in group

algebras. We begin this chapter by proving a result of Salvatore Siciliano

connecting this dimension to the sum of degrees of all irreducible complex

characters for semisimple group algebras. This sum is identical for all fields

such that the group algebra is semisimple. We use this result and some

classical and modern results about that sum within the character theory of

finite groups to bound this dimension like by the number of involutions,

by the order of the group, by the order of abelian subgroups and by the

maximal degree and determine this sum for several classes of groups

like for Frobenius groups, for direct products, for extra special p-groups, for

diverse linear groups, for ambivalent groups such as dihedral and symmetric

groups, for meta-cyclic groups, for p-groups, for nilpotent groups and for

minimal non-abelian p-groups.

Within chapter 7 we focus on the question whether the dimensions of the

maximal tori and of the Cartan subalgebras are unique for associated Lie

algebras of finite-dimensional associative unital algebras. For maximal tori

we give a positive answer to this question for associative algebras with sepa-

rable factor algebra by its nilradical by calculating this dimension explicitly.

The answer for the Cartan subalgebras is positive, too. In characteristic

zero we derive this result by using a classical result on Cartan subalgebras

over algebraically closed fields. In the modular case we begin the analysis by

proving the uniqueness for associated Lie algebras based on solvable finite-

dimensional associative algebras, for separable associative algebras and for

finite-dimensional associative algebras possessing a central nilradical. The

general case is derived by using a result of Premet (which was later proven

by Farnsteiner) for restricted Lie algebras over algebraically closed fields in

positive characteristic and by using the result on the dimension for maximal

tori. In general, the dimension of Cartan subalgebras can differ for restricted

Lie algebras. By using a second approach we extend our theorem for the

uniqueness of the dimension of Cartan subalgebras to the solvable and nilpo-

tency class. For this, we prove that all maximal tori and Cartan subalgebras

of Lie algebras associated to finite-dimensional associative algebras over an

arbitrary algebraically closed field are conjugated. We demonstrates these

three invariants dimension, nilpotency and solvable class by calculating

them for group algebras based on dihedral and quaternion groups.

Chapter 8 is an outlook on the second series about maximal nilpotent sub-

structures. We will focus on the solvable case of an associative algebra in

more details as in this first volume. For this, we will extend the topic to

all maximal nilpotent substructures and to the connection to the maximal

nilpotent subgroups of their group of unit. A graphic illustrates the prob-

lems analyzed in series II.

Within the appendix we classify a special class of algebras and analyze their

Lie nilpotency. This class of algebras was in focus of the diploma thesis of

Armin J¨

ollenbeck.

Chapter 1

Natural examples

This chapter has a preliminary function by summarizing those monoids,

groups, associative and Lie algebras which will arise in this work. They will

be used for examples of the proven theorems as well as for exercises in which

the reader shall apply the results.

Groups and monoids

Let n N, N be a set, M a monoid, G a group, A an associative unitary

algebra and q a prime power. We will focus on the following groups and

monoids:

· N - natural numbers

· N

- natural numbers containing zero

· (P (N ); ) - power set of N with operation

· (P (N ); ) - power set of N with operation - symmetric difference

· (P (M ); ·) - power set of M with complex product · as operation

· (P (G); ·) - power set of G with complex product · as operation

· D

- dihedral group of order 2n

· Q

- quaternion group of order 4n

· SD

- semi-dihedral group of order 2

· S

- symmetric group of degree n

· A

- alternating group of degree n

· GL(n, q) - general linear group of degree n over GF (q)

· SL(n, q) - special general linear group of degree n over GF (q)

· P SL(n, q) - projective special general linear group of degree n over GF (q)

· SP (2n, q) - symplectic group of degree 2n over GF (q)

· GSP (2n, q) - general similitudes group

· U (n, q) - unitary group of degree n over GF (q)

· C

or Z

- cyclic group of order n

· E(A) - group of units of A

· Q(A) - quasiregular group of A

· × - direct products of groups

- regular wreath product of groups

- semidirect product of groups.

General constructions of algebras

Let A be an algebra, K a field, G a group, I an ideal, M a monoid, n N

and T A. The following general constructions of algebras will be used:

· - tensor product of algebras

· × - direct products of algebras

· - direct sum of algebras

· A/I - factor algebra of A by the ideal I

· KG - group algebra of the group G and the field K

· KM - monoid algebra of the monoid M and the field K

· A

n×n

- algebra of n × n-matrices over A

· A

- associated Lie algebra of A

· T

- K-linear span of T

· T

- subalgebra generated by T

· T

- unital subalgebra generated by T

· A

- adjunction of a unit to A

· A

or A

- inverse or opposite algebra of A

· (A × A;

) - zero extension of A

· gl(n, K) - identical to (K

n×n

)

· eAe - identical to {eae | a A} for an idempotent e

· Aug(KG) - augmentation ideal of KG.

Commutative algebras

The following commutative algebras will appear:

· Z - the set of integers

· K[t] - polynomial algebra over K in one variable t.

Fields and skew fields

Let p be a prime number, n N and (K; L) a field extension. We will focus

on the following fields, skew fields and elements:

· Q - rational number field

· R - real number field

· C - complex number field

· H - real quaternion algebra

· GF (p

) - finite field with p

elements

· GF (q) - notation for GF (p

) and q = p

· A(a, b) - generalized quaternion algebra

· K(a) - smallest subfield in L containing a and K

- primitive dth root of unity

· cyclic division algebras.

(Central) - simple associative algebras

Let K be a field, D a division algebra and n N. We will use the following

(central)-simple associative algebras:

· K

n×n

- n × n-matrices over K

· D

n×n

- n × n-matrices over D

· A(a, b) - generalized quaternion algebra.

Semisimple associative algebras

We will use the following semisimple associative algebras:

· × - direct products of simple algebras

· A/rad(A) - the factor algebra by the nilradical of an associative algebra.

Nilpotent associative algebras

Let A be an associative algebra, K a field, p a prime number, n N and G

a p-group. We will focus on the following nilpotent associative algebras:

· rad(A) - nilradical of A

· J (A) - Jacobson radical of A

· s

u,n

- algebra of strict lower triangular matrices of K

n×n

· s

o,n

- algebra of strict upper triangular matrices of K

n×n

· Aug(KG) - augmentation ideal of KG based on a p-group G and char(K) =

Solvable associative algebras

Let n N, p a prime number, G a finite group and K be a field. We will

focus on the following solvable associative algebras:

· K

- Solomon-Tits algebra (see e.g. [76])

· D

- Solomon algebra in the case char(K) = 0 (see e.g. [4])

u,n

- algebra of lower triangular matrices of K

n×n

o,n

- algebra of upper triangular matrices of K

n×n

· KG - group algebra based on: char(K) = p and G possesses a normal

p-Sylow subgroup with an abelian p -Hall subgroup.

Chapter 2

Finite subgroups of fields

and division algebras

In this chapter we summarize some results of finite subgroups in unit groups

of fields and division algebras. For some of them we provide a proof, for the

others we reference the corresponding literature. We will use some of these

results in the next chapters. Therefor these results provide the reader a

deeper insight for understanding these results. In addition, this chapter is

included on personal interest of the author for the proofs of these results.

2.1

Finite subgroups of fields

By E(A) and K[t] we denote the group of units of an associative algebra A

and the algebra of polynomials over a ring K based on the single variable t.

For a group G and an element g of G let o(g) (more exact: o

(g)) the order

of g in G.

The following theorem is proven by various arguments within the litera-

ture. It is unknown which mathematician provided the first proof of this

result. Our variant is based on the main theorem on finite abelian groups.

Theorem 1 Every finite subgroup of the group of units of a field is cyclic.

In particular, the group of units of a finite field is cyclic.

Proof.

Let K be a field and U a finite subgroup of E(K). By using the

main theorem on finite abelian groups we decompose U in cyclic groups of

prime power order:

U = (G

1,1

× · · · × G

1,s

) × · · · × (G

r,1

× · · · × G

r,s

In this decomposition all groups G

i,j

are of prime power order with respect

to the prime number p

. We arrange the product such that G

i,1

is the great-

est factor within G

i,1

× · · · × G

i,r

. For every i let g

a generator of G

i,1

We focus on the element g := g

· · · g

. g is of order o(g) = o(g

) · · · o(g

)

because all prime numbers p

, · · · , p

are distinct. For every u U the

identity u

o(g)

= 1 is valid.

All elements of U are roots of the polynomial t

o(g)

- 1, and there are at

most o(g) distinct roots. Hence we derive | U | o(g). All o(g)-powers of g

are distinct. Therefor U is exactly the set of these powers of g. We conclude

that U is cyclic and generated by g.

2.2

Results of Wedderburn, Amitsur and Herstein

about division algebras

An unitary algebra is an algebra with a unit. An unital subalgebra of an

unitary algebra is a subalgebra containing the unit element of the global

algebra. Hence a unital subalgebra is unitary. An unitary subalgebra is a

subalgebra which is unitary as an algebra. An unitary subalgebra does not

need to be unital as its unitary unit could differ from the unit element of

the global algebra. Its unit element is an idempotent of the global algebra.

The center of A is denoted by Z(A).

Let G be a group, T a subset of G and g G. By

we symbolize the con-

jugation with g and by C

(T ) resp. N

(T ) the centralizer resp. normalizer

of T in G.

Our next focus is the proof of a theorem of Wedderburn about finite di-

vision algebras. For this proof we need the following two propositions.

Proposition 1 Let D be a K-division algebra and T be a unital finite-

dimensional subalgebra of D. Then T is a division algebra, too.

Proof.

Let t T and assume t = 0.

We consider the right and the

left multiplication with t on T . Both functions are injective because D is a

division algebra. Hence - using the finite dimension of T - they are surjective,

too. In particular, 1 has a pre-image with respect to these functions. Both

pre-images are the inverse of t and therefor contained in T .

Proposition 2 Let G be a finite group and U be a subgroup of G. The the

U and G are equal if and only if G is the union of all G-conjugate subgroups

of U .

Proof.

If U is a normal subgroup the statement is true. Let U be a non-

normal subgroup of G. Hence the statement G > N

(U ) is true. The

number of conjugates of U is exactly the index of the normalizer of U in

G which is

|G|

(U )|

. All conjugated of U have at least the unit element in

common. Therefor we conclude:

| 1 +

|G|

(U )|

· (| U | -1).

The right hand side is because of U N

(U ) not greater than

1+ | G | -

|G|

(U )|

By using G > N

(U ) we derive that this value is smaller than | G |.

We will prove the following theorem by usage of the theory of central-simple

associative division algebras. For this, let ind(D) (more exact: ind

(D)) the

index of a central-simple finite-dimensional associative unitary K-division al-

gebra which is the unique dimension of all maximal subfields of D. A good

introduction to this theory can be found [49] and in [39].

Theorem 2 (Wedderburn) Every finite division algebra is a field. In par-

ticular, its group of units is cyclic.

Proof.

Let D be a finite division algebra and K := Z(D). K is a field and

D a central-simple finite-dimensional associative unitary K-division algebra.

All maximal subfields have the same dimension ind

(D). Hence by using

the finiteness of D they are of the same order. Based on the finite field

theory we know that all maximal subfields are isomorphic. Now we use the

theorem of Skolem-Noether

and conclude that all maximal subfields are

conjugated. Every element d of D is contained in a maximal subfield of D

Thoralf Albert Skolem (born 23 May 1887, died 23 March 1963) was a Norwegian

mathematician who worked in mathematical logic and set theory. Although Skolem's fa-

ther was a primary school teacher, most of his extended family were farmers. Skolem

attended secondary school in Kristiania (later renamed Oslo), passing the university en-

trance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study

mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he

began working as an assistant to the physicist Kristian Birkeland, known for bombarding

magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first

publications were physics papers written jointly with Birkeland. In 1913, Skolem passed

the state examinations with distinction, and completed a dissertation titled Investigations

on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the

zodiacal light. He spent the winter semester of 1915 at the University of G¨

ottingen, at the

time the leading research center in mathematical logic, metamathematics, and abstract

algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research

fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathemat-

ics and was elected to the Norwegian Academy of Science and Letters. Skolem did not

at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary

in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theo-

rems about integral solutions to certain algebraic equations and inequalities. His notional

thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married

Edith Wilhelmine Hasvold. Skolem continued to teach at Det kongelige Frederiks Uni-

versitet (renamed the University of Oslo in 1939) until 1930 when he became a Research

Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to

conduct research free of administrative and teaching duties. However, the position also

required that he reside in Bergen, a city which then lacked a university and hence had no

because the subalgebra of D generated by {d, 1} is a subfield of D (see propo-

sition 1). Therefor D is the union of all maximal subfields of D. From this

we derive that E(D) is the union of all groups of units of all maximal sub-

fields and that these subgroups are conjugated. We can apply proposition 2

and conclude that D and one maximal subfield of D are identical. The proof

is complete and the add-on is a consequence of this result and of theorem 1.

Let V be a K-linear space and T a subset of V . By T

we denote the

K-linear span of T in V . GF (p

) resp. GF (q) symbolize a finite field of

order p

resp. q (Galois field).

By usage of our previous results we derive two theorems proven by Her-

stein:

Theorem 3 (Herstein) Every finite abelian subgroup of a division algebra

is cyclic.

Proof.

Let G be a finite abelian subgroup of a division algebra D. D is

a Z(D)-Algebra. We focus on the Z(D)-linear span of G in D. By using

proposition 1 we obtain that this span is a finite-dimensional unital Z(D)-

division algebra. G is commutative, and hence G

Z(D)

is a field and G a

finite subgroup of its groups of units. The proof is finished by using theorem

Theorem 4 (Herstein) Every finite subgroup of a division algebra in posi-

tive characteristics is cyclic.

Proof.

Let D be a division algebra, P the central prime subfield isomor-

phic to GF (p) and G a finite subgroup of E(D). We focus on the unital

P -subalgebra G

of the P -division algebra D. This division algebra is by

the finiteness of G finite-dimensional. Therefor proposition 1 implies that it

is a division algebra over P . P is finite and we conclude that this division

algebra is finite, too. By usage of theorem 2 of Wedderburn it is a field. The

corresponding theorem 1 for fields implies that G is as a finite subgroup

cyclic.

research library, so that he was unable to keep abreast of the mathematical literature. In

1938, he returned to Oslo to assume the Professorship of Mathematics at the university.

There he taught the graduate courses in algebra and number theory, and only occasionally

on mathematical logic. Skolem's Ph.D. student Øystein Ore went on to a career in the

USA. Skolem served as president of the Norwegian Mathematical Society, and edited the

Norsk Matematisk Tidsskrift (The Norwegian Mathematical Journal) for many years. He

was also the founding editor of Mathematica Scandinavica. After his 1957 retirement,

he made several trips to the United States, speaking and teaching at universities there.

He remained intellectually active until his sudden and unexpected death. For more on

Skolem's academic life, see Fenstad (1970).

Remark 1 The previous theorem 4 is wrong in characteristic zero. In the

real quaternion algebra the quaternion group of order 8 is a finite but non-

cyclic subgroup of the group of units.

Herstein and Amitsur

have classified the finite subgroups of division alge-

bras. A first results deals with so-called meta-cyclic groups. These groups

are characterized by possessing a cyclic normal subgroup whose factor group

is cyclic, too. A group having only cyclic Sylow subgroups is called a Z-

group. It can be proven that Z-groups are meta-cyclic.

Theorem 5 (Herstein) Every p-subgroup with respect to a prime number

p = 2 of the group of units of a division algebra is cyclic. In particular,

every subgroup of uneven order of the group of units of a division algebra is

a Z-group.

Proof.

We use theorem 5.3.7 in [63] to derive that a p-group of uneven

order is cyclic if it possesses exactly one subgroup of order p. This pre-

condition is with respect to 5.3.8 in [63] valid if every abelian subgroup is

cyclic. This was proven within theorem 3. The add-on follows as all Sylow

subgroups are cyclic.

Remark 2 The previous theorem 5 fails for p = 2. In the real quaternion

algebra the quaternion group of order 8 is a finite but non-cyclic subgroup

of the group of units. All of its subgroups are cyclic.

By C

or Z

we denote a cyclic group of order n N. If n, m are integers,

then let o

(m) := o

Z/nZ

(mZ).

We formulate the classification of finite

subgroups of division algebras (but we will not prove it here) in characteristic

zero:

Theorem 6 (Amitsur) Every finite subgroup of the group of units of a divi-

sion algebra in characteristic zero is isomorphic one of the following groups:

(i) C

Shimshon Avraham Amitsur (born August 26, 1921, died September 5, 1994) was

an Israeli mathematician. He is best known for his work in ring theory, in particular PI

rings, an area of abstract algebra. Amitsur was born in Jerusalem and studied at the

Hebrew University under the supervision of Jacob Levitzki. His studies were repeatedly

interrupted, first by World War II and then by the Israel's War of Independence. He

received his M.Sc. degree in 1946, and his Ph.D. in 1950. Later, for his joint work with

Levitzki, he received the first Israel Prize in Exact Sciences. He worked at the Hebrew

University until his retirement in 1989. Amitsur was a visiting scholar at the Institute

for Advanced Study from 1952 to 1954. He was an Invited Speaker at the ICM in 1970

in Nice. He was a member of the Israel Academy of Sciences, where he was the Head for

Experimental Science Section. He was one of the founding editors of the Israel Journal of

Mathematics, and the mathematical editor of the Hebrew Encyclopedia. Amitsur received

a number of awards, including the honorary doctorate from Ben-Gurion University in 1990.

His students included Avinoam Mann, Amitai Regev, Eliyahu Rips and Aner Shalev.

(ii) A Z-group of the form C

for which C

acts per inversion on C

and m is uneven.

(iii) A Z-group of the form T

× · · · × T

in which the orders of these factors

are pairwise prime to each other, T

is cyclic, every T

, i s is non-

cyclic of the form C

× · · · × C

), the prime numbers p, q

i r are distinct, for every i r the semidirect product C

is non-cyclic and is satisfying the following condition: if C

is the

kernel of the operation of C

on C

, then one of the following cases

are valid:

= 2, p -1 mod 4, c = 1) or

= 2, p -1 mod 4, 2

c+1

does not divide p

- 1) or

= 2, p 1 mod 4, 2

c+1

does not divide p - 1) or

> 2, q

c+1

does not divide p - 1.)

In addition, for every non-cyclic factor C

within every factor

the statement q

· o

(p) does not divide o

|T /T

(p) is valid.

(iv) C

in which m is uneven, an element of Q

of order 2

t-1

centralizes the group C

and an element of order 4 of Q

inverts the

group C

(v) Q

× Z in which Z is a Z-group of order m presented in (i), (ii) and

2 has uneven order in Z/Zm.

(vi) SL(2, 3) × Z in which Z is a Z-group of order m presented in (i), (ii)

and 2 has uneven order in Z/Zm.

(vii) The binary octahedral group of order 48.

(viii) The binary icosahedral group of order 120.

Proof.

see Amitsur [1], Herstein [20] and Lam [38]

2.3

Exercises

Excercise 1 Read the article [1] of Amitsur. Determine for all finite sub-

groups of division algebras suitable division algebra in which they are ap-

pearing!

Excercise 2 Define meta-abelian and supersolvable groups by a research in

the literature.

Excercise 3 Prove or disprove the following statements:

(i) Every cyclic group is meta-cyclic.

(ii) The converse of (i) is valid.

(iii) Every abelian group is meta-cyclic.

(iv) The converse of (i) is valid.

(v) Direct products of meta-cyclic groups are meta-cyclic.

(vi) Semidirect products of meta-cyclic groups are meta-cyclic.

(vii) Every meta-cyclic group is supersolvable.

(viii) Every meta-cyclic group is meta-abelian.

(ix) A group for which all Sylow subgroups are cyclic is meta-cyclic.

(x) A group of squarefree order is meta-cyclic.

(xi) Dihedral groups are meta-cyclic.

(xii) Quaternion groups are meta-cyclic.

(xiii) Semidihedral groups are meta-cyclic.

Excercise 4 Prove the following statements: An unitary algebra is an alge-

bra with a unit element. A unital subalgebra of an algebra A is a subalgebra

containing the unit element of A. A unital subalgebra is unitary. A uni-

tary subalgebra is a subalgebra which is unitary as an algebra. A unitary

subalgebra is not unital in general. (Tip: idempotent elements)

Excercise 5 By using an article [20] of Herstein prove the following state-

ments (p prime number, D a skew field and U a subgroup of E(D)):

(i) If U is of order p or p

, then U is cyclic.

(ii) If p = 2 and U is a p-group, then U is cyclic.

(iii) Is part (ii) true for p = 2?

(iv) If the order of U is uneven, then U is meta-cyclic.

Excercise 6 True or false: The unit group of an infinite field is cyclic. Is

it possible to characterize finite fields by characteristics of their unit group?

Excercise 7 Determine all finite subgroups of the multiplicative group of

complex numbers! How many non-isomorphic subgroups of order n are ex-

isting? Visualize them for n 8 on the complex plane!

Excercise 8 Are there finite subgroups in the additive group of complex

numbers? On what terms do finite subgroups of the additive group of a field

exist which are non-trivial? What is the answer for the multiplicative group?

Excercise 9 The additive group of a field is not isomorphic to the multi-

plicative group of a field.

Excercise 10 Focus on the exp-function from R to R

. Is it a homomor-

phism for the addition and multiplication?

Excercise 11 Is the exponential map related to the complex numbers a ho-

momorphism for the additive and multiplicative structure?

Excercise 12 Every finite generated subgroup of the multiplicative group of

rational numbers is cyclic.

Excercise 13 Determine the finite subgroups of the multiplicative group of

the real numbers.

Excercise 14 Determine the finite subgroups of the additive group of the

real numbers.

Excercise 15 Are there infinite subgroups of the additive and multiplicative

group of integers, rational, real and complex numbers?

Excercise 16 By a research of the literature determine all finite subgroups

of the additive and multiplicative group of integers, rational, real and complex

numbers. Are there subgroups which are simultaneously subgroups of the

additive and multiplicative group of these structures?

Excercise 17 True or false: Every finite subgroup of a division algebra is

cyclic! Analyze this question for the additive group of a division algebra.

Excercise 18 By a research in the literature determine all finite subgroups

of the real quaternion algebra.

Excercise 19 The additive factor group Q/Z is isomorphic to the group of

complex roots of unity.

Excercise 20 The additive factor group R/Z is isomorphic to the group of

complex roots of unity of absolute value 1.

Excercise 21 Determine the finite subgroups of the group of units of a

quaternion algebra in characteristic 2.

Excercise 22 Determine the finite subgroups of the group of units of a

quaternion algebra in characteristic p 3.

Excercise 23 Determine the finite abelian subgroups of the group of units

of a quaternion algebra in arbitrary characteristic.

Excercise 24 True or false: Every finite subgroup of the group of units of

a quaternion algebra in arbitrary characteristic is cyclic or abelian!

Chapter 3

Normal and subnormal

subgroups in unit groups of

division algebras

In this chapter we summarize some results of normal and subnormal sub-

groups in unit groups of division algebras. For some of them we provide a

proof, for the others we reference the corresponding literature. We will use

some of these results in the next chapters. Therefor these results provide

the reader a deeper insight for understanding these results. In addition, this

chapter is included on personal interest of the author for the proofs of these

results.

3.1

The theorem of Cartan-Brauer-Hua

For two elements a, b of a group we denote by [a, b] the commutator of a

and b. The theorem of Cartan-Brauer-Hua focus on the structure of the

normal subgroups in unit groups of division algebras. Its proof is based on

the so-called Hua-identity:

Proposition 3 (Hua-identity

) Let D be a division algebra, a, b E(D)

with a = 1. The following statements are valid:

(i) b

-1

(a - 1)b = b

-1

ab - 1

(ii) b

-1

ab = a[a, b]

(iii) (a - 1)[a - 1, b] = a[a, b] - 1

(iv) a([a, b] - [a - 1, b]) = 1 - [a - 1, b], and [a, b] - [a - 1, b] is not zero if

[a, b] = 1 is valid.

By using the identity in part (iv) we derive that every non-central element

is contained in the sub-(division)algebra of D generated by all commutators

of E(D):

a = (1 - [a - 1, b])(a([a, b] - [a - 1, b]))

-1

Proof.

The proof is an exercise for the reader (see exercise 26).

Theorem 7 (Cartan-Brauer-Hua) Let D be a division algebra and T an

unital divisions subalgebra of D. If E(T ) is a normal subgroup E(D), then

T is central or identical to D.

Add-on: If N is a normal subgroup of E(D), then N is central or the sub-

algebra generated by N (= N

) is exactly D.

Proof.

(Sysak [67]) We assume that T is not central. Let b be an element

of T \ Z(D). We focus on an element a D such that ab = ba is valid. E(T )

is a normal subgroup of E(D). Therefor we conclude:

1 = [a, b] = (a

-1

a)b T

and with an analog argument

1 = [a - 1, b] T .

By usage of proposition 3 we derive the identity

Hua Luogeng, or Hua Loo-Keng (born 12 November 1910; died 12 June 1985), was a

Chinese mathematician famous for his important contributions to number theory and for

his role as the leader of mathematics research and education in the People's Republic of

China. He was largely responsible for identifying and nurturing the renowned mathemati-

cian Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach

conjecture. In addition, Hua's later work on mathematical optimization and operations

research made an enormous impact on China's economy. Hua did not receive a formal

university education. Although awarded several honorary PhDs, he never got a formal

degree from any university. In fact, his formal education only consisted of six years of

primary school and three years of middle school. For that reason, Xiong Qinglai, after

reading one of Hua's early papers, was amazed by Hua's mathematical talent, and in 1931

Xiong invited him to study mathematics at Tsinghua University.

a = (1 - [a - 1, b])(a([a, b] - [a - 1, b]))

-1

T .

Every element of D not commutating with b is contained in T . Let c be

an element commutating with b. Then a + c is not commutating with b

(because a would be commutating with b). Hence a + c and a are contained

in T and therefor their difference c, too. We conclude the identity T = D.

The add-on is a direct consequence of this result.

3.2

The theorem of Scott

The theorem of Scott analyzes on what terms the group of unit of a division

algebra is solvable.

Theorem 8 (Scott) Let D be a division algebra. The factor group E(D)/E(Z(D))

possesses no non-trivial non-abelian normal subgroup. In addition, E(D) is

solvable if and only if K is a field.

Proof.

(Sysak [67]) We assume there is a non-central normal subgroup A

of E(D) such that A/Z(D) is abelian. The subalgebra of D generated by

A fulfills the preconditions of the theorem of Cartan-Brauer-Hua 7, and we

conclude that it is identical to D. Hence A is non-abelian and two elements

a, b A exist with 1 = [a, b] =: c Z(E(D)). We prove that Z(E(D)) is a

subgroup of order 2 of E(D) and a

, b

Z(E(D)) is valid. The identities

[1 + a, b]

[1 + a]

-1

(1 + a)b

(1 + a)

-1

(1 + ac)

are valid, and hence we conclude

[[1 + a, b]b]

[(1 + a)

-1

(1 + ac), b]

(1 + ac)

-1

(1 + a)(1 + ac)

-1

(1 + ac

)

(1 + a)(1 + ac)

-2

(1 + ac

With a similar argument we derive:

[[[1 + a, b], b], b]

(1 + a)

-1

(1 + ac)

(1 + ac

)

-3

(1 + ac

Because of [1 + a, b] A we get [[[1 + a, b], b], b] = 1. Hence

(1 + ac)

(1 + ac

)

(1 + a)(1 + ac

)

is valid. We conclude

a(3c + c

) + a

+ 3c

)

a(1 + 3c

) + a

(3c

+ c

)

which is equivalent to

(1 - c)

+ a

(-1 + c)

= 0.

By using 1 - c = 0 we derive

= c

-3

Z(E(D)).

A symmetric argument let us deduce b

Z(E(D)). For the element z

Z(E(D)) the identity (az)

= c

-3

is valid because of az A and [az, b] = c.

Hence z

= 1 is true and we conclude that the exponent and the order of

Z(E(D)) is 2. Z(D) is the prime field of order 3 in D and a, b elements of

finite order. The subgroup generated by {a, b} is nilpotent and hence as a

torsion group finite. By using theorem 4 the subgroup is cyclic. This is a

contradiction to the choice of a, b.

3.3

The theorem of Stuth

The next result which is formulated but not proven in details now general-

izes the previous results of Cartan-Brauer

-Hua and of Scott to the structure

of the subnormal subgroups and was proven by Stuth in [66]:

Richard Dagobert Brauer (born February 10, 1901, died April 17, 1977) was a leading

German and American mathematician. He worked mainly in abstract algebra, but made

important contributions to number theory. He was the founder of modular representation

theory. Alfred Brauer was Richards brother and seven years older. Alfred and Richard

were both interested in science and mathematics, but Alfred was injured in combat in

World War I. As a boy, Richard dreamt of becoming an inventor, and in February 1919

enrolled in Technische Hochschule Berlin-Charlottenburg. He soon transferred to Univer-

sity of Berlin. Except for the summer of 1920 when he studied at University of Freiburg,

he studied in Berlin, being awarded his Ph.D. on 16 March 1926. Issai Schur conducted

a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and

published a result. The problem also was solved by Heinz Hopf at the same time. Richard

wrote his thesis under Schur, providing an algebraic approach to irreducible, continu-

ous, finite-dimensional representations of real orthogonal (rotation) groups. Ilse Karger

also studied mathematics at the University of Berlin. She and Richard were married

17 September 1925. Their boys George Ulrich (1927) and Fred Gunther (1932) also be-

came mathematicians. Brauer began his teaching career in K¨

onigsberg (now Kaliningrad)

working as Konrad Knopp's assistant. Brauer expounded central division algebras over

a perfect field while in K¨

onigsberg: the isomorphism classes of such algebras form the

elements of the Brauer group he introduced. When the Nazi Party took over in 1933, the

Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer

and other Jewish scientists. Brauer was offered an assistant professorship at University

of Kentucky. Richard accepted the offer, and by the end of 1933 he was in Lexington,

Kentucky, teaching in English. Ilse followed the next year with George and Fred; brother

Alfred made it to the USA in 1939, but their sister Alice was killed in The Holocaust.

Theorem 9 (Stuth) Let D be a division algebra. The following statements

are valid:

(i) The central subgroups of E(D) are exactly the solvable subnormal sub-

groups of E(D).

(ii) For every non-central subnormal subgroup S of E(D) the identity C

E(D)

(S) =

Z(E(D)) is valid.

(iii) The intersection of two non-central subnormal subgroups of E(D) is

a non-central subnormal subgroup of E(D), too. In particular, there

is a smallest non-central subnormal subgroup of E(D) which is non-

solvable.

Hermann Weyl invited Richard to assist him at Princetons Institute for Advanced Study

in 1934. Richard and Nathan Jacobson edited Weyl's lectures Structure and Representa-

tion of Continuous Groups. Through the influence of Emmy Noether, Richard was invited

to University of Toronto to take up a faculty position. With his graduate student Cecil

J. Nesbitt he developed modular representation theory, published in 1937. Robert Stein-

berg, and Stephen Arthur Jennings were also Brauer's students in Toronto. Brauer also

conducted international research with Tadasi Nakayama on representations of algebras.

In 1941 University of Wisconsin hosted visiting professor Brauer. The following year he

visited the Institute for Advanced Study and Bloomington, Indiana where Emil Artin was

teaching. In 1948 Richard and Ilse moved to Ann Arbor, Michigan where he and Robert

M. Thrall contributed to the program in modern algebra at University of Michigan. With

his graduate student K. A. Fowler, Brauer proved the Brauer-Fowler theorem. Donald

John Lewis was another of his students at UM. In 1952 Brauer joined the faculty of Har-

vard University. Before retiring in 1971 he taught aspiring mathematicians such as Donald

Passman and I. Martin Isaacs. The Brauers frequently traveled to see their friends such

as Reinhold Baer, Werner Wolfgang Rogosinski, and Carl Ludwig Siegel.

3.4

Exercises

Excercise 25 Explain on what terms the theorem of Stuth (theorem 9) is

a generalization of the theorems of Cartan-Brauer-Hua and Scott.

Excercise 26 Prove all identities and statements within proposition 3!

Excercise 27 Study the article [67] with respect to the following topics:

definition of periodic groups, periodicity of the additive and multiplicative

group of Q, R and C, proof of the lemmas of Herstein and the theorem of

Jacobson that a division algebra with periodic group of unit is commutative.

In what way is this theorem a generalization of another theorem of Jacobson?

Excercise 28 On what terms is the unit group of a division algebra abelian?

Excercise 29 On what terms is the unit group of a division algebra nilpo-

tent? (Tip: Theorem of Scott)

Excercise 30 On what terms is the unit group of a division algebra solv-

able? (Tip: Theorem of Scott)

Excercise 31 On what terms is the unit group of a division algebra simple?

(Tip: center and theorem of Wedderburn)

Excercise 32 On what terms is the unit group of a division algebra cyclic?

(Tip: exercise 6)

Excercise 33 Let D be a division algebra. What do we know about finite

normal subgroups of E(D)?

Excercise 34 Let D be a division algebra. What do we know about finite

subnormal subgroups of E(D)?

Excercise 35 Do a research in literature and prove that a nilpotent and

finite-generated torsion group is finite. For what reason do we need this result

in this chapter? (Tip: Use the main theorem on finite-generated abelian

groups and apply it on the factor groups for the nilpotent groups with respect

to a central chain of finite length!)

Excercise 36 By using the tip in exercise 35 extend the results if possible

to solvable groups. Is this statement also valid for non-solvable groups?

Chapter 4

Nilradicals of Lie algebras

associated to associative

algebras

For every associative algebra the so-called associated Lie algebra can be de-

rived in a natural way. In this chapter we focus on the determination of

the nilradical the greatest nilpotent ideal of the associated Lie algebra.

We describe it by the structure of the associative algebra: it is the sum the

center and the nilradical of the associative algebra. In particular, the nil-

radical of the associated Lie algebra is an unital associative subalgebra. We

assume uneven characteristic of the ground field and the separability of the

factor algebra by the nilradical of the associative algebra. By assuming this

we can use the well-known theorem of Wedderburn-Malcev for the existence

and conjugacy of radical complements in the associative algebra.

Our strategy is to start with solvable associative algebras. For this, the gen-

eralized Jordan decomposition is the main tool for our analysis. We calculate

the nilradical for our standard examples (Solomon algebras, Solomon-Tits

algebras, group algebras and triangular matrices). In a next step we transfer

results of Herstein

about simple rings and its associated Lie ring to simple

Israel Nathan Herstein (born March 28, 1923, died February 9, 1988) was a math-

ematician, appointed as professor at the University of Chicago in 1951. He worked on

a variety of areas of algebra, including ring theory, with over 100 research papers and

over a dozen books. Herstein was born in Lublin, Poland, in 1923. His family emigrated

to Canada in 1926, and he grew up in a harsh and underprivileged environment where,

according to him, you either became a gangster or a college professor. During his school

years he played football, ice hockey, golf, tennis, and pool. He also worked as a steeplejack

and as a barker at a fair. He received his B.S. degree from the University of Manitoba and

his M.A. from the University of Toronto. He received his Ph.D from Indiana University

in 1948. His advisor was Max Zorn. He held positions at the University of Kansas, Ohio

State University, University of Pennsylvania, and Cornell University before permanently

settling at the University of Chicago in 1962. He was a Guggenheim Fellow for the aca-

demic year 1960/1961. He is known for his lucid style of writing, as exemplified by the

and semisimple algebras. The general case is studied in a last step using the

results for solvable and semisimple associative algebras.

We finish this chapter by applying the results to tensor products, adjunction

of a unit, matrix algebras, subalgebras etc. For this, we focus on describing

the nilradical by the underlying structure, e.g. by the factors of the tensor

product.

We use the term Lie nilpotent for the associated Lie algebra to pronounce

that the associated Lie algebra is nilpotent. In a similar matter we use the

term Lie solvable, Lie nilradical, Lie subalgebra, Lie ideal etc.

4.1

Solvable algebras

In this section we focus on solvable associative algebras and begin with the

definition of the most important tool for our analysis: the generalized Jordan

decomposition.

4.1.1

Generalized Jordan decomposition

Definitions and remarks 1 Let L be a K-Lie algebra and l L. We

define the multiplication with l also called adjoint representation by

ad(l) : L - L, x xl. The nilradical of L is the greatest nilpotent ideal

of L (if it is existing). The associated Lie algebra of an associative algebra

A with respect to the operation a b := ab - ba for all a, b A will be

symbolized by A

. If S, T are subsets of A

, then we denote by S T the

K-linear span of the set {s t | s S, t T }. Nilpotency and solvability is

used for groups and Lie algebras as usually.

By resp. we denote the left- resp. right-regular representation of an

associative algebra A. Based on definition 5.2.1 in [74] we call a polynomial

fully separable if it is squarefree and separable. If A is an associative unitary

K-algebra, then an element a A is fully separable if and only if its minimal

polynomial min

a,K

is fully separable. This is equivalent to the statement

that min

a,K

has no multiple roots in K[t] for every splitting field F of K.

Another description of this property is that min

a,K

and its formal deriva-

tion min

a,K

have no non-trivial common divisor: gcd(min

a,K

, min

a,K

) = 1.

classic and widely influential Topics in Algebra, an undergraduate introduction to abstract

algebra that was published in 1964, which dominated the field for 20 years. A more ad-

vanced classic text is his Noncommutative Rings in the Carus Mathematical Monographs

series. His primary interest was in noncommutative ring theory, but he also wrote papers

on finite groups, linear algebra, and mathematical economics. He had 30 Ph.D. students,

traveled and lectured widely, and spoke Italian, Hebrew, Polish, and Portuguese. He died

from cancer in Chicago, Illinois, in 1988. His doctoral students include Miriam Cohen,

Susan Montgomery, Karen Parshall, Claudio Procesi, Lance Small, and Murray Schacher.

It is straightforward to prove that a, a and a have the same minimal

polynomial. Thus a is nilpotent resp. fully separable if and only if a or a

has this property. Based on definition 5.1.4.1 in [74] a pair (r; s) A × A is

called a generalized Jordan decomposition of a A if a = r + s is valid, r

and s commute, r is nilpotent and s is fully separable. The nilradical of A

is denoted by rad(A). A is solvable if A/rad(A) the factor algebra by its

nilradical is commutative. If rad(A) possesses an complement T which is

a subalgebra, then we call T a radical complement: A = rad(A) T .

Lemma 1 Let K be a field, A an associative unitary finite-dimensional K-

algebra and a, r, s A. If (r; s) is a generalized Jordan decomposition of a,

then (ad(r); ad(s)) is one of ad(a).

Proof.

Step 1: It is straightforward to prove that a, (a) and (a) possess

the same minimal polynomial and ad(a) = ad(r + s) = ad(r) + ad(s) is valid.

Step 2: We prove now that ad(r) and ad(s) commute: A is associative.

We conclude that for all x, y A the functions (x) and (y) commute.

Because of rs = sr the functions (r) and (s) as well as (r) and (s)

commute. We conclude:

ad(r)ad(s)

((r) - (r))((s) - (s))

(r)(s) - (r)(s) - (r)(s) + (r)(s)

(s)(r) - (s)(r) - (s)(r) + (s)(r)

((s) - (s))((r) - (r))

ad(s)ad(r).

Step 3: By usage of Step 1 we derive that r and hence also (r) and (r)

are nilpotent. A is associative and therefor (r) (r) = (r) (r). As a con-

sequence the subalgebra generated by {(r), (r)} is commutative and by

proposition 5 in [74] even nilpotent. Thus ad(r) = (r) - (r) is nilpotent,

too.

Step 4: By usage of Step 1 we derive that r and hence also (r) and (r)

are fully separable. A is associative and therefor (r) (r) = (r) (r). As a

consequence the subalgebra generated by {(r), (r)} is commutative and by

proposition 5 in [74] separable. Thus ad(r) = (r) - (r) is fully separable,

too.

4.1.2

The nilradical

In this section we determine the nilradical of a Lie algebra associated to

a solvable associative algebra. For this we will use our results about the

Jordan decomposition.

Definitions and remarks 2 An associative finite-dimensional unitary com-

mutative algebra with separable factor algebra by the nilradical possesses

exactly one radical complement: the set of all fully separable elements. If

the algebra is not commutative but solvable and T is a radical complement,

then T Z(A) is a radical complement of the center of A. This set coincides

with the intersection of all radical complements of rad(A) in A (see e.g. [74],

chapter 5).

Let L be a Lie algebra. A subalgebra M of L is called maximal nilpotent if

M is not contained in a proper nilpotent subalgebra of L.

Theorem 10 Let K be a field, A an associative unitary finite-dimensional

solvable K-algebra with separable factor algebra by the nilradical and Z the

radical complement of the center of A. The nilradical of A

is rad(A) Z =

rad(A) + Z(A). In particular, the nilradical of A

is an unital associative

subalgebra of A and the only maximal nilpotent subalgebra of A

containing

rad(A).

Proof.

Both ideals rad(A) and Z of A

are nilpotent, Hence by using

a theorem of Fitting their sum is nilpotent, too. Thus the nilpotent ideal

rad(A)Z is contained in the nilradical of A

. Let N be the nilradical of A

and n N . By the theorem of Wedderburn-Malcev a radical complement T

exists. Let r rad(A) and t T such that n = r + t is valid. As rad(A) is

contained in N we derive t N . N is a nilpotent Lie algebra and we conclude

that ad(t) is a nilpotent endomorphism of N . In particular, ad(t)

|rad(A)

is a

nilpotent endomorphism of rad(A). A/rad(A) is separable and commutative

and by usage of theorem 5.3.1 in [74] every element of T is fully separable.

Lemma 1 implies that ad(t) is fully separable, too. In particular, ad(t)

|rad(A)

is fully separable. We derive that ad(t)

|rad(A)

is nilpotent and fully separable.

Lemma 1 implies ad(t)

|rad(A)

= 0.

We conclude that t centralizes rad(A).

T is commutative and thus we get t T Z(A). This subalgebra is exactly

Z which is proven in 5.1.4 in [74].

The identity Z(A) = rad(Z(A)) + Z is valid. For an associative solvable

algebra the nilradical is exactly the set of all nilpotent elements (see e.g. [4]

or the section about reduced algebras in this volume). Therefor rad(A)+Z =

rad(A) + rad(Z(A)) + Z = rad(A) + Z(A) is valid

The add-on is a consequence of the observation that every subalgebra of A

containing rad(A) is because of A A rad(A) an ideal of A

and the

nilradical coincide with the sum of an ideal and a subalgebra of A.

Every element possesses at most one Jordan decomposition (see e.g. chapter 5 in [74]).

4.2

Standard examples of solvable associative al-

gebras

4.2.1

Solvable group algebras

In this section we determine the nilradical of the associated Lie algebra

(KG)

for a modular solvable group algebra KG with respect to dihedral

and quaternion groups. We need the following general insight about solvable

group algebras:

Preliminary remark 1 Let K be a field and G a finite group. By using

3.2.20 in [74] the group algebra KG is solvable if and only if G is abelian or

char(K) = p is valid and G the derived subgroup of G is a p-group (p a

prime number). If G is abelian, then (KG)

is nilpotent. Let char(K) = p

and G a p-group. Then there exists a p-Sylow subgroup P containing the

derived subgroup. Hence P is a normal p-Sylow subgroup of G, and therefor

its the only p-Sylow subgroup by using Sylow's theorem. The theorem of

Schur-Zassenhaus assured a complement H of P in G. If is the lineariza-

tion of the natural epimorphism from G onto the factor group G/P , then

the kernel of is well-known: Kern = KGAug(KP ) = Aug(KP )KG,

and Aug(KP ) is the augmentation ideal of KP . By a theorem of Wallace

Aug(KP ) is nilpotent, and thus Kern is nilpotent, too. The factor alge-

bra KG modulo Kern is isomorphic to K(G/P ) which is isomorphic to

KH. KH is using Maschke's theorem semisimple, and we conclude by

1.9.4 in [74] that it is even separable. We derive rad(KG) = KG Aug(KP ),

and KH is a separable radical complement in KG. In particular, the di-

mension of the nilradical is | G | - | H |. By using results in [74], chapter

3, the intersection of all radical complements is the radical complement of

the center of KG. In particular, the radical complement of the center of

KG is contained in KH. The nilradical of (KG)

is determined by using

theorem 10 as the direct sum of the radical complement of the center of

KG and the nilradical (=KGAug(KP )) of KG. By using the same theorem

it is also the sum of the radical of KG with the center of KG, and the lat-

ter subalgebra is K-spanned by the conjugacy class sums of G. The radical

complement of the center of KG can be constructed as follows: calculate the

generalized Jordan decomposition for a basis of Z(KG). The K-linear span

of the calculated fully separable parts is the desired radical complement of

the center of KG. This calculation is done within remark 3.

Let G be a group, K a field and T a finite subset of G. By T we denote the

sum of all elements of T in KG. C(G) symbolizes the set of all conjugacy

classes of G and k(G) the number of these classes. If U is a subgroup of G,

then the core of U in G is the greatest normal subgroup of G contained in

U . Its exactly the intersection of all conjugates of H in G and is symbolized

by core

(U ) =

. By using the preliminary remark we prove now:

Theorem 11 (Lie nilradical of solvable group algebras) Let G be a

finite group and K a field with char(K) = p > 0 such that KG is solvable.

If H is a p -Hall subgroup and P is the normal p-Sylow subgroup of G, then

(Aug(KP )KG) K(Z(G) H) is the nilradical of (KG)

. In particular,

the dimension of the Lie nilradical is | G | - | H | + | Z(G) H |.

Proof.

We use some results of theorem 16 and the preliminary remark 1.

KH is a radical complement and Z(KG) KH is the radical complement of

Z(KG). We have to prove that this radical complement is exactly K(Z(G)

H). Then the proof is finished by using theorem 10. Surely, K(Z(G) H)

is contained in the other substructure. Let x :=

h an element of

Z(KG)KH. The center of KG is spanned by the conjugacy class sums as a

K-space, and we present x as

CC(G)

C. Let h H with k

= 0, then there

exists a conjugacy class C of G and an element c C with h = c and k

= 0.

All conjugates of c which is the set C are contained in H: the coefficient

of all elements of C is exactly k

and the conjugacy classes partition G.

Therefor all conjugates of c satisfy the identity

h =

CC(G)

C and

are contained in H. We have proven h

= C H. By using h

H we

derive h

. This set is the core of H in G. This normal subgroup has

trivial intersection with P , and thus every element of the core commutates

with all elements of P . H is abelian and G = P H is valid. This implies

that the core of H in G is central in G. We have proven h Z(G) H as

desired.

Corollary 1 Let p be a prime number, K a field with char(K) = p 3,

n N and G a group of order 2 · p

. The following conditions are valid:

(i) If a 2-Sylow subgroup is central, then (KG)

is nilpotent.

(ii) If a 2-Sylow subgroup is not central, then rad(KG) K · 1

is the

nilradical of (KG)

of dimension 2p

- 1.

Proof.

The p-Sylow subgroup is a normal subgroup of order p

and of

index 2. The complement H in the preliminary remark 1 is of order 2. Thus

the radical complement KH is of dimension 2, and the radical complement

of the center of KG is KH or K · 1

(using again the preliminary remark

1). Now the proof is finished using again the preliminary remark 1.

For the dihedral groups we conclude:

Corollary 2 Let p be a prime number, K a field with char(K) = p 3,

n N and G = D

. rad(KG) K · 1

is the nilradical of (KG)

dimension 2p

- 1.

For the quaternion groups we derive:

Corollary 3 Let p be a prime number, K a field with char(K) = p 3,

n N

and G = Q

. rad(KG) KZ(G) is the nilradical of (KG)

dimension 4p

- 2.

Proof.

Let a, b G with o(a) = 2p

, o(b) = 4, z = b

= a

such that G is

generated by a, b, the center of G is generated by z and a

= a

-1

z is valid.

We use the results of the preliminary remark 1. The derived subgroup of G

is generated by a

and is thus a p-group of order p

. One of its complement

H is generated by b and is of order 4. 1 and z are fully separable because

p = 2 is valid, and H is non-central. Therefor the radical complement of the

center of KG is contained in KH and possesses the dimension 2 or 3. By

usage of theorem 11 its exactly K(Z(G) H) and we conclude that it is of

dimension 2.

At the end of this section we determine (as announced earlier in this sec-

tion) the generalized Jordan decomposition of the conjugacy classes within

the center Z(KG) of KG for solvable KG:

Remark 3 (Generalized Jordan decomposition of the conjugacy

class sums) Let K be a field with char(K) = p = 0, G a finite group with

normal p-Sylow subgroup P and abelian Hall complement H. Z(KG) =

(rad(KG) Z(KG)) K(Z(G) H) is valid using theorem 11. rad(KG) is

K-linear spanned by the elements (p - 1)h with p = 1, p P, h H (using

the preliminary remark 1).

Let g

be a conjugacy class of G for an element g which is not central

in G. Thus the length of g

is at least 2.

We begin to focus on the case g = p P . Let C(p) be a set of conjugators

of p in G which is one of the minimal sets M in G with p

= p

. The

following equation is valid:

xC(p)

((p

- 1) + 1)

(

xC(p)

- 1))+ | p

| ·1

The second summand which is a multiple of 1 is surely central. Thus the first

summand which lies in the nilradical of KG is central, too. As a consequence

we have found the decomposition. The fully separable part is zero (the class

sum consequently nilpotent) if and only if the length of the conjugacy class

is divided by p. For example, this is the case if P is abelian.

Let g = h H with h

H. Following the proof of theorem 11 the element

h is central in G. h

is no subset of H. Let C(h) be a set of conjugators of

h in G. Then | C(h) | = | h

| is valid. For every g C(h) elements h

and p

P exist with g = h

. As H is abelian, we derive h

= h

This element can be re-written to (p

-1

- 1)hp

+ h(p

- 1) + h. Thus there

exists an element j rad(KG) with h

= j+ | h

·h. H is abelian, and

therefor H is contained in C

(h). We conclude that | h

| is a power of p

or h ist central. By our assumption h is not central, and we derive that h

is nilpotent (because of char(K) = p).

Let g = ph with p = 1 = h. We begin our analysis by assuming that p is

central. Thus h is not central and (ph)

= h

· p is valid. By our analysis so

far we deduct the nilpotency of h

and the one of (hp)

. Now let h central

and p not central. We calculate (ph)

= p

· h. The conjugates of ph in

G are closely connected to the ones of p in G, because h is central. There

exist elements j rad(KG) and k K with h

= j + k · 1

. We conclude

(ph)

= h · j + k · h. Because h Z(G) H is valid, the element h · j is

central in KG and contained in rad(KG). In this case we have determined

the Jordan decomposition. Now let p and h be not central. We proceed as

in the case for h

. Let C(ph) be a set of conjugators of ph in G. For every

x C(ph) let p

P and h

H such that x = h

is valid. We derive:

(ph)

xC(ph)

- 1)h

xC(ph)

The first sum is contained in rad(KG). For the second sum we calculate

further:

xC(ph)

-1

) + h(p

- 1) +

xC(ph)

Thus there exists an element j rad(KG) such that

xC(ph)

(ph)

= j+ | C(ph) |

·h

is valid. This decomposition is unique as direct sum of elements of rad(KG)

and KH.

xC(ph)

(hp)

decomposes in Z(KG) as a sum of elements of

Z(KG) rad(KG) and of Z(KG) KH. This sum is K-direct. There-

for both decompositions are identically. h is not central, and thus | C(ph) |

must be divided by p and is zero. In particular, the whole conjugacy class

sum is nilpotent.

Only the case g Z(G) is remaining now. For this we prove that P Z(G)

is the normal p-Sylow subgroup with abelian complement Z(G)H in Z(G).

If g Z(G), then there exist p Z(G) P and h Z(G) H such that

g = ph = (p - 1)h + h is valid. This is the desired decomposition. The prime

divisors of | Z(G) | are ones of | G |, too. As an abelian group Z(G) is the

direct product of its Sylow subgroups. The theorem of Sylow implies that

we can conjugate them into the corresponding ones of G. But the center is

central and the conjugation on the center is the identity. As a consequence

they are contained in every corresponding Sylow subgroup of G.

As another application we determine within the next sections the Lie nil-

radical for our standard examples: the Solomon-Tits algebras, the Solomon

algebras and the algebras of upper and lower triangular matrices.

4.2.2

Triangular matrices

Let K be a field and n N. The algebra of lower triangular matrices are

examples of solvable associative K-algebras with separable factor algebra by

its nilradical. The nilradical of the algebra of lower triangular matrices of

n×n

- denoted by

u,n

- is the subalgebra of strict lower triangular matrices

- symbolized by s

u,n

. The nilradical has the dimension

n-1

i=1

i =

(n - 1)n.

u,n

is central, its center is of dimension 1. By theorem 10 we derive:

Corollary 4 For the Lie nilradical of the algebra of lower triangular ma-

trices the following identities are valid:

(i) nil(

u,n

) = s

u,n

K · 1

(ii) dim

(nil(

u,n

)) = 1 +

(n - 1)n.

Let K be a field and n N. The algebra of upper triangular matrices are

examples of solvable associative K-algebras with separable factor algebra by

its nilradical. The nilradical of the algebra of upper triangular matrices of

n×n

- denoted by

o,n

- is the subalgebra of strict upper triangular matrices

- symbolized by s

o,n

. The nilradical has the dimension

n-1

i=1

i =

(n - 1)n.

o,n

is central, its center is of dimension 1. By theorem 10 we derive:

Corollary 5 For the Lie nilradical of the algebra of upper triangular ma-

trices the following identities are valid:

(i) nil(

o,n

) = s

u,n

K · 1

(ii) dim

(nil(

o,n

)) = 1 +

(n - 1)n.

4.2.3

Solomon algebras in characteristic zero

Let K be a field of characteristic zero and n N. By D

we denote the

Solomon algebra. It is the K-linear span of all class sums of so-called defect

classes in KS

: if S

, then D() := {i | i > (i + 1)}. The Solomon

algebra is the K-linear span of {

D()=D

| D n - 1

}. The surprising

insight of Luis Solomon was that products of two defect class sums is a

linear combination of defect class sums. Thus the linear span of the defect

class sums is indeed an associative algebra. The Solomon algebra possesses

the dimension 2

n-1

, and its factor algebra by the nilradical is of dimension

p(n) - the number of partitions of n. D

is a solvable associative algebra with

separable factor algebra by its nilradical. Its center is semisimple. Thus the

intersection of the center with the nilradical is trivial. For even resp. uneven

n the center is of dimension 3 resp. 2. For details the reader may study

the dissertation of Thorsten Bauer [4], in particular chapter 3, in which the

descriptions of the radical is contained, too. By theorem 10 we derive:

Corollary 6 For the Lie nilradical of the Solomon algebra in characteristic

zero the following identities are valid:

(i) nil(D

) = rad(D

) Z(D

)

(ii) dim

(nil(D

)) = 2

n-1

- p(n) + 3, n even.

(iii) dim

(nil(D

)) = 2

n-1

- p(n) + 2, n uneven.

4.2.4

Solomon-Tits algebras

Let K be a field and n N. By S(n, k) we denote the so-called Stirling

numbers with resp. to k n

. Its the number of unordered set partitions of

n consisting of exactly k subsets. In [76] the Solomon-Tits algebra K

analyzed in details for several questions and problems based on a paper by

Manfred Schocker. The algebra is defined as the monoid algebra K

with

resp. to the monoid

. This monoid consists of all ordered set partitions

of n. If (P

, · · · , P

) and (Q

, · · · Q

) are two elements of this monoid, then

their product is defined by

, · · · , P

)

, · · · , Q

) :=

, P

, · · · , P

, P

, · · · , P

)

The symbol

signalizes that empty sets are deleted from this tuple. K

is again an example for a solvable associative algebra with separable factor

algebra by its nilradical.

The nilradical of K

is described in several ways which are not presented

here (see e.g. chapter 2 in [76]), and its dimension is dim

(rad(K

)) =

k=0

(k! - 1) S(n, k) (see corollary 8 in [76]).

The center of K

is one-

dimensional because K

is central (see theorem 13 in [76]). Therefor we

can apply theorem 10 and conclude:

Corollary 7 For the Lie nilradical of the Solomon-Tits algebra the follow-

ing identities are valid:

(i) nil(K

) = rad(K

) K · 1

(ii) dim

(nil(K

)) = 1 +

k=0

(k! - 1) S(n, k).

Details

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Type of Edition: Erstausgabe
Publication Year: 2016
ISBN (PDF): 9783960676034
ISBN (Softcover): 9783960671039
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Language: English
Publication date: 2016 (December)
Keywords: Cartan subalgebra Nilradical Lie algebra Associative algebra Maximal nilpotent Group algebra Exercise Lie nilpotency Salvatore Siciliano

Author

Sven Bodo Wirsing (Author)

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.

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

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Sven Bodo Wirsing (Author)