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

Wirsing, Sven Bodo

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

by Dr. rer. nat. Sven Bodo Wirsing (Author)

Summary

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

Excerpt

Wirsing, Sven Bodo: Maximal nilpotent subalgebras II: A correspondence theorem

within solvable associative algebras. With 242 exercises, Hamburg, Anchor Academic

Publishing 2017

Buch-ISBN: 978-3-96067-196-1

PDF-eBook-ISBN: 978-3-96067-696-6

Druck/Herstellung: Anchor Academic Publishing, Hamburg, 2017

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Contents

Introduction

Standard examples, symbols and notations

Radical algebras and the theorem of Xiankun Du

2.1

Radical algebras and central chains . . . . . . . . . . . . . . .

2.2

Results of Stephen Arthur Jennings, Hartmut Laue and Xi-

ankun Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Standard examples . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1

The algebras of upper and lower triangular matrices .

2.3.2

Solomon algebras in characteristic zero . . . . . . . . .

2.3.3

Solomon-Tits algebras . . . . . . . . . . . . . . . . . .

2.4

Open-ended questions and exercises

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

Solvability

3.1

Solvability of the associated Lie algebra . . . . . . . . . . . .

3.2

Solvability of the group of units . . . . . . . . . . . . . . . . .

3.3

The theorem of Sophus Lie and Borel subalgebras . . . . . . .

3.4

Standard examples . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Group algebras . . . . . . . . . . . . . . . . . . . . . .

3.4.2

The algebras of upper and lower triangular matrices .

3.4.3

Solomon algebras in characteristic zero . . . . . . . . .

3.4.4

Solomon-Tits algebras . . . . . . . . . . . . . . . . . .

3.5

Open-ended questions and exercises

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

Carter subgroups of the group of units of solvable associative

algebras

4.1

The pendant within the associated Lie algebra: Cartan sub-

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

The determination of the Carter subgroups by Thorsten Bauer 60

4.3

Further connections in the case of finite fields . . . . . . . . .

4.4

Standard examples . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1

Group algebras . . . . . . . . . . . . . . . . . . . . . .

4.4.2

The algebras of upper and lower triangular matrices .

4.4.3

Solomon algebras in characteristic zero . . . . . . . . .

4.4.4

Solomon-Tits algebras . . . . . . . . . . . . . . . . . .

4.5

Open-ended questions and exercises

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

The Fitting subgroup of the group of units of a solvable

associative algebra

5.1

The pendant within the associated Lie algebra: the nilradical

5.2

The partner within the group of units: the Fitting subgroup .

5.3

Standard examples . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

The algebras of upper and lower triangular matrices .

5.3.2

Solomon algebras in characteristic zero . . . . . . . . .

5.3.3

Solomon-Tits algebras . . . . . . . . . . . . . . . . . .

5.3.4

Group algebras . . . . . . . . . . . . . . . . . . . . . .

5.4

Open-ended questions and exercises

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

Maximal nilpotency in Lie algebras associated to solvable

associative algebras

6.1

Associativity

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

6.2

Manifold centralizers . . . . . . . . . . . . . . . . . . . . . . .

6.3

Futile algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1

Futility and radical complements . . . . . . . . . . . . 103

6.3.2

Futility and unital subalgebras . . . . . . . . . . . . . 110

6.4

Finiteness of the number of isomorphism classes . . . . . . . . 114

6.5

Cardinalities

. . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.6

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.8

Open-ended questions and exercises

. . . . . . . . . . . . . . 124

A correspondence theorem between maximal nilpotent sub-

groups and Lie subalgebras

131

7.1

The correspondence theorem

. . . . . . . . . . . . . . . . . . 131

7.2

Open-ended questions and exercises

. . . . . . . . . . . . . . 135

Maximal nilpotency in unit groups of solvable associative

algebras

137

8.1

A direct decomposition . . . . . . . . . . . . . . . . . . . . . . 137

8.2

Manifold centralizers . . . . . . . . . . . . . . . . . . . . . . . 138

8.3

Finiteness of the number of isomorphism classes . . . . . . . . 148

8.4

Cardinalities

. . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.5

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.7

Open-ended questions and exercises

. . . . . . . . . . . . . . 157

Fischer subgroups, nilpotent projectors and injectors

163

9.1

Fischer subgroups

. . . . . . . . . . . . . . . . . . . . . . . . 163

9.2

The pendant for Lie algebras: Fischer subalgebras

. . . . . . 164

9.3

Nilpotent projectors . . . . . . . . . . . . . . . . . . . . . . . 165

9.4

The pendant for Lie algebras: nilpotent Lie projectors . . . . 166

9.5

Nilpotent injectors . . . . . . . . . . . . . . . . . . . . . . . . 167

9.6

The pendant for Lie algebras: nilpotent Lie injectors . . . . . 168

9.7

Open-ended questions and exercises

. . . . . . . . . . . . . . 170

10 Outlook on series III

173

List of tables

175

List of figures

177

Bibliography

177

Index

184

Introduction

Within series I we have focussed on the following two main topics: the de-

termination of the Cartan subalgebras and of the nilradical of the associated

Lie algebra A

based on a finite-dimensional associative unitary algebra A.

Both Lie substructures are maximal nilpotent in A

: Cartan subalgebras

with respect to the subalgebra lattice and the nilradical with respect to the

ideal lattice of A

. If the factor algebra by the nilradical of A is separable,

then by using the theorem of Wedderburn-Malcev a radical complement

T of rad(A) in A exists. Based on this radical complement we were able to

determine within series I the Cartan subalgebras and the nilradical of A

for several classes of algebras A. In particular, if A is solvable (which is the

case of A/rad(A) and T being commutative) we have proven that the cen-

tralizers of the radical complements denoted by C

(T ) are exactly the

Cartan subalgebras of A

. This results was proven originally by Thorsten

Bauer within his dissertation [4]. In particular, all Cartan subalgebras of

are associative subalgebras of A. The theorem of Wedderburn-Malcev

is used to prove further that all Cartan subalgebras of A

are conjugated

under the group 1 + rad(A). If we focus on the central part of T in A

which is Z(A)

T we have derived in series I additionally that this part is

separable and that the nilradical of A

is the inner direct sum of rad(A) and

Z(A)

T . Cartan subalgebras are maximal Lie nilpotent subalgebras. If A is

solvable, then the nilradical of A

is a maximal Lie nilpotent subalgebra, too.

Within this series we will enhance this theory of maximal nilpotent sub-

algebras of A

in the solvable case of A further. The following questions

are the guidelines of this series related to the associated Lie algebra A

and

also to the group of units E(A) of A:

· In what way can we determine all maximal nilpotent Lie subalgebras of

· Does a special or extremal position of the nilradical and the Cartan sub-

algebras exist among all maximal nilpotent Lie subalgebras of A

· In what way can we determine the Carter subgroups and the Fitting sub-

group of E(A)? Is the Fitting subgroup a maximal nilpotent subgroup?

· In what way can we determine all maximal nilpotent subgroups of E(A)?

· Does a special or extremal position of the Fitting subgroup and the Carter

subgroups exist among all maximal nilpotent subgroups of E(A)?

· Do structural connections exist between maximal nilpotent subalgebras of

and maximal nilpotent subgroups of E(A)?

The intention of chapter 1 is to summarize special structures like group al-

gebra, the Solomon algebra or the Solomon Tits algebra. These algebras are

used to visualize the results within this work and to guide the reader within

the exercises to a deeper insight of the proven results.

For the analysis of structural connections between maximal nilpotent sub-

groups and Lie subalgebras we will use the main result of chapter 2 frequently

in this work: the theorem of Xiankun Du proven in 1992 based on radical

algebras comprised that the upper central chain of the associated Lie algebra

coincide with the upper central chain of the quasi regular group or also

called star or circle group (which is a generalization of the group of units)

in every step. In particular, the class of nilpotency of both structures

is identical. This result was conjectured by Stephen Arthur Jennings 40

years ago and partly proven by Hartmut Laue in the eighties. Oftentimes,

it is simpler to do calculations in the Lie algebra and not within the circle

group. For example, radicals of associative algebras are radical algebras. In

the context of maximal nilpotent substructures we use the result to derive a

connection between the nilpotency classes of maximal nilpotent Lie subalge-

bras and maximal nilpotent subgroups. As an excursus at the end of chapter

2 we derive another application of the theorem of Xiankun Du. If we focus

on the upper central chain of the circle group of a radical algebra and here

on the factor groups of the (n + 1)-th modulo the n-th center, then these

factor groups are by definition of the upper central chain abelian groups.

In the case of a radical algebra based on a field of positive characteristic p

we can derive by using the theorem of Xiankun Du that these factor

groups are indeed of exponent p. Applied to the group algebra for which

Adalbert Bovdi has published this result the reader may prove this result

within the exercises and experience the transfer of group theoretic questions

to Lie theory.

As aforementioned, the guidelines of this work are connected to solvable

associative algebras. The main focus will be to analyze structural proper-

ties of and connections between the associative and the associated Lie as

well as the derived group structure in form of the group of units concerning

maximal nilpotency. For the solvability itself a connection between these

three structures is existing: we will prove within chapter 3 that the solv-

ability for the associative algebra, its associated Lie algebra and its group of

units based on a finite-dimensional associative unitary K-algebra (for a field

K possessing at least 5 elements and char(K) = 2) are equivalent. This

result was one incentive for our guidelines. As an excursus we focus at the

end of chapter 3 on a connection between maximal solvable Lie subalgebras

and maximal solvable subgroups: the so-called Borel subalgebras of A

which are maximal solvable Lie subalgebras are indeed associative unital

subalgebras of A based for fields of characteristic zero. For proving this, we

need a theorem of Sophus Lie and a result of Hartmut Laue concerning the

associative algebra span. The group of units of the Borel subalgebras are

solvable groups. Unfortunately, the proof that they are maximal solvable

subgroups which are so-called Borel subgroups was not possible to per-

form. But we could prove that each Borel subalgebra leads to a new group of

units. The reason is that the K-space generated by the group of units is the

whole algebra. This approach creating the group of units and the K-space

generated by them will often be useful within this work for describing and

analyzing the connections between subalgebras and subgroups.

Thorsten Bauer has already analyzed one guideline of this work within his

dissertation [4]: the determination of the Carter subgroups of the group of

units of an unital finite-dimensional associative solvable algebra possessing

a separable factor algebra by its nilradical. He has proven that the Carter

subgroups for a field possessing at least three elements are exactly the

group of units of the Cartan subalgebras of the associated Lie algebra. The

assumption for the field is necessary to ensure that the algebra is generated

by its group of units. Thus, the result of Thorsten Bauer can be reformulated

as follows: the K-space generated by the Carter subgroups are exactly the

Cartan subalgebras. Again, the concept of creating the group of units

and creating the

K-space generated by the group of units arise.

Within the article [5] of Thorsten Bauer and Salvatore Siciliano concerning

the determination of the Carter subgroups a result is proven which will be

of significant importance later on in this work, too: the K-space generated

of a nilpotent subgroup based on a finite-dimensional associative solvable

K-algebra is Lie nilpotent.

The phenomenon of connecting Cartan subalgebras and Carter subgroups

arise for the nilradical and the Fitting subgroup, too. We will prove within

chapter 5 that both structures are connected via creating the group of units

and the creating the K-space based on the group of units. The result of

Thorsten Bauer and Salvatore Siciliano concerning the K-space generated

by a nilpotent subgroup will be of significant importance for proving this

connection.

The previous chapters have focussed on special and prominent examples

of maximal nilpotent substructures within the group of units and the as-

sociated Lie algebra. In this chapter we analyze more generally the con-

struction, determination and characterization of all maximal nilpotent Lie

subalgebras. In a first step we prove in analogy to the Borel subalge-

bras stated earlier (but based on a completely different argumentation)

that maximal Lie nilpotent subalgebras are unital associative subalgebras.

Thus, we are able to use results of series I concerning these special associa-

tive subalgebras: the inner structure of these associative subalgebras M of

A is presentable as the inner direct sum of its nilradical rad(M ) (which is

contained in rad(A) by using the solvability of A) and the unique and cen-

tral radical complement V SEP (M ) consisting of fully separable elements:

M = rad(M )

V SEP (M). The theorem of Wedderburn-Malcev lets us

derive that V SEP (M ) is contained in a radical complement T of rad(A) in

A. Based on the inner structure of M and the radical complement T we can

prove that a Lie nilpotent associative subalgebra M is maximal Lie nilpotent

if and only if the centralizer conditions C

rad

(A)

(V SEP (M )) = rad(M ) and

(rad(M )) = V SEP (M ) are valid. A simple but remarkable consequence

is that maximal nilpotent Lie subalgebras satisfy the double-centralizer con-

ditions C

rad

(A)

(rad(M ))) = rad(M ) and C

rad

(A)

(V SEP (M ))) =

V SEP (M ). For determining all maximal Lie nilpotent subalgebras we use

these centralizer and double-centralizer properties: we start with an unital

subalgebra C of T and calculate the double-centralizer C

rad

(A)

(C)). We

proceed by calculating the double-centralizer of the double-centralizer again

and again. This process must be stable of finite many steps because of the

finite dimension of A. If the process is stable, then the resulting subalgebra

C in T combined with the direct summand C

rad

(A)

( ^

C) is maximal Lie nilpo-

tent. The dual process beginning with a subalgebra of rad(A) leads also

to maximal Lie nilpotent subalgebras, but not to new ones. A natural ques-

tion is to determine the number of steps after which the double-centralizing

is stable. The answer is simple: not from the beginning but after the first

double-centralizing. Thus, we have to use the double-centralizing on the

lattice of unital subalgebras of T resp. the lattice of subalgebras of rad(A)

once and construct as already described all maximal Lie nilpotent subalge-

bras. The nilradical and the Cartan subalgebras have an extremal position

among all maximal Lie nilpotent subalgebras. The component of the nilrad-

ical resp. Cartan subalgebras in T is central in A resp. the whole radical

complement. Within the nilradical its extremely large resp. small (and

therefore dual). For all other maximal nilpotent subalgebras the part in T

resp. rad(A) is situated between these two values. By using another radical

complement only isomorphic copies of maximal nilpotent Lie subalgebras

arise (based on the theorem of Wedderburn-Malcev). Hence, all isomorphic

classes of maximal nilpotent subalgebras can be bounded by the number of

unital subalgebras of a fixed radical complement T . This number is finite

because T is separable and commutative: T is a so-called futile algebra.

We prove this statement within a separate section and estimate this num-

ber by the upper bound B(dim

(T )) which are the so-called Bell numbers.

In chapter 7 we present a bijective connection between maximal nilpotent Lie

subalgebras and maximal nilpotent subgroups. It becomes apparent that

as already stated for the Cartan subalgebras and the Carter subgroups resp.

the nilradical and the Fitting subgroup there is a general connection be-

tween maximal nilpotent substructures: the group of units of maximal Lie

nilpotent subalgebra (which is indeed an unital associative subalgebra) is a

maximal nilpotent subgroup and the K-space generated by a maximal nilpo-

tent subgroup is a maximal nilpotent Lie subalgebra (Here we will use the

already mentioned result of T. Bauer and S. Siciliano again.). In addition,

this connection is bijective: the functions E(

·) creating the group of units

and

creating the K-space generated by the group of units are

inverse to each other. By using the theorem of Xiankun Du we derive the

more deeper insight that the classes of nilpotency of two connected maxi-

mal nilpotent substructures are identical. The results presented in chapter 6

can be transferred by using this connection to maximal nilpotent subgroups

which is the content of chapter 8.

Thus, in analogy to chapter 6 we describe within chapter 8:

· the inner structure of the maximal nilpotent subgroups as direct products

of unipotent and central, fully separable elements,

· the characterization of maximal nilpotent subgroups by manifold central-

izers,

· the determination of all maximal nilpotent subgroups by double-centralizing

all subgroups of E(T ) and combining the centralized unipotent part to it,

· the dual determination of all maximal nilpotent subgroups by double-

centralizing all subgroups of 1 + rad(A) and combining the centralized

fully-separable part to it,

· the extremal position of the Carter subgroups and the Fitting subgroup

among all maximal nilpotent subgroups of E(A),

· the behavior of the maximal nilpotent subgroups by changing the radical

complement and

· the finiteness of the number of isomorphic classes of maximal nilpotent

subgroups which can be bonded by Bell numbers.

The last chapter is dedicated to other prominent maximal nilpotent sub-

groups which are the so-called nilpotent injectors, nilpotent projectors and

the Fischer subgroups. We will prove that they coincide with the Carter

subgroups resp. the Fitting subgroup. Afterwards these prominent maxi-

mal nilpotent substructures are also defined for Lie algebras (nilpotent Lie

injectors, nilpotent Lie projectors and Fischer subalgebras), and we will

prove that they coincide with the Cartan subalgebras and the Lie nilradi-

cal. Posthumous, we derive the result that the group of units of the Fischer

subalgebras, the nilpotent Lie projectors and the nilpotent Lie injectors are

exactly the Fischer subgroups, the nilpotent projectors and injectors. Vice

versa, the K-space generated by them is exactly the Fischer subalgebras,

the nilpotent Lie projectors and the nilpotent Lie injectors.

As stated earlier we illustrate our results by using standard examples. These

are mainly group algebras, the algebras of upper and lower triangular matri-

ces over a field, the Solomon algebra in characteristic zero and the Solomon-

Tits algebra. Within the first chapters these examples are investigated on a

high detailed level, but within the last four chapters we use them only ex-

emplary. A detailed analysis needs a deeper insight, and the author decided

not to disconnect the reader from the general theory too far, but to do this

analysis in series III.

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 can experience 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 diagrams to visualize

the main results of this work.

Excercise 1

What are the answers for the guidelines of this work?

Chapter 1

Standard examples, symbols

and notations

This chapter has a preliminary function by summarizing those monoids,

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

They will be used as examples for the proven theorems as well as for the

exercises in which the reader shall apply the general results to them. In

addition, we list the symbols and notations used in this series.

Sets and numbers

Let A, B, T be sets and i, n, k

. We use the following symbols linked to

set and number theoretical topics:

· - the empty set

· A B - intersection of A, B

· A B - union of A, B

· A \ B - difference of A, B

· A × B - cartesian product of A, B

· P (A) - power set of A

· n - the first n natural numbers

· n

- the first n natural numbers including zero

· p(n) - the number of partitions of n

· n! - factorial of n

· | A | - number of elements of A

· S(n, k) - the kth-Stirling number of n

· B(n) - the nth-Bell number

- n choose k

- the set of subsets of order i of T

· - equivalent

· mod - modulo.

Groups and monoids

Let p

P, n N, N be a set, M a monoid, G a group, N a normal subgroup

of G, a, b

G, U, V G, A an associative unitary K-algebra, c K and

q a prime power number. The following monoids, groups and symbols are

used:

· st(G) - solvable class of G

· cl(G) - nilpotency class of G

· (Z

(G))

- ascending central chain of G

· (G

(n)

)

- descending central chain of G

· (

(G))

, (G

[n]

)

- commutator or derived series of G

· ,

- star or circle composition

· [a, b] - commutator of a, b

· [U, V ] - commutator of U, V

· a

-1

- inverse element of a

· a

- conjugate element of a with b

· 1 + rad(A) - normalized units

· C

(V ) - centralizer of V in U

· N

(V ) - normalizer of V in U

· G - class of groups

· G - derived subgroup of G

· (F

(G))

-Fitting series of G

· G/N - factor group of G by N

· SU

(A)

- set of solvable subgroups of E(A)

- ordered set partitions

- special composition on

· exp(G) - exponent of G

· U

- unipotent factor of U

· U

- fully-separable factor of U

· O

(G) - p-core of G

· G(U), U

(T )

- subgroups of E(T ) possessing the double-centralizer prop-

erty

· U

- order of

G(U)

· G

- subgroups of 1 + rad(A) possessing the double-centralizer property

· G

U T

- maximal nilpotent subgroups of E(A) linked to E(T )

· G

- maximal nilpotent subgroups E(A)

· E(A)

- maximal nilpotent subgroups of E(A)

· a

- quasiregular inverse element

· x

(a)

- quasiregular conjugate element

· N - natural numbers

· N

- natural numbers containing zero

· 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)

· 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

- semidirect product of groups.

General algebra constructions

Let A be an algebra, S

A, S finite, K a field, G a group, I an ideal,

M a monoid, n

N, S

and T

A. The following general algebra

constructions and symbols are used:

· dim

(A) - K-dimension of A

· A - class of associative algebras

· A

- class of associative unitary algebras

· st(A) - solvable class of A

· cl(A) - class of nilpotency of A

· D() - defect class of

· a

nil

- nilpotent part of the generalized Jordan decomposition

· a

vsep

- fully-separable part of the generalized Jordan decomposition

· S - sum of all elements of S

· A - derivation of A

· A

- nth associative power of A

· C

(T ) - centralizer of T in A

· a

- sum of a

, i

T , T n, a

· · · a

· - tensor product of algebras

· × - direct products of algebras

· - direct sum of algebras

- semidirect product 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

- 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 an unit to A

· A

or A

- opposite or inverse algebra of A

· (A × A; ) - zero extension of A

· gl(n, K) - identical to (K

×n

)

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

· Aug(KG) - augmentation ideal of KG.

Commutative associative algebras

Let n

N and K be a field. The following commutative associative algebras

and symbols are used:

· D(n, K) the set of diagonal matrices in K

×n

· e

· · · , e

- primitive orthogonal idempotents in D(n, K)

· e

- sum of all e

, i

T n

· K

- n-tuple space over the field K

· K[a] - smallest subalgebra containing a and K

· V SEP (·) - set of fully-separable elements

· char(K) - characteristic of the field K

· n

- is identical to

· Z - the set of integers

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

· K[t

, . . . , t

] - polynomial algebra over K in the variables t

, . . . , t

Fields and skew fields

Let p be a prime number, n

N and (K; L) a field extension. The following

fields, skew fields and symbols are used:

· Q - rational number field

· R - real number field

· C - complex number field

· H - real quaternion algebra

· GF (p

) - finite field possessing 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

· K(t) - field of fractions over K in one variable t

· K(t

, . . . , t

) - field of fractions over K in the variables t

, . . . , t

(Central-) simple associative algebras

Let K be a field, D a division algebra and n

N. The following (central-)

simple associative algebras and symbols are used:

· e

- base matrices of K

×n

· det - determinant function

· tr - trace function

· K

×n

- n

× n-matrices over K

· D

×n

- n

× n-matrices over D

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

Semisimple associative algebras

The following semisimple associative algebras are used:

· right artian associative algebras A for which rad(A) = 0 is valid

· × - direct products of simple algebras

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

artian algebra.

Nilpotent associative algebras

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

N and G

a group. The following nilpotent associative algebras are used:

· rad(A) - nilradical of A

· J(A) - Jacobson radical of A

· s

u,n

- algebra of strict lower triangular matrices of K

×n

· s

o,n

- algebra of strict upper triangular matrices of K

×n

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

Solvable associative algebras

Let n

N and K be a field. The following solvable associative algebras are

used:

· K

- Solomon-Tits algebra (see e.g. [77])

· D

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

u,n

- algebra of lower triangular matrices of K

×n

o,n

- algebra of upper triangular matrices of K

×n

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

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

Lie algebras

Let n

N, K be a field, L a Lie algebra, H L, x L and A an associative

algebra. The following Lie algebras and symbols are used:

· - associated Lie composition

· A

- associated Lie algebra

· ad(x) - adjoint representation x

· L

(ad(H)) - Fitting null component of L with resp. to ad(H)

· B

- Borel subalgebras of A

· st(L) - solvable class of L

· cl(L) - class of nilpotency of L

· (Z

(L))

- ascending central chain of L

· (L

(n)

)

- descending central chain of L

· (L

[n]

)

- descending commutator or derived series of L

· L - class of Lie algebras

· nil(L) - nilradical of L

· M(T ), A

- subalgebras of T possessing the double-centralizer property

· m

- order of

M(T )

· A

- subalgebras of rad(A) possessing the double-centralizer property

· A

M T

- maximal nilpotent Lie subalgebras of A

linked to T

· A

- maximal nilpotent Lie subalgebras of A

· L - derivation of L

· t

, u

- unit group and K-space creation

· J

(I) - special sequence of subalgebras and subgroups in rad(A)

· T

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 applica-

tions 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 subalge-

bras and subgroups.

2.1

Radical algebras and central chains

Definition and remark 1

Let A be an associative K-algebra. By rad(A)

resp. J (A) we symbolize the nilradical resp. the Jacobson radical of A. If

A is right or left artian, then Gottfried K¨

othe has proven that both radicals

coincide and are nilpotent.

The associative nilpotency class is symbolized

by cl(A). A is called radical algebra, if A = J (A) is valid. A is a nil algebra,

if A = rad(A) is true. rad(A) and J (A) are radical algebras.

The Lie algebra associated to A is symbolized by A

equipped with the

multiplication a

b := ab - ba for all a, b A. The upper central chain

of A

is defined recursively by Z

) :=

{0} and Z

) :=

{z | z

a A : z a Z

-1

)

} for all n N. A is Lie nilpotent, if a nat-

ural number n

N exists such that A = Z

) is valid. The minimal n

possessing this property is called the class of nilpotency of A

symbolized

by cl(A

). The lower central chain is defined recursively by (A

)

(0)

:= A

and (A

)

(n)

:= (A

)

(n-1)

A for all n N. A is Lie nilpotent if and only if

the lower central chains reaches the null space after finite many steps. The

minimal number of these steps is again the class of nilpotency. Sufficient

and often used within applications for the Lie nilpotency is the associative

nilpotency, e.g. for rad(A) if A is right artian. In [40] it is proven by Hart-

mut Laue that all members of the upper Lie central chain are associative

subalgebras.

For all a, b

A we define (as original defined by Bartel Leendert van der

Waerden) a

b := a + b + ab, and we call

the circle or star composition

on A. A is a monoid based on the composition

, and 0 is its unit ele-

ment. The group of units based on this monoid is called the star group or

quasi regular group of A symbolized by Q(A). The elements of Q(A) are

Gottfried Maria Hugo K¨

othe (born 25 December 1905 in Graz; died 30 April 1989

in Frankfurt) was an Austrian mathematician working in abstract algebra and functional

analysis. In 1923 K¨

othe enrolled in the University of Graz. He started studying chemistry,

but switched to mathematics a year later after meeting the philosopher Alfred Kastil. In

1927 he submitted his thesis Contributions to Finslers foundations of set theory and was

awarded a doctorate. After spending a year in Z¨

urich working with Paul Finsler, K¨

othe

received a fellowship to visit the University of G¨

ottingen, where he attended the lectures of

Emmy Noether and Bartel van der Waerden on the emerging subject of abstract algebra.

He began working in ring theory and in 1930 published the K¨

othe conjecture stating that a

sum of two left nil ideals in an arbitrary ring is a nil ideal. By a recommendation of Emmy

Noether, he was appointed an assistant of Otto Toeplitz in Bonn University in 1929 to

1930. During this time he began transition to functional analysis. He continued scientific

collaboration with Toeplitz for several years afterward. K¨

othes Habilitationsschrift Skew

fields of infinite rank over the center, was accepted in 1931. He became Privatdozent at

University of M¨

unster under Heinrich Behnke. During World War II he was involved in

coding work. In 1946 he was appointed the director of the Mathematics Institute at the

University of Mainz and he served as a dean (1948 to 1950) and a rector of the university

(1954 to 1956). In 1957 he became the founding director of the Institute for Applied

Mathematics at the University of Heidelberg and served as a rector of the university (1960

to 1961). K¨

othes best known work has been in the theory of topological vector spaces.

In 1960, volume 1 of his seminal monograph topological vector spaces was published (the

second edition was translated into English in 1969). It was not until 1979 that volume

2 appeared, this time written in English. He also made contributions to the theory of

lattices.

called quasi regular, the inverse of a

Q(A) will be denoted by a

, and the

conjugated to a by b is symbolized by a

(b)

:= b

b. Every nilpotent

element is quasi regular, and thus for every nil associative algebra A the

identity A = Q(A) = rad(A) is valid. If Q(A) = A is true, then we use the

symbol A for Q(A). rad(A) is a group based on . The Jacobson radical

is a group based on , too. The upper central chain of Q(A) is recursively

defined by Y

(Q(A)) :=

{0} and Y

(Q(A)) :=

{y | y Q(A), a Q(A) :

[y, a]

-1

(Q(A))

} for all n N. The commutator [y, a] is defined by

(a)

for all y, a

Q(A). Q(A) is nilpotent, if the upper central chain

of Q(A) reaches Q(A) in finite many steps. The minimal number of these

steps is called the class of nilpotency of Q(A) symbolized by cl(Q(A)).

The lower central chain of Q(A) is defined recursively by Q(A)

(0)

= Q(A)

and Q(A)

(n)

:= [Q(A)

(n-1)

, Q(A)] for all n

N. Q(A) is nilpotent if and

only if the lower central chain reaches the trivial subgroup after finite many

steps. The minimal number of these steps is again the class of nilpotency.

Within the literature

(G) is used for the k-th member of the lower central

chain of a group G.

The group Q(A) acts per conjugation on the additive group of A. For this op-

eration another central chain can be defined recursively by X

(Q(A)) :=

{0}

and X

(Q(A)) :=

{x | x Q(A), a Q(A) : x

(a)

- x X

-1

(Q(A))

} for

all n

In addition, we define for a radical algebra A inductively W

(A) :=

{0}

and W

(A) :=

{w | w A, a A : a

(w)

- a W

-1

(A)

} for all n N.

Now we demonstrate how the nilpotency and the upper central chains of

A and A

are connected for radical algebras.

2.2

Results of Stephen Arthur Jennings, Hartmut

Laue and Xiankun Du

A first insight about a nilpotent connection between the group A the Lie

algebra A

was proven by Stephen Arthur Jennings in 1955 (see [30]) for

radical algebras:

Theorem 1

(Jennings, 1955) Let A be a radical algebra. A is nilpotent if

and only if A

is nilpotent.

Stephen Arthur Jennings

conjectured for radical algebras A that the nilpo-

tency classes of A and A

coincide. In 1984 Hartmut Laue has proven some

Stephen Arthur Jennings (May 11, 1915 to February 2, 1979) was a mathematician

who made significant breakthroughs in the study of modular representation theory (1941).

His advisor was Richard Brauer, and his student Rimhak Ree discovered two infinite

aspects of this conjecture in [41]:

Theorem 2

(Laue, 1984) Let A be an associative K-algebra. The following

statements are valid:

(i) If A = Q(A) and A is Lie-nilpotent, then A is nilpotent and cl(A )

cl(A

) is valid.

(ii) Let A be nil, K a field and A

nilpotent. Z

) = Y

(A ) is valid for

all n

(iii) If Q(A) = A is valid, then Z

) = Y

(A ) is true.

Hartmut Laue conjectured that for a radical algebra A the ascending central

chains of A and A

are identical in every step. Xiankun Du has proven

this conjecture in 1992 (see [14]). Thus, the conjecture of Stephen Arthur

Jennings was proven, too.

Theorem 3

(Du, 1992) Let A be a radical algebra. For all n

the

identity Z

) = Y

(A ) is valid.

In particular, the conjecture of Jennings is true: A is nilpotent if and only

if A

is nilpotent. If A or A

is nilpotent, then cl(A

) = cl(A ) is valid.

Further analysis performed by Karsten Scholz and Hartmut Laue (see [40]

and [58]) yields to the following main theorem:

Main theorem 1

(Laue, Scholz, 1996) Let A be a radical algebra. For all

the identity Z

) = Y

(A ) = X

(A) = W

(A) is valid.

In particular, every member of these four central chains are additive closed,

associative subalgebras of A, Lie ideals of A

and normal subgroups of A ,

and they are invariant under all automorphism of A and A

and under all

A -module automorphism of the additive group of A.

series of finite simple groups known as the Ree groups. He was an editor of Mathematics

Magazine and an acting president of the University of Victoria. Stephen was born in

Walthamstow, England and immigrated to Canada with his family in 1928. He had been

receiving scholarships in England and these were transferred to Canada. He finished his

high school education in Toronto and in September 1932 he went to University College

in Toronto. In 1939 he received his PHD from the University of Toronto. He married

Dorothy Freeda Rintoul (University of Western University - B.A. and University of Toronto

M.A. Psychology) in 1939. On November 14, 1942 he became a member of the Zeta Psi

fraternity. When he was made a professor, he was the youngest professor ever appointed in

Canada. In 1944 Stephen was appointed 2nd Lieutenant (Paymaster) in Canadian Army.

While in Vancouver, teaching at the University of British Columbia, Stephen and Dorothy

established their family, two children, Judith Anne Jennings and James Stephen Jennings.

Stephen was Dean of Graduate Studies at the University of Victoria and the Head of the

Math Department there.

We close this section by proving an application based on the theorem of

Xiankun Du concerning the factor groups along the ascending central chain

of the circle group of a radical algebra in positive characteristic p: they

are elementary-p-abelian. The proof demonstrates the transfer of group-

theoretic topics to questions within Lie algebras.

Let T be a set and i

. By

we denote the set of all subsets of

T possessing exactly i elements. With respect to the natural order on

| T |

and T exactly one monotone bijection from

| T | onto T exists, and we sym-

bolize this map by

The next proposition is straightforward to be proven by an induction ar-

gument and therefor maybe handled by the reader as an exercise:

Proposition 1

Let A be an associative K-algebra, n

N and x

, . . . , x

A. The statement x

· · · x

=1 T

(

)

. . . x

)

is valid.

Corollary 1

Let A be an associative K-algebra, n

N and a A. The

following statements are valid:

(i) a

· · · a

-fold

i K

(ii) If p is a prime number and char(K) = p is valid, then

· · · a

-fold

= a

)

is true.

Proof

ad(i): By using proposition 1 we derive

· · · a

-fold

=1 T

(

)

i K

ad(ii): For all i

- 1 it is well-known (see e.g. [18])that the prime

number p = char(K) is a divisor of

. Thus part (ii) is a consequence of

part (i).

The proof of the following proposition is straightforward to be executed

and thus maybe done as an exercise by the reader:

Proposition 2

If A is an associative K-algebra and x, y

A, the following

statements are valid:

(i)

n N : x y · · · y

-fold

k K

(

-1

)

-k

(ii) If p is a prime number and char(K) = p is valid, then the following

identity is true: x

y · · · y

-fold

= x

Theorem 4

Let p be a prime number, K a field, char(K) = p and A a

K-radical algebra. For all n

N the identity exp(Z

)/Z

)) = p is

valid.

Proof

By using the theorem of Xiankun Du (see theorem 3) for all n

the sets Z

) and Z

) are identical. Based on corollary 1 for all a

the identity a

· · · a

-fold

= a

is valid. The theorem is proven by using propo-

sition 2.

We remark that the exponent of the center is not p in general. The structure

of the center is analyzed by the author within [76] for a p-group and a (finite)

field of characteristic p. Within this context theorem 4 is applicable, and,

in addition, the theorem of Xiankun Du can be used to calculate the class

of Lie nilpotency for the nilradical. Some examples are included within the

exercises in which the reader can experience the connection between group

and Lie theory.

In the next chapter we apply the latter results to our standard examples.

2.3

Standard examples

2.3.1

The algebras of upper and lower triangular matrices

Let K be a field and n

N. The algebras of upper and lower triangular

matrices are examples of solvable associative K-algebras possessing a factor

algebra by its nilradical which is isomorphic to a n-fold product of the base

field. In addition, it can be proven that every radical complement is self-

centralizing.

The algebra of lower triangular matrices of K

×n

- symbolized by

u,n

possesses as nilradical the subalgebra of strict lower triangular matrices -

symbolized by s

u,n

. The nilradical is of dimension

-1

i =

- 1)n. The

set of diagonal matrices D(n, K) is a radical complement of dimension n,

and it is self-centralizing.

The algebra of upper triangular matrices of K

×n

- symbolized by

u,n

possesses as nilradical the subalgebra of strict upper triangular matrices -

symbolized by s

u,n

. The nilradical is of dimension

-1

i =

-1)n. Again,

the set of diagonal matrices D(n, K) is a radical complement.

It is well-known (see e.g. [38]) that for the associative nilpotency classes

the following identities are valid: cl(s

o,n

) = cl(s

u,n

) = n.

Within [77] it is proven for an associative algebra possessing a self-centralizing

radical complement and a radical factor algebra isomorphic to a n-fold di-

rect product of the base field that the associative powers of the nilradical are

identical to the members of the descending central chain of the associated

Lie algebra. Thus, the Lie nilpotency class is identical to the associative

nilpotency class of the associative nilradical. By using this result and the

theorem of Xiankun Du (see theorem 3) we derive:

Theorem 5

Let K be a field and n

N. The following identities are valid:

(i) cl(s

o,n

) = cl(s

u,n

) = n

(ii) cl((s

o,n

)

) = cl((s

u,n

)

) = n

(iii) cl((s

o,n

) ) = cl((s

u,n

) ) = n .

2.3.2

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 defined by the K-span of class sums of so-called defect

classes of KS

: if

is valid, then we define D() :=

{i | i > (i+1)}.

Solomons algebra is the K-span of

{

()=D

| D n - 1}. The surprising

insight of Louis Solomon was that the product of two defect class sums is

a K-linear sum of defect class sums. The Solomon algebra is of dimension

-1

, its radical factor algebra of dimension p(n) - the number of parti-

tions of n and is isomorphic to K

(n)

. All radical complements are self-

centralizing. A deep insight in this theory is included in the dissertation of

Thorsten Bauer [4] (especially in chapter 3.)

Within [77] it is proven for an associative algebra possessing a self-centralizing

radical complement and a radical factor algebra isomorphic to a n-fold di-

rect product of the base field that the associative powers of the nilradical

are identical to the members of the descending central chain of the associ-

ated Lie algebra. By using this result we derive by using the theorem of

Xiankun Du (see theorem 3) and a result of Michael D. Atkinson for the

class of nilpotency of rad(D

) (see [1], cl(rad(D

)) = n

- 1) for the classes

of nilpotency:

Theorem 6

Let K be a field of characteristic zero and n

N. The identity

cl(rad(D

)) = cl(rad(D

)

) = cl(rad(D

) ) = n

- 1 is valid.

2.3.3

Solomon-Tits algebras

Let K be a field and n

N. By S(n, k) we denote the so-called Stirling

number of k

. This number is the quantity of all unordered set partitions

of n possessing exactly k subsets of n. In [57] and [77] the Solomon-Tits

algebra K

is presented and analyzed in details by Manfred Schocker and

by the author. K

is defined based on the monoid

which consists of

all ordered set partitions of n. If (P

· · · , P

) and (Q

· · · , Q

) are two of

these ordered partitions, then their product

is defined by

· · · , P

)

· · · , Q

) :=

, P

· · · , P

, P

· · · , P

)

symbolizes that empty sets are deleted from this tuple. Again, K

is an

example of an associative algebra possessing a factor algebra by its nilradical

which is isomorphic to a r-fold product of the base field. In addition, every

radical complement is self-centralizing.

The nilradical of K

is described within chapter 2 in [77]. Its dimen-

sion is dim

(rad(K

)) =

(k!

- 1) S(n, k) (see corollary 8 in [77]). The

factor algebra by the nilradical is of dimension B(n) the so-called nth Bell

number.

Again, by results in [77] the associative powers of the nilradical are identical

to the members of the descending central chain of the associated Lie algebra.

Manfred Schocker (see [57]) has proven that the associative nilpotency class

of the nilradical of K

is exactly n. We conclude by using theorem 3 the

theorem of Xiankun Du:

Theorem 7

Let K be a field and n

N. The following identity is valid:

cl(rad(K

)) = cl(rad(K

)

) = cl(rad(K

) ) = n.

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 al-

gebra 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!

Excercise 5

Let K be a field and n

N. Summarize the classes of nilpo-

tency within theorem 5 for n

20. In addition, calculate the dimension of

the algebra, the nilradical and the factor algebra by the nilradical for n

20.

Are these numbers convergent for n

Excercise 6

Let K be a field and n

N. Summarize the classes of nilpo-

tency within theorem 7 for n

20. In addition, calculate the dimension of

the algebra, the nilradical and the factor algebra by the nilradical for n

20.

Are these numbers convergent for n

Excercise 7

Let K be a field, char(K) = 0 and n

N. Summarize the

classes of nilpotency within theorem 6 for n

20. In addition, calculate

the dimension of the algebra, the nilradical and the factor algebra by the

nilradical for n

20. Are these numbers convergent for n ?

Excercise 8

Let A be an associative K-algebra and c

K. We define

b := a + b + cab. A is equipped with the composition

a monoid

possessing the neutral element 0. If c, d are units in (K;

·), then the monoids

(A;

) and (A;

) are isomorphic. Are these monoids isomorphic to (A; )?

If A = K is a finite field, then find two examples of c, d such that the monoids

(A;

) and (A;

) are not isomorphic.

Excercise 9

If A is an associative unitary K-algebra, then the monoids

(A;

·) and (A; ) are isomorphic. What is the consequence for their groups

of units?

Excercise 10

Determine Q(A) for the following cases of an associative K-

algebra A:

(i) A =

(ii) A is a field.

(iii) A is a division algebra.

(iv) A = K

2×2

(v) A is nilpotent.

(vi) A is nil.

(vii) A is a radical algebra.

(viii) The inverse algebra of A.

(ix) A is a direct product of algebras.

(x) A is unitary.

(xi) A is semidirect decomposition of the nilradical and a radical comple-

ment.

Which of these algebras are radical algebras? On what terms are these alge-

bras radical algebras?

Excercise 11

Let A be an associative K-algebra and z

A. On what terms

are the sets

{x - xz | x A} and A identical? Answer the same question

for the set

{x - zx | x A}!

Excercise 12

Let A be a radical K-algebra and c an unit of K. Prove for

all n

the identity Y

((A; )) = Y

((A;

)). (Tip: theorem of Xiankun

Du and exercise 8)

Excercise 13

Let K be a field and G a finite group. The center of KG is

K-linear generated by the conjugacy class sums of G.

Excercise 14

Let p be a prime number, K a field of characteristic p and

P a p-group. Analyze the following statements:

(i) The nilradical rad(KP ) is the augmentation ideal Aug(KP ) =

{g-1 |

P }

. (Tip: The elements g

- 1 are nilpotent and generate an

ideal. Apply a theorem of Joseph Wedderburn to this result!)

(ii) K1

is a central radical complement.

(iii) (KG)

is nilpotent, and its class of nilpotency is identical to the one

of rad(KG)

(iv) E(KG) is the direct product of 1 + rad(KG) and (K

\ {0})1

(v) What is the quasi-regular group of KG?

(vi) G is contained in 1 + rad(KG).

(vii) If K is finite, then determine the order of 1 + rad(KG).

(viii) Let K be finite. Focus on the factor groups along the ascending central

chain of 1 + rad(KG) and describe the structure of these factor groups.

(ix) In what way is it possible to apply the theorem of Xiankun Du to the

previous part?

Excercise 15

Let K be a finite field of characteristic 2 and G = Q

G = D

. Determine the ascending central chain of rad(KG)

, calculate the

class of nilpotency of 1 + rad(KG) and determine the structure of the center

and the factor groups along the ascending central chains of 1 + rad(KG).

(Tip: exercise 14, theorem of Xiankun Du and theorem 4)

Excercise 16

Prove the following statement:

For all i

- 1 the prime number p = char(K) is a divisor of

What is the importance of this result within this chapter?

Excercise 17

Let A, B be associative radical algebras. Prove that A

× B is

a radical algebra. Apply the theorem of Xiankun Du to A

×B and determine

the ascending central chain and class of nilpotency for (A

×B)

. Is the latter

Lie algebra identical to (A

)

× (B

)? Analyze star groups of direct products

of radical algebras, too.

Excercise 18

(eAe) Let A be an associative K-algebra and e an idempotent

of A. If A is a radical algebra, then eAe is a radical algebra. (Tip: Do a

research in the literature and determine the nilradical of eAe!)

Excercise 19

(zero-extension) Let A be a K-algebra based on a composition

·. On the K-space B := A × A a multiplication

is defined by (a, x)

(b, y) := (ab, ay + xb). True or false: B is a radical algebra if and only if A

is a radical algebra.

Excercise 20

Let K be a field. Determine the ascending central chain of

)

and (s

)

. For K being finite of characteristic p > 0 determine

the structure of the center and the factor groups along the ascending central

chains of these algebras. Is there a conjecture for arbitrary n?

Excercise 21

Let K be a field. Determine the ascending central chain of

)

and (s

)

. For K being finite of characteristic p > 0 determine

the structure of the center and the factor groups along the ascending central

chains of these algebras. Is there a conjecture for arbitrary n?

Excercise 22

Determine the defect classes of S

and S

Calculate all

binary products of defect classes (also the squares). Is it possible to represent

each product as a sum of defect class sums possessing coefficients in

N, Z

Excercise 23

Let A be an associative nilpotent K-algebra. A

and A are

nilpotent, and the identity cl(A)

cl(A

) = cl(A ) is valid. (Tip: bound the

powers of A by the members of the descending central chain, use the theorem

of Xiankun Du)

Excercise 24

Let K be a field and A a finite-dimensional associative K-

algebra possessing a self-centralizing radical complement which is isomor-

phic to a n-fold product of the base field K. The identity cl(rad(A)) =

cl(rad(A)

) = cl(rad(A) ) is valid. (Tip: [77], calculate the descending

central chain, theorem of Xiankun Du)

Excercise 25

Let A be an associative K-algebra, x

, . . . , x

A and n N.

Calculate x

· · ·x

for n

3. For the algebra of real quaternions determine

j and i

Excercise 26

Within exercise 25 calculate x

· · · x

-fold

. What is the result

of this n-fold star product within the quaternion algebra for x

{i, j, k}?

Is the calculation dependent on the base field K?

Excercise 27

Let K := GF (3). Within K

2×2

determine two non-commuting

matrices A, B and calculate A

B, A B B and A B B B. Calculate

the same products by switching A and B. What are the results for GF (2)?

Excercise 28

What results can be deduced, if within exercise 27 the com-

position

is replaced by ?

Excercise 29

What results can be derived by corollary 1, if the composition

is replaced by

? What is the answer if c is an unit (see exercise 8)?

Excercise 30

If A is an associative algebra, then every nilpotent element

is quasi-regular.

Excercise 31

Let A be an associative K-algebra. A can be extended to an

unitary associative K-algebra A

(see e.g. series I or [75] for an exact

definition and construction of A

). Is there a connection between the quasi

regular groups of A and A

and the group of units of A

? Is A a radical

algebra if and only if A

is a radical algebra?

Excercise 32

Prove that for a finite-dimensional associative unitary al-

gebra every element is an unit or a zero divisor (see [77]). What is the

definition of a zero divisor?

Excercise 33

By using exercise 31 try to transfer the results of exercise 32

to non-unitary algebras.

Excercise 34

Let p be a prime number, K be a finite field of characteristic

p and A be an associative finite-dimensional K-algebra. Let us focus on the

associative powers of the nilradical rad(A) which is the series (rad(A)

)

Prove the following statements

(i) There are only finite many associative powers of the radical.

(ii) For all i

N the factor algebra rad(A)

/rad(A)

is a zero-algebra.

(iii) For all i

N the group (rad(A)

/rad(A)

) is elementary-p-abelian.

Excercise 35

In view of exercise 34 find the decomposition into cyclic

groups of order p for the star groups (rad(A)

/rad(A)

)

for all i

with respect to the following algebras based on a finite field of characteristic

p > 0:

(i) The algebra of upper triangular matrices.

(ii) The algebra of lower triangular matrices.

(iii) The Solomon-Tits algebra (see the article of Manfred Schocker [57] for

the associative powers of the nilradical).

(iv) Let G be a cyclic p-group generated by z. Then KG decomposes into

rad(KG) and K1

. The radical is exactly KG(1

- z) (see e.g. [76]).

Excercise 36

Prove theorem 4 in details!

Excercise 37

Why is within the proof of theorem 4 the statement not valid

for the center itself ?

Chapter 3

Solvability

Within this chapter we analyze for a finite-dimensional associative unitary

K-algebra the connection of solvability between the associative algebra, its

group of units and their associated Lie algebra. We will prove that for ade-

quate fields these three conditions are equivalent.

An analogue to the theorem of Xiankun Du for the class of solvability is

not known by the author. But within our examples a close connection for

the class of solvability between the associative algebra, its group of units

and their associated Lie algebra is obtained.

3.1

Solvability of the associated Lie algebra

Definition and remark 2

Let A be an associative K-algebra. A is called

solvable, if A/rad(A) is commutative.

For a Lie algebra L resp. for a group G let (L

[n]

)

resp. (G

[n]

)

the

descending sequence of derived subalgebras resp. subgroups (also called

commutator subalgebras resp. subgroups) of L resp. G. L resp. G is called

solvable, if the sequence of derivations is reaching the trivial subalgebra

resp. subgroup after finite many steps. The minimal number of these steps

is called the class of solvability denoted by st(L) resp. st(G).

For an associative algebra A the class of solvability can be defined, too.

For this, we focus on the ideal of A generated by A

A. This ideal is called

the derived subalgebra of A and is denoted by A

[1]

or by A . For every

N we define the nth derived subalgebra or commutator ideal of A by

[n+1]

:= (A

[n]

) . A is solvable as associative algebra if and only if an ele-

ment m

N exists such that the ideal of A generated by A

[m]

is zero. The

minimal m possessing this condition is called the class of solvability of A

denoted by st(A). It is straightforward to prove that for A being solvable

the Lie algebra A

is solvable, too, and that st(A)

st(A

) is valid.

Proposition 3

If A is an associative right artian solvable K-algebra, then

is solvable.

Proof

A is solvable, and hence A

rad(A) is valid. By using the

associative nilpotency of rad(A) we conclude the Lie nilpotency of rad(A)

rad(A)

is a subalgebra of A

, and we conclude that A

is nilpotent.

Hence A

is solvable.

To prove the opposition implication of proposition 3 we use a well-known

strategy within the theory of associative algebras: we begin the proof for

central division algebras, afterwards for division algebras, then for simple

and semi-simple algebras and at the end for an arbitrary algebra. The next

two remarks are related to this strategy.

Remark 1

Let K be a field and A, B associative unitary K-algebras. The

following statements are valid:

(i) For all a, c

A and b, d B: (ab)(cd) = (ac)bd+ca(bd).

(ii) If A

is solvable and B

is abelian, then (A

is solvable.

Proof

ad(i): This can be verified by a straightforward calculation.

ad(ii): B

is abelian, and thus for all x, y

A and c, d B we derive

by (i): (

) (x c) (y d) = (x y) (cd).

Let T := A

B. By using (

) and an induction argument we conclude for

all m

N that the identity (T

)

[m]

)

[m]

B is valid. A

is solvable,

and thus part (ii) is proven by using the solvability of A

Within the second remark we analyze full matrix algebras concerning solv-

ability.

Example 1

(i) Let K be a field. A := K

2×2

is a simple associative K-

algebra. We prove that A

is solvable if and only if char(K) = 2 is valid.

For this, let B :=

, e

} be the standard basis of A. (e

is de-

fined to be the matrix such that only the (i; j)-entry is different from zero

and in addition equal to 1.) It is straightforward to calculate the identities

= e

, e

-e

, e

= 0

, e

= e

- e

= e

and e

-e

. Hence, (A

)

[1]

= e

, e

- e

11 K

is valid. In addition, e

= e

- e

, e

- e

) =

-2

and

- e

) =

-2

are valid. If char(K) = 2 is assumed, then

)

[2]

= e

- e

11 K

and (A

)

[3]

} (and thus st(A

) = 3) are true.

In the other case (A

)

[1]

= (A

)

[2]

is valid and A

is not solvable.

(ii) Let K be a field and n

. We prove that (K

×n

)

is not solv-

able. Let

| 1 i, j n} the standard basis of K

×n

as used in (i). It

is sufficient to prove that (K

3×3

)

is not solvable because (K

×n

)

is con-

taining a subalgebra isomorphic to (K

3×3

)

. The identities e

= e

, e

-e

, e

= e

, e

= e

, e

-e

= e

- e

and e

= e

- e

are valid.

In addition,

= e

- e

, e

= e

- e

, e

= e

, e

= e

, e

= e

, e

= e

and e

= e

are true.

We derive e

, e

- e

, e

- e

33 K

((K

3×3

)

[n]

for

all n

N. Hence, (K

3×3

)

is not solvable.

(iii) Let K be a field, char(K) = 2 and A a 4-dimensional central-simple

associative unitary algebra. By using [36] a K-basis B :=

, i, j, k

} of A

and elements a

K, b K \ {0

} exist such that i

+ i = a1

, j

= b1

and ij = k = j(i + 1

) are valid. These algebras are called generalized

quaternion algebras in characteristic two. We prove that A

is solvable.

The identities i

j = ij + ji = j, i k = ik + ki = i(ij) + (ji + j)i =

(i + a1

)j + j(i + a1

) + ji = ij + aj + ji + aj + ji = ij = k and

k = jk + kj = j(ji + j) + ij

= j

i + j

+ ij

= bi + j

+ bi = j

= b1

are valid. We conclude (A

)

[1]

= j, k, b1

A K

. Hence, (A

)

[2]

= b1

A K

and

)

[3]

} are true.

The following lemma is the key element related to our strategy:

Lemma 1

Let K be a field and D a central-simple and finite-dimensional

associative K-division algebra. If char(K) = 2 is valid, then let the K-

dimension of D be not 4. The following statements are equivalent:

(i) D is solvable.

(ii) D = K1

(iii) D

is solvable.

Proof

Because of rad(D) =

{0} part (i) is equivalent to part (ii). By

using proposition 3 part (ii) results in part (iii).

We assume part (iii). Let L be a maximal subfield of D and n = dim

(L) =

dim

(D). Then (see e.g. [50]) the statement D

×n

is valid.

Thus, n

= dim

(D) is true, too. Remark 1 implies that (L

×n

)

is solv-

able as K-algebra. If n = 1 is valid, then part (ii) is true. We assume

. If n = 2 is valid, then the K-algebra L

×n

contains a subalgebra

T isomorphic to K

3×3

. Thus, T

is a K-subalgebra of (L

×n

)

and solvable,

too. This is a contradiction to remark 1. For n = 2 we have to analyze

only the case char(K) = 2 (due to our assumptions). The K-algebra L

×n

Details

Pages
Type of Edition: Erstausgabe
Publication Year: 2017
ISBN (PDF): 9783960676966
ISBN (Softcover): 9783960671961
File size: 12.8 MB
Language: English
Publication date: 2017 (November)
Keywords: Cartan subalgebra Carter subgroup Lie algebra Associative algebra Maximal nilpotent Group algebra Exercise Lie nilpotency Nilradical Fitting subgroup Soluble

Author

Dr. rer. nat. Sven Bodo Wirsing (Author)

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

Summary

Excerpt

Table Of Contents

Details

Author

Dr. rer. nat. Sven Bodo Wirsing (Author)