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

©2017 Textbook 193 Pages

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

Table Of Contents


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
7
1
Standard examples, symbols and notations
13
2
Radical algebras and the theorem of Xiankun Du
21
2.1
Radical algebras and central chains . . . . . . . . . . . . . . .
21
2.2
Results of Stephen Arthur Jennings, Hartmut Laue and Xi-
ankun Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3
Standard examples . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.1
The algebras of upper and lower triangular matrices .
26
2.3.2
Solomon algebras in characteristic zero . . . . . . . . .
27
2.3.3
Solomon-Tits algebras . . . . . . . . . . . . . . . . . .
28
2.4
Open-ended questions and exercises
. . . . . . . . . . . . . .
31
3
Solvability
37
3.1
Solvability of the associated Lie algebra . . . . . . . . . . . .
37
3.2
Solvability of the group of units . . . . . . . . . . . . . . . . .
41
3.3
The theorem of Sophus Lie and Borel subalgebras . . . . . . .
43
3.4
Standard examples . . . . . . . . . . . . . . . . . . . . . . . .
45
3.4.1
Group algebras . . . . . . . . . . . . . . . . . . . . . .
45
3.4.2
The algebras of upper and lower triangular matrices .
46
3.4.3
Solomon algebras in characteristic zero . . . . . . . . .
46
3.4.4
Solomon-Tits algebras . . . . . . . . . . . . . . . . . .
46
3.5
Open-ended questions and exercises
. . . . . . . . . . . . . .
53
4
Carter subgroups of the group of units of solvable associative
algebras
59
4.1
The pendant within the associated Lie algebra: Cartan sub-
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
The determination of the Carter subgroups by Thorsten Bauer 60
4.3
Further connections in the case of finite fields . . . . . . . . .
65
4.4
Standard examples . . . . . . . . . . . . . . . . . . . . . . . .
67
4.4.1
Group algebras . . . . . . . . . . . . . . . . . . . . . .
67
4.4.2
The algebras of upper and lower triangular matrices .
70
3

4
4.4.3
Solomon algebras in characteristic zero . . . . . . . . .
71
4.4.4
Solomon-Tits algebras . . . . . . . . . . . . . . . . . .
71
4.5
Open-ended questions and exercises
. . . . . . . . . . . . . .
74
5
The Fitting subgroup of the group of units of a solvable
associative algebra
77
5.1
The pendant within the associated Lie algebra: the nilradical
77
5.2
The partner within the group of units: the Fitting subgroup .
77
5.3
Standard examples . . . . . . . . . . . . . . . . . . . . . . . .
79
5.3.1
The algebras of upper and lower triangular matrices .
79
5.3.2
Solomon algebras in characteristic zero . . . . . . . . .
79
5.3.3
Solomon-Tits algebras . . . . . . . . . . . . . . . . . .
80
5.3.4
Group algebras . . . . . . . . . . . . . . . . . . . . . .
80
5.4
Open-ended questions and exercises
. . . . . . . . . . . . . .
83
6
Maximal nilpotency in Lie algebras associated to solvable
associative algebras
85
6.1
Associativity
. . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6.2
Manifold centralizers . . . . . . . . . . . . . . . . . . . . . . .
89
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
7
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
8
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

5
9
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
A
(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
A
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
A
?
· 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?
7

8
· 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
A
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

9
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-

10
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
C
T
(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)
(C
T
(rad(M ))) = rad(M ) and C
T
(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
T
(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-

11
ber by the upper bound B(dim
K
(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
·
K
­ 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

12
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
N
0
. 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
0
- 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
13

14
· S(n, k) - the kth-Stirling number of n
· B(n) - the nth-Bell number
·
n
k
- n choose k
·
T
i
- 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
n
(G))
n
N
- ascending central chain of G
· (G
(n)
)
n
N
- descending central chain of G
· (
n
(G))
n
N
, (G
[n]
)
n
N
- commutator or derived series of G
· ,
c
- star or circle composition
· [a, b] - commutator of a, b
· [U, V ] - commutator of U, V
· a
-1
- inverse element of a
· a
b
- conjugate element of a with b
· 1 + rad(A) - normalized units
· C
U
(V ) - centralizer of V in U
· N
U
(V ) - normalizer of V in U
· G - class of groups
· G - derived subgroup of G
· (F
n
(G))
n
N
-Fitting series of G

15
· G/N - factor group of G by N
· SU
E
(A)
- set of solvable subgroups of E(A)
·
n
- ordered set partitions
·
n
- special composition on
n
· exp(G) - exponent of G
· U
U
- unipotent factor of U
· U
V
- fully-separable factor of U
· O
p
(G) - p-core of G
· G(U), U
E
(T )
- subgroups of E(T ) possessing the double-centralizer prop-
erty
· U
T
- order of
G(U)
· G
DJ
- subgroups of 1 + rad(A) possessing the double-centralizer property
· G
U T
- maximal nilpotent subgroups of E(A) linked to E(T )
· G
U
- maximal nilpotent subgroups E(A)
· E(A)
M
- maximal nilpotent subgroups of E(A)
· a
-
- quasiregular inverse element
· x
(a)
- quasiregular conjugate element
· N - natural numbers
· N
0
- natural numbers containing zero
· D
2n
- dihedral group of order 2n
· Q
4n
- quaternion group of order 4n
· SD
2
n
- semi-dihedral group of order 2
n
· S
n
- symmetric group of degree n
· A
n
- alternating group of degree n
· GL(n, q) - general linear group of degree n over GF (q)
· C
n
or Z
n
- cyclic group of order n
· E(A) - group of units of A

16
· 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
n
and T
A. The following general algebra
constructions and symbols are used:
· dim
K
(A) - K-dimension of A
· A - class of associative algebras
· A
1
- 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
n
- nth associative power of A
· C
A
(T ) - centralizer of T in A
· a
T
- sum of a
i
, i
T , T n, a
1
,
· · · a
n
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

17
· A
n
×n
- algebra of n
× n-matrices over A
· A
- associated Lie algebra of A
· T
K
- K-linear span of T
· T
A
- subalgebra generated by T
· T
A
1
- unital subalgebra generated by T
· A
K
- adjunction of an unit to A
· A
op
or A
-
- opposite or inverse 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 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
×n
· e
1
,
· · · , e
n
- primitive orthogonal idempotents in D(n, K)
· e
T
- sum of all e
i
, i
T n
· K
n
- 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
K
- is identical to
n
i
=1
1
K
· Z - the set of integers
· K[t] - polynomial algebra over K in one variable t
· K[t
1
, . . . , t
n
] - polynomial algebra over K in the variables t
1
, . . . , t
n
.

18
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
n
) - finite field possessing p
n
elements
· GF (q) - notation for GF (p
n
) and q = p
n
· A(a, b) - generalized quaternion algebra
· K(a) - smallest subfield in L containing a and K
·
d
- primitive dth root of unity
· K(t) - field of fractions over K in one variable t
· K(t
1
, . . . , t
n
) - field of fractions over K in the variables t
1
, . . . , t
n
.
(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
ij
- base matrices of K
n
×n
· det - determinant function
· tr - trace function
· 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
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.

19
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
×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) =
p.
Solvable associative algebras
Let n
N and K be a field. The following solvable associative algebras are
used:
· K
n
- Solomon-Tits algebra (see e.g. [77])
· D
n
- 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 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
0
(ad(H)) - Fitting null component of L with resp. to ad(H)
· B
A
- Borel subalgebras of A
· st(L) - solvable class of L

20
· cl(L) - class of nilpotency of L
· (Z
n
(L))
n
N
- ascending central chain of L
· (L
(n)
)
n
N
- descending central chain of L
· (L
[n]
)
n
N
- descending commutator or derived series of L
· L - class of Lie algebras
· nil(L) - nilradical of L
· M(T ), A
DT
- subalgebras of T possessing the double-centralizer property
· m
T
- order of
M(T )
· A
DJ
- subalgebras of rad(A) possessing the double-centralizer property
· A
M T
- maximal nilpotent Lie subalgebras of A
linked to T
· A
M
- maximal nilpotent Lie subalgebras of A
· L - derivation of L
· t
M
, u
M
- unit group and K-space creation
· J
i
(I) - special sequence of subalgebras and subgroups in rad(A)
· T
i
(C) - special sequence of subalgebras and subgroups in 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
21

22
coincide and are nilpotent.
1
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
(A
) :=
{0} and Z
n
(A
) :=
{z | z
A,
a A : z a Z
n
-1
(A
)
} for all n N. A is Lie nilpotent, if a nat-
ural number n
N exists such that A = Z
n
(A
) 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
1
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.

23
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
-
a
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
0
(Q(A)) :=
{0} and Y
n
(Q(A)) :=
{y | y Q(A), a Q(A) :
[y, a]
Y
n
-1
(Q(A))
} for all n N. The commutator [y, a] is defined by
y
-
y
(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
k
(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
0
(Q(A)) :=
{0}
and X
n
(Q(A)) :=
{x | x Q(A), a Q(A) : x
(a)
- x X
n
-1
(Q(A))
} for
all n
N.
In addition, we define for a radical algebra A inductively W
0
(A) :=
{0}
and W
n
(A) :=
{w | w A, a A : a
(w)
- a W
n
-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
2
conjectured for radical algebras A that the nilpo-
tency classes of A and A
coincide. In 1984 Hartmut Laue has proven some
2
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

24
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
n
(A
) = Y
n
(A ) is valid for
all n
N
0
.
(iii) If Q(A) = A is valid, then Z
2
(A
) = Y
2
(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
N
0
the
identity Z
n
(A
) = Y
n
(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
n
N
0
the identity Z
n
(A
) = Y
n
(A ) = X
n
(A) = W
n
(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.

25
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
N
0
. By
T
i
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
T
.
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
1
, . . . , x
n
A. The statement x
1
· · · x
n
=
n
i
=1 T
(
n
i
)
x
(1
T
)
. . . x
(i
T
)
is valid.
Corollary 1
Let A be an associative K-algebra, n
N and a A. The
following statements are valid:
(i) a
· · · a
n
-fold
=
n
i
=1
n
i K
a
i
(ii) If p is a prime number and char(K) = p is valid, then
a
· · · a
p
n
-fold
= a
(p
n
)
is true.
Proof
.
ad(i): By using proposition 1 we derive
a
· · · a
n
-fold
=
n
i
=1 T
(
n
i
)
a
i
=
n
i
=1
|
n
i
|
K
a
i
=
n
i
=1
n
i K
a
i
.
ad(ii): For all i
p
n
- 1 it is well-known (see e.g. [18])that the prime
number p = char(K) is a divisor of
p
n
i
. 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
n
-fold
=
n
k
=0
n
k K
(
-1
K
)
k
y
k
xy
n
-k
.

26
(ii) If p is a prime number and char(K) = p is valid, then the following
identity is true: x
y · · · y
p
-fold
= x
y
p
.
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
n
+1
(A
)/Z
n
(A
)) = p is
valid.
Proof
.
By using the theorem of Xiankun Du (see theorem 3) for all n
N
the sets Z
n
(A
) and Z
n
(A
) are identical. Based on corollary 1 for all a
A
the identity a
· · · a
p
-fold
= a
p
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
×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
n
-1
i
=1
i =
1
2
(n
- 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
×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
n
-1
i
=1
i =
1
2
(n
-1)n. Again,

27
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
n
we denote the
Solomon algebra. It is defined by the K-span of class sums of so-called defect
classes of KS
n
: if
S
n
is valid, then we define D() :=
{i | i > (i+1)}.
Solomons algebra is the K-span of
{
D
()=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
2
n
-1
, its radical factor algebra of dimension p(n) - the number of parti-
tions of n ­ and is isomorphic to K
p
(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
n
) (see [1], cl(rad(D
n
)) = 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
n
)) = cl(rad(D
n
)
) = cl(rad(D
n
) ) = n
- 1 is valid.

28
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
n
0
. 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
n
is presented and analyzed in details by Manfred Schocker and
by the author. K
n
is defined based on the monoid
n
which consists of
all ordered set partitions of n. If (P
1
,
· · · , P
l
) and (Q
1
,
· · · , Q
k
) are two of
these ordered partitions, then their product
n
is defined by
(P
1
,
· · · , P
l
)
n
(Q
1
,
· · · , Q
k
) :=
(P
1
Q
1
, P
1
Q
2
,
· · · , P
1
Q
k
,
· · · , P
l
Q
1
, P
l
Q
2
,
· · · , P
l
Q
k
)
.
symbolizes that empty sets are deleted from this tuple. Again, K
n
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
n
is described within chapter 2 in [77]. Its dimen-
sion is dim
K
(rad(K
n
)) =
n
k
=0
(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
n
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
n
)) = cl(rad(K
n
)
) = cl(rad(K
n
) ) = n.



31
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
?

32
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
a
c
b := a + b + cab. A is ­ equipped with the composition
c
­ a monoid
possessing the neutral element 0. If c, d are units in (K;
·), then the monoids
(A;
c
) and (A;
d
) 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;
c
) and (A;
d
) 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 =
Z
(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}!

33
Excercise 12
Let A be a radical K-algebra and c an unit of K. Prove for
all n
N
0
the identity Y
n
((A; )) = Y
n
((A;
c
)). (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 |
g
P }
K
. (Tip: The elements g
- 1 are nilpotent and generate an
ideal. Apply a theorem of Joseph Wedderburn to this result!)
(ii) K1
G
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
G
.
(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
8
or
G = D
8
. 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
p
n
- 1 the prime number p = char(K) is a divisor of
p
n
i
.
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.

34
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
(s
u,
3
)
and (s
u,
4
)
. 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
(s
o,
3
)
and (s
o,
4
)
. 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
3
and S
4
!
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
or
Q?
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
1
, . . . , x
n
A and n N.
Calculate x
1
· · ·x
n
for n
3. For the algebra of real quaternions determine
i
j and i
j
k.
Excercise 26
Within exercise 25 calculate x
1
· · · x
1
n
-fold
. What is the result
of this n-fold star product within the quaternion algebra for x
1
{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)?

35
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
c
? 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
K
(see e.g. series I or [75] for an exact
definition and construction of A
K
). Is there a connection between the quasi
regular groups of A and A
K
and the group of units of A
K
? Is A a radical
algebra if and only if A
K
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)
n
)
n
N
.
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)
i
/rad(A)
i
+1
is a zero-algebra.
(iii) For all i
N the group (rad(A)
i
/rad(A)
i
+1
) 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)
i
/rad(A)
i
+1
)
for all i
N
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
G
. 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]
)
n
N
resp. (G
[n]
)
n
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
N we define the nth derived subalgebra or commutator ideal of A by
A
[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
37

38
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
A
is solvable.
Proof
.
A is solvable, and hence A
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
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
K
B)
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
K
B. By using (
) and an induction argument we conclude for
all m
N that the identity (T
)
[m]
(A
)
[m]
K
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
11
, e
12
, e
21
, e
22
} be the standard basis of A. (e
ij
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
22
e
21
= e
21
, e
22
e
12
=
-e
12
, e
22
e
11
= 0
A
, e
21
e
12
= e
22
- e
11
,
e
21
e
11
= e
21
and e
12
e
11
=
-e
12
. Hence, (A
)
[1]
= e
21
, e
12
, e
22
- e
11 K
is valid. In addition, e
21
e
12
= e
22
- e
11
, e
21
(e
22
- e
11
) =
-2
K
e
21
and
e
12
(e
22
- e
11
) =
-2
K
e
12
are valid. If char(K) = 2 is assumed, then
(A
)
[2]
= e
22
- e
11 K
and (A
)
[3]
=
{0
A
} (and thus st(A
) = 3) are true.
In the other case (A
)
[1]
= (A
)
[2]
is valid and A
is not solvable.

39
(ii) Let K be a field and n
N
3
. We prove that (K
n
×n
)
is not solv-
able. Let
{e
ij
| 1 i, j n} the standard basis of K
n
×n
as used in (i). It
is sufficient to prove that (K
3×3
)
is not solvable because (K
n
×n
)
is con-
taining a subalgebra isomorphic to (K
3×3
)
. The identities e
11
e
12
= e
12
,
e
11
e
13
= e
13
, e
11
e
31
=
-e
31
, e
21
e
22
= e
21
, e
23
e
33
= e
23
, e
32
e
33
=
-e
32
,
e
12
e
21
= e
11
- e
22
and e
13
e
31
= e
11
- e
33
are valid.
In addition,
e
12
e
21
= e
11
- e
22
, e
13
e
31
= e
11
- e
33
, e
12
e
23
= e
13
, e
13
e
32
= e
12
,
e
32
e
21
= e
31
, e
21
e
13
= e
23
, e
23
e
31
= e
21
and e
31
e
12
= e
32
are true.
We derive e
12
, e
13
, e
21
, e
23
, e
31
, e
32
, e
11
- e
22
, e
11
- 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 :=
{1
A
, i, j, k
} of A
and elements a
K, b K \ {0
K
} exist such that i
2
+ i = a1
A
, j
2
= b1
A
and ij = k = j(i + 1
A
) 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
A
)j + j(i + a1
A
) + ji = ij + aj + ji + aj + ji = ij = k and
j
k = jk + kj = j(ji + j) + ij
2
= j
2
i + j
2
+ ij
2
= bi + j
2
+ bi = j
2
= b1
A
are valid. We conclude (A
)
[1]
= j, k, b1
A K
. Hence, (A
)
[2]
= b1
A K
and
(A
)
[3]
=
{0
A
} 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
A
(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
K
(L) =
dim
L
(D). Then (see e.g. [50]) the statement D
K
L
=
A
1
L
n
×n
is valid.
Thus, n
2
= dim
K
(D) is true, too. Remark 1 implies that (L
n
×n
)
is solv-
able as K-algebra. If n = 1 is valid, then part (ii) is true. We assume
n
N
2
. If n = 2 is valid, then the K-algebra L
n
×n
contains a subalgebra
T isomorphic to K
3×3
. Thus, T
is a K-subalgebra of (L
n
×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
×n

Details

Pages
Type of Edition
Erstausgabe
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
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