Maximal nilpotent subalgebras I: Nilradicals and Cartan subalgebras in associative algebras. With 428 exercises
©2016
Textbook
242 Pages
Summary
During the author’s doctorate time at the ChristianAlbrechtsUniversitat to Kiel, Salvatore Siciliano gave a stimulating talk in the upper seminar algebra theory about Cartan subalgebras in Lie algebra associates to associative algebra. This talk was the incentive for the author to analyze maximal nilpotent substructures of the Lie algebra associated to associative algebras. In the present work Siciliano's theory about Cartan subalgebras is worked off and expanded to different special associative algebra classes. In addition, a second maximal nilpotent substructure is analyzed: the nilradical. Within this analysis the main focus is to describe these substructure with the associative structure of the underlying algebra. This is successfully realized in this work. Numerous examples (like group algebras and Solomon (Tits) algebras) illustrate the results to the reader. Within the numerous exercises these results can be applied by the reader to get a deeper insight in this theory.
Excerpt
Table Of Contents
4.6.1
Subalgebras . . . . . . . . . . . . . . . . . . . . . . . .
53
4.6.2
Right and left ideals . . . . . . . . . . . . . . . . . . .
53
4.6.3
Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.6.4
Factor algebras . . . . . . . . . . . . . . . . . . . . . .
54
4.6.5
The opposite algebra . . . . . . . . . . . . . . . . . . .
54
4.6.6
Matrix algebras . . . . . . . . . . . . . . . . . . . . . .
54
4.6.7
Adjunction of a unit . . . . . . . . . . . . . . . . . . .
55
4.6.8
Tensor products
. . . . . . . . . . . . . . . . . . . . .
56
4.7
The group algebra once again . . . . . . . . . . . . . . . . . .
56
4.8
Openended questions and exercises
. . . . . . . . . . . . . .
59
5
Cartan subalgebras in Lie algebras associated to associative
algebras
65
5.1
Cartan subalgebras are associative solvable subalgebras
. . .
65
5.1.1
Associative structure of Cartan subalgebras . . . . . .
66
5.1.2
Openended questions and exercises
. . . . . . . . . .
68
5.2
Maximal tori and Cartan subalgebras
. . . . . . . . . . . . .
71
5.2.1
Maximal tori . . . . . . . . . . . . . . . . . . . . . . .
71
5.2.2
Maximal tori of group algebras based on dihedral and
quaternion groups . . . . . . . . . . . . . . . . . . . .
72
5.2.3
Cartan subalgebras . . . . . . . . . . . . . . . . . . . .
73
5.2.4
Cartan subalgebras of group algebras based on dihe
dral and quaternion groups . . . . . . . . . . . . . . .
75
5.2.5
Maximal tori and Cartan subalgebras of Solomon al
gebras, SolomonTits algebras and triangular matrices
76
5.2.6
Openended questions and exercises
. . . . . . . . . .
79
5.3
Division algebras . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.3.1
Central division algebras . . . . . . . . . . . . . . . . .
81
5.3.2
Noncentral division algebras . . . . . . . . . . . . . .
82
5.3.3
Openended questions and exercises
. . . . . . . . . .
85
5.4
Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . .
86
5.4.1
The case characteristic = 2 . . . . . . . . . . . . . . .
86
5.4.2
Examples for the associative conjugacy
. . . . . . . .
86
5.4.3
Remark on the Lie conjugacy . . . . . . . . . . . . . .
87
5.4.4
The case characteristic 2 . . . . . . . . . . . . . . . . .
87
5.4.5
Openended questions and exercises
. . . . . . . . . .
90
5.5
Solvable algebras . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.5.1
Unitary solvable algebras . . . . . . . . . . . . . . . .
91
5.5.2
The quasi regular group and nonunitary solvable as
sociative algebras . . . . . . . . . . . . . . . . . . . . .
93
5.5.3
Solvable group algebras . . . . . . . . . . . . . . . . .
96
5.5.4
Solvable group algebras of dihedral groups . . . . . . .
97
5.5.5
Solvable group algebras of quaternion groups . . . . . 100
5.5.6
Splitting solvable algebras . . . . . . . . . . . . . . . . 101
5.5.7
Openended questions and exercises
. . . . . . . . . . 105
5.6
Lie nilpotent associative algebras . . . . . . . . . . . . . . . . 107
5.6.1
Lie nilpotency
. . . . . . . . . . . . . . . . . . . . . . 107
5.6.2
Nilpotent group of units . . . . . . . . . . . . . . . . . 110
5.6.3
Group algebras and Lie nilpotency . . . . . . . . . . . 112
5.6.4
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7
Simple, semisimple and separable algebras . . . . . . . . . . . 116
5.7.1
Centralsimple algebras . . . . . . . . . . . . . . . . . 116
5.7.2
Simple algebras . . . . . . . . . . . . . . . . . . . . . . 119
5.7.3
Semisimple and separable algebras . . . . . . . . . . . 119
5.7.4
Openended questions and exercises
. . . . . . . . . . 122
5.8
Basic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.8.1
Characterizations of reduced algebras
. . . . . . . . . 125
5.8.2
Maximal solvable subalgebras . . . . . . . . . . . . . . 126
5.8.3
Cartan subalgebras . . . . . . . . . . . . . . . . . . . . 127
5.8.4
Maximal nilpotent subalgebras . . . . . . . . . . . . . 128
5.8.5
Reduced group algebras: the semisimple case . . . . . 129
5.8.6
Reduced group algebras: the modular case
. . . . . . 133
5.8.7
An example of Benjamin Steinberg . . . . . . . . . . . 134
5.8.8
Openended questions and exercises
. . . . . . . . . . 137
5.9
Associative algebras with separable factor algebra by its nil
radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.9.1
A description by radical complements . . . . . . . . . 143
5.9.2
A strategy for the determination of Cartan subalgebras 144
5.9.3
Solvable algebras revised . . . . . . . . . . . . . . . . . 144
5.9.4
Group algebras for dihedral groups . . . . . . . . . . . 144
5.9.5
Openended questions and exercises
. . . . . . . . . . 147
5.10 Natural determination of Cartan subalgebras . . . . . . . . . 148
5.10.1 Sub and factor structures . . . . . . . . . . . . . . . . 148
5.10.2 Direct products . . . . . . . . . . . . . . . . . . . . . . 150
5.10.3 Adjunction of a unit . . . . . . . . . . . . . . . . . . . 150
5.10.4 Cyclic algebras . . . . . . . . . . . . . . . . . . . . . . 151
5.10.5 Tensor products and extension of the base field . . . . 151
5.10.6 Matrix algebras . . . . . . . . . . . . . . . . . . . . . . 151
5.10.7 Group algebras . . . . . . . . . . . . . . . . . . . . . . 152
5.10.8 Openended questions and exercises
. . . . . . . . . . 154
6
Summation formulas for the dimension of maximal tori in
group algebras
157
6.1
The semisimple case . . . . . . . . . . . . . . . . . . . . . . . 157
6.2
Upper and lower bounds . . . . . . . . . . . . . . . . . . . . . 158
6.3
Special classes of groups . . . . . . . . . . . . . . . . . . . . . 160
6.4
The modular case . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.5
Openended questions and exercises
. . . . . . . . . . . . . . 178
7
Invariants
189
7.1
Dimension formula for maximal tori
. . . . . . . . . . . . . . 189
7.2
Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . 191
7.3
Openended questions and exercises
. . . . . . . . . . . . . . 206
8
Outlook on series II
213
A Derived algebras
215
A.1 Definition and initial properties . . . . . . . . . . . . . . . . . 215
A.2 Structural properties . . . . . . . . . . . . . . . . . . . . . . . 217
A.3 Lie nilpotency . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.4 Isomorphism of derived algebras
. . . . . . . . . . . . . . . . 218
A.5 Openended questions and exercises
. . . . . . . . . . . . . . 223
List of figures
227
Bibliography
227
Index
234
Introduction
Maximal nilpotent are Cartansubalgebras
as well as the nilradical.
Both will be studied in the magnificient sal
of associated Lie algebras.
(Sven Wirsing, December 2015)
Within the theory of Lie algebras Cartan subalgebras play an important role
for the classification of semisimple Lie algebras as well as within the theory
of symmetric spaces.
During my time of studying at the ChristianAlbrechtsUniversit¨
at of Kiel
Salvatore Siciliano presented his researches in the Oberseminar Algebren
theorie about Cartan subalgebras in Lie algebras associated to associative
algebras. His presentation was the starting point for me to study maximal
nilpotent substructures in associated Lie algebras of associative algebras. In
this work we will present his theory of Cartan subalgebras and enhance it
to some special associative algebras (e.g. basic algebras, division algebras,
algebras with separable factor algebra by its nilradical). In addition, a sec
ond maximal nilpotent substructure is analyzed, its the socalled nilradical
of a Lie algebra.
The first chapter introduces some special associative and Lie algebras, monoids
and groups. They will be important to visualize and illustrate the general
theorems proven within this work. Some applications are also transferred to
the exercises at the end of each section or chapter. There are some exercises
included enhancing the theory presented so far such that the reader gets a
deeper insight. In addition, at the beginning of each exercise series some
openended topics are included which can be used by the reader and also
by the author to do additional researches within this theory. The author
has included some (manually created) graphics mostly so called Hasse di
agrams to visualize the results of each section or chapter.
Within chapter 2 basic results about finite subgroups of fields and divi
7
8
sion algebras are summarized. Some will be proven in details, others will be
just presented without a proof. They will play a role later on in the next
chapters of this work such that their understanding leads to a better insight
of the latter results. In addition, the author includes some proofs of these
basic results because of personal interest on the proofs itself. The summary
will include the proof that finite subgroups of fields are cyclic, the theorem
of Wedderburn about finite division algebras as well as results of Herstein
and Amitsur about the classification of finite subgroups of division algebras.
Likewise structured is chapter 3. This chapter focusses on the normal and
subnormal subgroup structure of division algebras. We will prove the the
orem of CartanBrauerHua about normal subgroups of division algebras
and the theorem of Scott about solvable group of units of division alge
bras. Finally, the theorem of Stuth about subnormal subgroups is presented
(without proving it) enhancing the theorem of Scott.
For an associative algebra the associated Lie algebra can be derived in a
natural way. In chapter 4 we analyze the nilradical the greatest nilpotent
ideal of that Lie algebra and focus our analysis on its associative structure.
For this, the center and the nilradical of the associative algebra are of im
portance: the nilradical is the sum of these two associative substructures. In
particular, its an associative subalgebra. For this theorem we assume that
the factor algebra by the nilradical of the associative algebra is separable
and thus we can use the theorem of WedderburnMalcev within the proof.
The analysis begins by determining the nilradical in the case of a solvable
associative algebra.
For this, results of the socalled generalized Jordan
decomposition are used. We demonstrate the theorem on special solvable
group algebras based on dihedral and quaternion groups, on the Solomon
algebras in characteristic zero, on the SolomonTits algebras and on the al
gebras of upper and lower triangular matrices over an arbitrary field.
In a second step some results of Herstein about simple rings and their as
sociated Lie ring are transferred to simple and semisimple algebras: we will
prove that the nilradical is identical to the greatest solvable Lie ideal the
socalled solvable radical. Both structures are identical to the center of the
associative algebra. For the semisimple case it is proven that the Lie nil
radical of direct products is the direct product of the Lie nilradicals of the
corresponding components: there are no diagonals possible.
Both results for solvable and semisimple associative algebras are used to
determine the nilradical for arbitrary associative algebras.
The chapter is finalized to apply and enhance the theorem for algebra con
structions like the tensor product, the adjunction of a unit and matrix al
gebras over algebras. The idea is to determine the Lie nilradical by the
components of the algebra constructions, like by the factors for the tensor
product. We will give proofs or counterexamples for these constructions
9
with respect to this question.
In the previous chapter we have deeply analyzed the Lie nilradical of an
associative algebra with respect to its associative structure. The Lie nilrad
ical is a maximal nilpotent substructure, and the Cartan subalgebras are
maximal nilpotent, too. They are in focus of the next chapter. They are
defined as being nilpotent and selfnormalizing Lie subalgebras. The aim
of this chapter is the same as for Lie nilradical: their determination and
the description of their associative structure. Some results of this chapter
are based on an article of Salvatore Siciliano [59], others are enhancements
of his theory to other classes of associative algebras like division algebras,
simple, semisimple and separable associative algebras, reduced associative
algebras or associative algebras with separable factor algebra by their nil
radical. Standard examples are investigated in details, in particular group
algebras, lower and upper triangular matrices and Solomon(Tits) algebras
for illustrating the developed theory.
The main result of this chapter is the 1:1connection between maximal
tori (maximal commutative separable subalgebras) und Cartan subalgebras.
Centralizing maximal tori is a bijection between these structures. The in
verse calculates for every Cartan subalgebras a maximal torus by creating
the set of fully separable elements of the Cartan subalgebra.
In some cases both sets maximal tori and Cartan subalgebras are iden
tical, like for separable associative algebras. Central divisions algebras are
separable, too, and we prove a theorem of Salvatore Siciliano (in a different
way) that maximal tori and Cartan subalgebras are exactly the maximal
separable subfields. We enhance the theorem by proving that these are ex
actly the separable maximal subfields which is also an alternative proof of a
theorem of Emmy Noether. In particular, it is proven that all maximal tori
= Cartan subalgebras have the same dimension and are isomorphic as Lie
algebras. This theorem is transferred to noncentral division algebras.
Solvable associative algebras have the property that maximal tori are exactly
the radical complements if the factor algebra by its associative nilradical is
separable. This result proven by Thorsten Bauer in his dissertation [4] and
by Salvatore Siciliano in [59] is proven by a different approach and revised
later on in the second to last section of this chapter. As a consequence of our
main theorem about Cartan subalgebras and the theorem of Wedderburn
Malcev all maximal tori and Cartan subalgebras are conjugated, and the
Cartansubalgebras are exactly the centralizers of the radical complements.
For basic algebras we transfer the determination of Cartan subalgebras to
Cartan subalgebras of maximal solvable substructures. These maximal ones
are describable as direct sums of maximal tori and the associative nilradical.
The centralizers of the maximal tori of the underlying algebra are identical
to the centralizer within these maximal solvable subalgebras. Afterwards we
focus on reduced group algebras. In the modular case the terms basic and
10
solvable are equivalent. For semisimple group algebras the situation is more
complex: the group is hamiltonian and the equation a
2
+ b
2
+ 1 = 0 has
no solution in special field extension based on roots of unities. Finally, we
determine the dimension of the Cartan subalgebras for these group algebras
based on the results of chapter 6.
In the second to last section we analyze how the determination of Cartan
subalgebras can be done based on separable radical complements. The max
imal tori of the radical complement and of the whole algebra are identically.
For separable radical complements maximal tori and Cartan subalgebras are
identically, too. The centralizers of them are exactly the Cartan subalgebra
of the underlying algebra. Based on this result a strategy is developed for
determining Cartan subalgebras. For solvable algebras this strategy is used
and the determination of Cartan subalgebras is revised in a more transpar
ent way. We apply this strategy also on group algebras of dihedral groups.
The chapter is finalized to apply and enhance the theorem for Cartan sub
algebras for algebra constructions like the tensor product, the adjunction
of a unit and matrix algebras over algebras. The idea is to determine the
Lie nilradical by the components of the algebra constructions, like by the
factors for the tensor product. We will give proofs or counterexamples for
these constructions with respect to this question.
The next chapter is dedicated to the dimension of maximal tori in group
algebras. We begin this chapter by proving a result of Salvatore Siciliano
connecting this dimension to the sum of degrees of all irreducible complex
characters for semisimple group algebras. This sum is identical for all fields
such that the group algebra is semisimple. We use this result and some
classical and modern results about that sum within the character theory of
finite groups to bound this dimension like by the number of involutions,
by the order of the group, by the order of abelian subgroups and by the
maximal degree and determine this sum for several classes of groups
like for Frobenius groups, for direct products, for extra special pgroups, for
diverse linear groups, for ambivalent groups such as dihedral and symmetric
groups, for metacyclic groups, for pgroups, for nilpotent groups and for
minimal nonabelian pgroups.
Within chapter 7 we focus on the question whether the dimensions of the
maximal tori and of the Cartan subalgebras are unique for associated Lie
algebras of finitedimensional associative unital algebras. For maximal tori
we give a positive answer to this question for associative algebras with sepa
rable factor algebra by its nilradical by calculating this dimension explicitly.
The answer for the Cartan subalgebras is positive, too. In characteristic
zero we derive this result by using a classical result on Cartan subalgebras
over algebraically closed fields. In the modular case we begin the analysis by
proving the uniqueness for associated Lie algebras based on solvable finite
11
dimensional associative algebras, for separable associative algebras and for
finitedimensional associative algebras possessing a central nilradical. The
general case is derived by using a result of Premet (which was later proven
by Farnsteiner) for restricted Lie algebras over algebraically closed fields in
positive characteristic and by using the result on the dimension for maximal
tori. In general, the dimension of Cartan subalgebras can differ for restricted
Lie algebras. By using a second approach we extend our theorem for the
uniqueness of the dimension of Cartan subalgebras to the solvable and nilpo
tency class. For this, we prove that all maximal tori and Cartan subalgebras
of Lie algebras associated to finitedimensional associative algebras over an
arbitrary algebraically closed field are conjugated. We demonstrates these
three invariants dimension, nilpotency and solvable class by calculating
them for group algebras based on dihedral and quaternion groups.
Chapter 8 is an outlook on the second series about maximal nilpotent sub
structures. We will focus on the solvable case of an associative algebra in
more details as in this first volume. For this, we will extend the topic to
all maximal nilpotent substructures and to the connection to the maximal
nilpotent subgroups of their group of unit. A graphic illustrates the prob
lems analyzed in series II.
Within the appendix we classify a special class of algebras and analyze their
Lie nilpotency. This class of algebras was in focus of the diploma thesis of
Armin J¨
ollenbeck.
Chapter 1
Natural examples
This chapter has a preliminary function by summarizing those monoids,
groups, associative and Lie algebras which will arise in this work. They will
be used for examples of the proven theorems as well as for exercises in which
the reader shall apply the results.
Groups and monoids
Let n N, N be a set, M a monoid, G a group, A an associative unitary
algebra and q a prime power. We will focus on the following groups and
monoids:
· N  natural numbers
· N
0
 natural numbers containing zero
· (P (N ); )  power set of N with operation
· (P (N ); )  power set of N with operation
· (P (N ); )  power set of N with operation  symmetric difference
· (P (M ); ·)  power set of M with complex product · as operation
· (P (G); ·)  power set of G with complex product · as operation
· D
2n
 dihedral group of order 2n
· Q
4n
 quaternion group of order 4n
· SD
2
n
 semidihedral group of order 2
n
· S
n
 symmetric group of degree n
· A
n
 alternating group of degree n
13
14
· GL(n, q)  general linear group of degree n over GF (q)
· SL(n, q)  special general linear group of degree n over GF (q)
· P SL(n, q)  projective special general linear group of degree n over GF (q)
· SP (2n, q)  symplectic group of degree 2n over GF (q)
· GSP (2n, q)  general similitudes group
· U (n, q)  unitary group of degree n over GF (q)
· C
n
or Z
n
 cyclic group of order n
· E(A)  group of units of A
· Q(A)  quasiregular group of A
· ×  direct products of groups
·
 regular wreath product of groups
·
 semidirect product of groups.
General constructions of algebras
Let A be an algebra, K a field, G a group, I an ideal, M a monoid, n N
and T A. The following general constructions of algebras will be used:
·  tensor product of algebras
· ×  direct products of algebras
·  direct sum of algebras
· A/I  factor algebra of A by the ideal I
· KG  group algebra of the group G and the field K
· KM  monoid algebra of the monoid M and the field K
· A
n×n
 algebra of n × nmatrices over A
· A
 associated Lie algebra of A
· T
K
 Klinear span of T
· T
A
 subalgebra generated by T
· T
A
1
 unital subalgebra generated by T
· A
K
 adjunction of a unit to A
15
· A
op
or A

 inverse or opposite algebra of A
· (A × A;
)  zero extension of A
· gl(n, K)  identical to (K
n×n
)
· eAe  identical to {eae  a A} for an idempotent e
· Aug(KG)  augmentation ideal of KG.
Commutative algebras
The following commutative algebras will appear:
· Z  the set of integers
· K[t]  polynomial algebra over K in one variable t.
Fields and skew fields
Let p be a prime number, n N and (K; L) a field extension. We will focus
on the following fields, skew fields and elements:
· Q  rational number field
· R  real number field
· C  complex number field
· H  real quaternion algebra
· GF (p
n
)  finite field with 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
· cyclic division algebras.
(Central)  simple associative algebras
Let K be a field, D a division algebra and n N. We will use the following
(central)simple associative algebras:
· K
n×n
 n × nmatrices over K
· D
n×n
 n × nmatrices over D
· A(a, b)  generalized quaternion algebra.
16
Semisimple associative algebras
We will use the following semisimple associative algebras:
· ×  direct products of simple algebras
· A/rad(A)  the factor algebra by the nilradical of an associative algebra.
Nilpotent associative algebras
Let A be an associative algebra, K a field, p a prime number, n N and G
a pgroup. We will focus on the following nilpotent associative algebras:
· rad(A)  nilradical of A
· J (A)  Jacobson radical of A
· s
u,n
 algebra of strict lower triangular matrices of K
n×n
· s
o,n
 algebra of strict upper triangular matrices of K
n×n
· Aug(KG)  augmentation ideal of KG based on a pgroup G and char(K) =
p.
Solvable associative algebras
Let n N, p a prime number, G a finite group and K be a field. We will
focus on the following solvable associative algebras:
· K
n
 SolomonTits algebra (see e.g. [76])
· 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
pSylow subgroup with an abelian p Hall subgroup.
Chapter 2
Finite subgroups of fields
and division algebras
In this chapter we summarize some results of finite subgroups in unit groups
of fields and division algebras. For some of them we provide a proof, for the
others we reference the corresponding literature. We will use some of these
results in the next chapters. Therefor these results provide the reader a
deeper insight for understanding these results. In addition, this chapter is
included on personal interest of the author for the proofs of these results.
2.1
Finite subgroups of fields
By E(A) and K[t] we denote the group of units of an associative algebra A
and the algebra of polynomials over a ring K based on the single variable t.
For a group G and an element g of G let o(g) (more exact: o
G
(g)) the order
of g in G.
The following theorem is proven by various arguments within the litera
ture. It is unknown which mathematician provided the first proof of this
result. Our variant is based on the main theorem on finite abelian groups.
Theorem 1 Every finite subgroup of the group of units of a field is cyclic.
In particular, the group of units of a finite field is cyclic.
Proof.
Let K be a field and U a finite subgroup of E(K). By using the
main theorem on finite abelian groups we decompose U in cyclic groups of
prime power order:
U = (G
1,1
× · · · × G
1,s
1
) × · · · × (G
r,1
× · · · × G
r,s
r
).
In this decomposition all groups G
i,j
are of prime power order with respect
to the prime number p
i
. We arrange the product such that G
i,1
is the great
est factor within G
i,1
× · · · × G
i,r
i
. For every i let g
i
a generator of G
i,1
.
17
18
We focus on the element g := g
1
· · · g
r
. g is of order o(g) = o(g
1
) · · · o(g
r
)
because all prime numbers p
1
, · · · , p
r
are distinct. For every u U the
identity u
o(g)
= 1 is valid.
All elements of U are roots of the polynomial t
o(g)
 1, and there are at
most o(g) distinct roots. Hence we derive  U  o(g). All o(g)powers of g
are distinct. Therefor U is exactly the set of these powers of g. We conclude
that U is cyclic and generated by g.
2.2
Results of Wedderburn, Amitsur and Herstein
about division algebras
An unitary algebra is an algebra with a unit. An unital subalgebra of an
unitary algebra is a subalgebra containing the unit element of the global
algebra. Hence a unital subalgebra is unitary. An unitary subalgebra is a
subalgebra which is unitary as an algebra. An unitary subalgebra does not
need to be unital as its unitary unit could differ from the unit element of
the global algebra. Its unit element is an idempotent of the global algebra.
The center of A is denoted by Z(A).
Let G be a group, T a subset of G and g G. By
g
we symbolize the con
jugation with g and by C
G
(T ) resp. N
G
(T ) the centralizer resp. normalizer
of T in G.
Our next focus is the proof of a theorem of Wedderburn about finite di
vision algebras. For this proof we need the following two propositions.
Proposition 1 Let D be a Kdivision algebra and T be a unital finite
dimensional subalgebra of D. Then T is a division algebra, too.
Proof.
Let t T and assume t = 0.
We consider the right and the
left multiplication with t on T . Both functions are injective because D is a
division algebra. Hence  using the finite dimension of T  they are surjective,
too. In particular, 1 has a preimage with respect to these functions. Both
preimages are the inverse of t and therefor contained in T .
Proposition 2 Let G be a finite group and U be a subgroup of G. The the
U and G are equal if and only if G is the union of all Gconjugate subgroups
of U .
Proof.
If U is a normal subgroup the statement is true. Let U be a non
normal subgroup of G. Hence the statement G > N
G
(U ) is true. The
number of conjugates of U is exactly the index of the normalizer of U in
G which is
G
N
G
(U )
. All conjugated of U have at least the unit element in
common. Therefor we conclude:
19

gG
U
g
 1 +
G
N
G
(U )
· ( U  1).
The right hand side is because of U N
G
(U ) not greater than
1+  G  
G
N
G
(U )
.
By using G > N
G
(U ) we derive that this value is smaller than  G .
We will prove the following theorem by usage of the theory of centralsimple
associative division algebras. For this, let ind(D) (more exact: ind
K
(D)) the
index of a centralsimple finitedimensional associative unitary Kdivision al
gebra which is the unique dimension of all maximal subfields of D. A good
introduction to this theory can be found [49] and in [39].
Theorem 2 (Wedderburn) Every finite division algebra is a field. In par
ticular, its group of units is cyclic.
Proof.
Let D be a finite division algebra and K := Z(D). K is a field and
D a centralsimple finitedimensional associative unitary Kdivision algebra.
All maximal subfields have the same dimension ind
K
(D). Hence by using
the finiteness of D they are of the same order. Based on the finite field
theory we know that all maximal subfields are isomorphic. Now we use the
theorem of SkolemNoether
1
and conclude that all maximal subfields are
conjugated. Every element d of D is contained in a maximal subfield of D
1
Thoralf Albert Skolem (born 23 May 1887, died 23 March 1963) was a Norwegian
mathematician who worked in mathematical logic and set theory. Although Skolem's fa
ther was a primary school teacher, most of his extended family were farmers. Skolem
attended secondary school in Kristiania (later renamed Oslo), passing the university en
trance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study
mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he
began working as an assistant to the physicist Kristian Birkeland, known for bombarding
magnetized spheres with electrons and obtaining auroralike effects; thus Skolem's first
publications were physics papers written jointly with Birkeland. In 1913, Skolem passed
the state examinations with distinction, and completed a dissertation titled Investigations
on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the
zodiacal light. He spent the winter semester of 1915 at the University of G¨
ottingen, at the
time the leading research center in mathematical logic, metamathematics, and abstract
algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research
fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathemat
ics and was elected to the Norwegian Academy of Science and Letters. Skolem did not
at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary
in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theo
rems about integral solutions to certain algebraic equations and inequalities. His notional
thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married
Edith Wilhelmine Hasvold. Skolem continued to teach at Det kongelige Frederiks Uni
versitet (renamed the University of Oslo in 1939) until 1930 when he became a Research
Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to
conduct research free of administrative and teaching duties. However, the position also
required that he reside in Bergen, a city which then lacked a university and hence had no
20
because the subalgebra of D generated by {d, 1} is a subfield of D (see propo
sition 1). Therefor D is the union of all maximal subfields of D. From this
we derive that E(D) is the union of all groups of units of all maximal sub
fields and that these subgroups are conjugated. We can apply proposition 2
and conclude that D and one maximal subfield of D are identical. The proof
is complete and the addon is a consequence of this result and of theorem 1.
Let V be a Klinear space and T a subset of V . By T
V
we denote the
Klinear span of T in V . GF (p
n
) resp. GF (q) symbolize a finite field of
order p
n
resp. q (Galois field).
By usage of our previous results we derive two theorems proven by Her
stein:
Theorem 3 (Herstein) Every finite abelian subgroup of a division algebra
is cyclic.
Proof.
Let G be a finite abelian subgroup of a division algebra D. D is
a Z(D)Algebra. We focus on the Z(D)linear span of G in D. By using
proposition 1 we obtain that this span is a finitedimensional unital Z(D)
division algebra. G is commutative, and hence G
Z(D)
is a field and G a
finite subgroup of its groups of units. The proof is finished by using theorem
1.
Theorem 4 (Herstein) Every finite subgroup of a division algebra in posi
tive characteristics is cyclic.
Proof.
Let D be a division algebra, P the central prime subfield isomor
phic to GF (p) and G a finite subgroup of E(D). We focus on the unital
P subalgebra G
P
of the P division algebra D. This division algebra is by
the finiteness of G finitedimensional. Therefor proposition 1 implies that it
is a division algebra over P . P is finite and we conclude that this division
algebra is finite, too. By usage of theorem 2 of Wedderburn it is a field. The
corresponding theorem 1 for fields implies that G is as a finite subgroup
cyclic.
research library, so that he was unable to keep abreast of the mathematical literature. In
1938, he returned to Oslo to assume the Professorship of Mathematics at the university.
There he taught the graduate courses in algebra and number theory, and only occasionally
on mathematical logic. Skolem's Ph.D. student Øystein Ore went on to a career in the
USA. Skolem served as president of the Norwegian Mathematical Society, and edited the
Norsk Matematisk Tidsskrift (The Norwegian Mathematical Journal) for many years. He
was also the founding editor of Mathematica Scandinavica. After his 1957 retirement,
he made several trips to the United States, speaking and teaching at universities there.
He remained intellectually active until his sudden and unexpected death. For more on
Skolem's academic life, see Fenstad (1970).
21
Remark 1 The previous theorem 4 is wrong in characteristic zero. In the
real quaternion algebra the quaternion group of order 8 is a finite but non
cyclic subgroup of the group of units.
Herstein and Amitsur
2
have classified the finite subgroups of division alge
bras. A first results deals with socalled metacyclic groups. These groups
are characterized by possessing a cyclic normal subgroup whose factor group
is cyclic, too. A group having only cyclic Sylow subgroups is called a Z
group. It can be proven that Zgroups are metacyclic.
Theorem 5 (Herstein) Every psubgroup with respect to a prime number
p = 2 of the group of units of a division algebra is cyclic. In particular,
every subgroup of uneven order of the group of units of a division algebra is
a Zgroup.
Proof.
We use theorem 5.3.7 in [63] to derive that a pgroup of uneven
order is cyclic if it possesses exactly one subgroup of order p. This pre
condition is with respect to 5.3.8 in [63] valid if every abelian subgroup is
cyclic. This was proven within theorem 3. The addon follows as all Sylow
subgroups are cyclic.
Remark 2 The previous theorem 5 fails for p = 2. In the real quaternion
algebra the quaternion group of order 8 is a finite but noncyclic subgroup
of the group of units. All of its subgroups are cyclic.
By C
n
or Z
n
we denote a cyclic group of order n N. If n, m are integers,
then let o
n
(m) := o
Z/nZ
(mZ).
We formulate the classification of finite
subgroups of division algebras (but we will not prove it here) in characteristic
zero:
Theorem 6 (Amitsur) Every finite subgroup of the group of units of a divi
sion algebra in characteristic zero is isomorphic one of the following groups:
(i) C
n
2
Shimshon Avraham Amitsur (born August 26, 1921, died September 5, 1994) was
an Israeli mathematician. He is best known for his work in ring theory, in particular PI
rings, an area of abstract algebra. Amitsur was born in Jerusalem and studied at the
Hebrew University under the supervision of Jacob Levitzki. His studies were repeatedly
interrupted, first by World War II and then by the Israel's War of Independence. He
received his M.Sc. degree in 1946, and his Ph.D. in 1950. Later, for his joint work with
Levitzki, he received the first Israel Prize in Exact Sciences. He worked at the Hebrew
University until his retirement in 1989. Amitsur was a visiting scholar at the Institute
for Advanced Study from 1952 to 1954. He was an Invited Speaker at the ICM in 1970
in Nice. He was a member of the Israel Academy of Sciences, where he was the Head for
Experimental Science Section. He was one of the founding editors of the Israel Journal of
Mathematics, and the mathematical editor of the Hebrew Encyclopedia. Amitsur received
a number of awards, including the honorary doctorate from BenGurion University in 1990.
His students included Avinoam Mann, Amitai Regev, Eliyahu Rips and Aner Shalev.
22
(ii) A Zgroup of the form C
m
C
4
for which C
4
acts per inversion on C
m
and m is uneven.
(iii) A Zgroup of the form T
0
× · · · × T
s
in which the orders of these factors
are pairwise prime to each other, T
0
is cyclic, every T
i
, i s is non
cyclic of the form C
p
a
(C
q
b1
1
× · · · × C
q
br
r
), the prime numbers p, q
i
,
i r are distinct, for every i r the semidirect product C
p
a
C
q
bi
i
is noncyclic and is satisfying the following condition: if C
p
c
is the
kernel of the operation of C
q
bi
i
on C
p
a
, then one of the following cases
are valid:
(q
i
= 2, p 1 mod 4, c = 1) or
(q
i
= 2, p 1 mod 4, 2
c+1
does not divide p
2
 1) or
(q
i
= 2, p 1 mod 4, 2
c+1
does not divide p  1) or
(q
i
> 2, q
c+1
i
does not divide p  1.)
In addition, for every noncyclic factor C
p
a
C
q
bi
i
within every factor
T
j
the statement q
i
· o
p
c
(p) does not divide o
T /T
i

(p) is valid.
(iv) C
m
Q
2
t
in which m is uneven, an element of Q
2
t
of order 2
t1
centralizes the group C
m
and an element of order 4 of Q
2
t
inverts the
group C
m
.
(v) Q
8
× Z in which Z is a Zgroup of order m presented in (i), (ii) and
2 has uneven order in Z/Zm.
(vi) SL(2, 3) × Z in which Z is a Zgroup of order m presented in (i), (ii)
and 2 has uneven order in Z/Zm.
(vii) The binary octahedral group of order 48.
(viii) The binary icosahedral group of order 120.
Proof.
see Amitsur [1], Herstein [20] and Lam [38]
2.3
Exercises
Excercise 1 Read the article [1] of Amitsur. Determine for all finite sub
groups of division algebras suitable division algebra in which they are ap
pearing!
Excercise 2 Define metaabelian and supersolvable groups by a research in
the literature.
Excercise 3 Prove or disprove the following statements:
(i) Every cyclic group is metacyclic.
23
(ii) The converse of (i) is valid.
(iii) Every abelian group is metacyclic.
(iv) The converse of (i) is valid.
(v) Direct products of metacyclic groups are metacyclic.
(vi) Semidirect products of metacyclic groups are metacyclic.
(vii) Every metacyclic group is supersolvable.
(viii) Every metacyclic group is metaabelian.
(ix) A group for which all Sylow subgroups are cyclic is metacyclic.
(x) A group of squarefree order is metacyclic.
(xi) Dihedral groups are metacyclic.
(xii) Quaternion groups are metacyclic.
(xiii) Semidihedral groups are metacyclic.
Excercise 4 Prove the following statements: An unitary algebra is an alge
bra with a unit element. A unital subalgebra of an algebra A is a subalgebra
containing the unit element of A. A unital subalgebra is unitary. A uni
tary subalgebra is a subalgebra which is unitary as an algebra. A unitary
subalgebra is not unital in general. (Tip: idempotent elements)
Excercise 5 By using an article [20] of Herstein prove the following state
ments (p prime number, D a skew field and U a subgroup of E(D)):
(i) If U is of order p or p
2
, then U is cyclic.
(ii) If p = 2 and U is a pgroup, then U is cyclic.
(iii) Is part (ii) true for p = 2?
(iv) If the order of U is uneven, then U is metacyclic.
Excercise 6 True or false: The unit group of an infinite field is cyclic. Is
it possible to characterize finite fields by characteristics of their unit group?
Excercise 7 Determine all finite subgroups of the multiplicative group of
complex numbers! How many nonisomorphic subgroups of order n are ex
isting? Visualize them for n 8 on the complex plane!
Excercise 8 Are there finite subgroups in the additive group of complex
numbers? On what terms do finite subgroups of the additive group of a field
exist which are nontrivial? What is the answer for the multiplicative group?
24
Excercise 9 The additive group of a field is not isomorphic to the multi
plicative group of a field.
Excercise 10 Focus on the expfunction from R to R
>0
. Is it a homomor
phism for the addition and multiplication?
Excercise 11 Is the exponential map related to the complex numbers a ho
momorphism for the additive and multiplicative structure?
Excercise 12 Every finite generated subgroup of the multiplicative group of
rational numbers is cyclic.
Excercise 13 Determine the finite subgroups of the multiplicative group of
the real numbers.
Excercise 14 Determine the finite subgroups of the additive group of the
real numbers.
Excercise 15 Are there infinite subgroups of the additive and multiplicative
group of integers, rational, real and complex numbers?
Excercise 16 By a research of the literature determine all finite subgroups
of the additive and multiplicative group of integers, rational, real and complex
numbers. Are there subgroups which are simultaneously subgroups of the
additive and multiplicative group of these structures?
Excercise 17 True or false: Every finite subgroup of a division algebra is
cyclic! Analyze this question for the additive group of a division algebra.
Excercise 18 By a research in the literature determine all finite subgroups
of the real quaternion algebra.
Excercise 19 The additive factor group Q/Z is isomorphic to the group of
complex roots of unity.
Excercise 20 The additive factor group R/Z is isomorphic to the group of
complex roots of unity of absolute value 1.
Excercise 21 Determine the finite subgroups of the group of units of a
quaternion algebra in characteristic 2.
Excercise 22 Determine the finite subgroups of the group of units of a
quaternion algebra in characteristic p 3.
Excercise 23 Determine the finite abelian subgroups of the group of units
of a quaternion algebra in arbitrary characteristic.
Excercise 24 True or false: Every finite subgroup of the group of units of
a quaternion algebra in arbitrary characteristic is cyclic or abelian!
Chapter 3
Normal and subnormal
subgroups in unit groups of
division algebras
In this chapter we summarize some results of normal and subnormal sub
groups in unit groups of division algebras. For some of them we provide a
proof, for the others we reference the corresponding literature. We will use
some of these results in the next chapters. Therefor these results provide
the reader a deeper insight for understanding these results. In addition, this
chapter is included on personal interest of the author for the proofs of these
results.
3.1
The theorem of CartanBrauerHua
For two elements a, b of a group we denote by [a, b] the commutator of a
and b. The theorem of CartanBrauerHua focus on the structure of the
normal subgroups in unit groups of division algebras. Its proof is based on
the socalled Huaidentity:
25
26
Proposition 3 (Huaidentity
1
) Let D be a division algebra, a, b E(D)
with a = 1. The following statements are valid:
(i) b
1
(a  1)b = b
1
ab  1
(ii) b
1
ab = a[a, b]
(iii) (a  1)[a  1, b] = a[a, b]  1
(iv) a([a, b]  [a  1, b]) = 1  [a  1, b], and [a, b]  [a  1, b] is not zero if
[a, b] = 1 is valid.
By using the identity in part (iv) we derive that every noncentral element
is contained in the sub(division)algebra of D generated by all commutators
of E(D):
a = (1  [a  1, b])(a([a, b]  [a  1, b]))
1
.
Proof.
The proof is an exercise for the reader (see exercise 26).
Theorem 7 (CartanBrauerHua) Let D be a division algebra and T an
unital divisions subalgebra of D. If E(T ) is a normal subgroup E(D), then
T is central or identical to D.
Addon: If N is a normal subgroup of E(D), then N is central or the sub
algebra generated by N (= N
K
) is exactly D.
Proof.
(Sysak [67]) We assume that T is not central. Let b be an element
of T \ Z(D). We focus on an element a D such that ab = ba is valid. E(T )
is a normal subgroup of E(D). Therefor we conclude:
1 = [a, b] = (a
1
b
1
a)b T
and with an analog argument
1 = [a  1, b] T .
By usage of proposition 3 we derive the identity
1
Hua Luogeng, or Hua LooKeng (born 12 November 1910; died 12 June 1985), was a
Chinese mathematician famous for his important contributions to number theory and for
his role as the leader of mathematics research and education in the People's Republic of
China. He was largely responsible for identifying and nurturing the renowned mathemati
cian Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach
conjecture. In addition, Hua's later work on mathematical optimization and operations
research made an enormous impact on China's economy. Hua did not receive a formal
university education. Although awarded several honorary PhDs, he never got a formal
degree from any university. In fact, his formal education only consisted of six years of
primary school and three years of middle school. For that reason, Xiong Qinglai, after
reading one of Hua's early papers, was amazed by Hua's mathematical talent, and in 1931
Xiong invited him to study mathematics at Tsinghua University.
27
a = (1  [a  1, b])(a([a, b]  [a  1, b]))
1
T .
Every element of D not commutating with b is contained in T . Let c be
an element commutating with b. Then a + c is not commutating with b
(because a would be commutating with b). Hence a + c and a are contained
in T and therefor their difference c, too. We conclude the identity T = D.
The addon is a direct consequence of this result.
3.2
The theorem of Scott
The theorem of Scott analyzes on what terms the group of unit of a division
algebra is solvable.
Theorem 8 (Scott) Let D be a division algebra. The factor group E(D)/E(Z(D))
possesses no nontrivial nonabelian normal subgroup. In addition, E(D) is
solvable if and only if K is a field.
Proof.
(Sysak [67]) We assume there is a noncentral normal subgroup A
of E(D) such that A/Z(D) is abelian. The subalgebra of D generated by
A fulfills the preconditions of the theorem of CartanBrauerHua 7, and we
conclude that it is identical to D. Hence A is nonabelian and two elements
a, b A exist with 1 = [a, b] =: c Z(E(D)). We prove that Z(E(D)) is a
subgroup of order 2 of E(D) and a
2
, b
2
Z(E(D)) is valid. The identities
[1 + a, b]
=
[1 + a]
1
b
1
(1 + a)b
=
(1 + a)
1
(1 + ac)
are valid, and hence we conclude
[[1 + a, b]b]
=
[(1 + a)
1
(1 + ac), b]
=
(1 + ac)
1
(1 + a)(1 + ac)
1
(1 + ac
2
)
=
(1 + a)(1 + ac)
2
(1 + ac
2
).
With a similar argument we derive:
[[[1 + a, b], b], b]
=
(1 + a)
1
(1 + ac)
3
(1 + ac
2
)
3
(1 + ac
3
).
Because of [1 + a, b] A we get [[[1 + a, b], b], b] = 1. Hence
(1 + ac)
3
(1 + ac
3
)
=
(1 + a)(1 + ac
2
)
3
=
0
28
is valid. We conclude
a(3c + c
3
) + a
3
(c
3
+ 3c
5
)
=
a(1 + 3c
2
) + a
3
(3c
4
+ c
6
)
which is equivalent to
(1  c)
3
+ a
2
c
3
(1 + c)
3
= 0.
By using 1  c = 0 we derive
a
2
= c
3
Z(E(D)).
A symmetric argument let us deduce b
2
Z(E(D)). For the element z
Z(E(D)) the identity (az)
2
= c
3
is valid because of az A and [az, b] = c.
Hence z
2
= 1 is true and we conclude that the exponent and the order of
Z(E(D)) is 2. Z(D) is the prime field of order 3 in D and a, b elements of
finite order. The subgroup generated by {a, b} is nilpotent and hence as a
torsion group finite. By using theorem 4 the subgroup is cyclic. This is a
contradiction to the choice of a, b.
3.3
The theorem of Stuth
The next result which is formulated but not proven in details now general
izes the previous results of CartanBrauer
2
Hua and of Scott to the structure
of the subnormal subgroups and was proven by Stuth in [66]:
2
Richard Dagobert Brauer (born February 10, 1901, died April 17, 1977) was a leading
German and American mathematician. He worked mainly in abstract algebra, but made
important contributions to number theory. He was the founder of modular representation
theory. Alfred Brauer was Richards brother and seven years older. Alfred and Richard
were both interested in science and mathematics, but Alfred was injured in combat in
World War I. As a boy, Richard dreamt of becoming an inventor, and in February 1919
enrolled in Technische Hochschule BerlinCharlottenburg. He soon transferred to Univer
sity of Berlin. Except for the summer of 1920 when he studied at University of Freiburg,
he studied in Berlin, being awarded his Ph.D. on 16 March 1926. Issai Schur conducted
a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and
published a result. The problem also was solved by Heinz Hopf at the same time. Richard
wrote his thesis under Schur, providing an algebraic approach to irreducible, continu
ous, finitedimensional representations of real orthogonal (rotation) groups. Ilse Karger
also studied mathematics at the University of Berlin. She and Richard were married
17 September 1925. Their boys George Ulrich (1927) and Fred Gunther (1932) also be
came mathematicians. Brauer began his teaching career in K¨
onigsberg (now Kaliningrad)
working as Konrad Knopp's assistant. Brauer expounded central division algebras over
a perfect field while in K¨
onigsberg: the isomorphism classes of such algebras form the
elements of the Brauer group he introduced. When the Nazi Party took over in 1933, the
Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer
and other Jewish scientists. Brauer was offered an assistant professorship at University
of Kentucky. Richard accepted the offer, and by the end of 1933 he was in Lexington,
Kentucky, teaching in English. Ilse followed the next year with George and Fred; brother
Alfred made it to the USA in 1939, but their sister Alice was killed in The Holocaust.
29
Theorem 9 (Stuth) Let D be a division algebra. The following statements
are valid:
(i) The central subgroups of E(D) are exactly the solvable subnormal sub
groups of E(D).
(ii) For every noncentral subnormal subgroup S of E(D) the identity C
E(D)
(S) =
Z(E(D)) is valid.
(iii) The intersection of two noncentral subnormal subgroups of E(D) is
a noncentral subnormal subgroup of E(D), too. In particular, there
is a smallest noncentral subnormal subgroup of E(D) which is non
solvable.
Hermann Weyl invited Richard to assist him at Princetons Institute for Advanced Study
in 1934. Richard and Nathan Jacobson edited Weyl's lectures Structure and Representa
tion of Continuous Groups. Through the influence of Emmy Noether, Richard was invited
to University of Toronto to take up a faculty position. With his graduate student Cecil
J. Nesbitt he developed modular representation theory, published in 1937. Robert Stein
berg, and Stephen Arthur Jennings were also Brauer's students in Toronto. Brauer also
conducted international research with Tadasi Nakayama on representations of algebras.
In 1941 University of Wisconsin hosted visiting professor Brauer. The following year he
visited the Institute for Advanced Study and Bloomington, Indiana where Emil Artin was
teaching. In 1948 Richard and Ilse moved to Ann Arbor, Michigan where he and Robert
M. Thrall contributed to the program in modern algebra at University of Michigan. With
his graduate student K. A. Fowler, Brauer proved the BrauerFowler theorem. Donald
John Lewis was another of his students at UM. In 1952 Brauer joined the faculty of Har
vard University. Before retiring in 1971 he taught aspiring mathematicians such as Donald
Passman and I. Martin Isaacs. The Brauers frequently traveled to see their friends such
as Reinhold Baer, Werner Wolfgang Rogosinski, and Carl Ludwig Siegel.
31
3.4
Exercises
Excercise 25 Explain on what terms the theorem of Stuth (theorem 9) is
a generalization of the theorems of CartanBrauerHua and Scott.
Excercise 26 Prove all identities and statements within proposition 3!
Excercise 27 Study the article [67] with respect to the following topics:
definition of periodic groups, periodicity of the additive and multiplicative
group of Q, R and C, proof of the lemmas of Herstein and the theorem of
Jacobson that a division algebra with periodic group of unit is commutative.
In what way is this theorem a generalization of another theorem of Jacobson?
Excercise 28 On what terms is the unit group of a division algebra abelian?
Excercise 29 On what terms is the unit group of a division algebra nilpo
tent? (Tip: Theorem of Scott)
Excercise 30 On what terms is the unit group of a division algebra solv
able? (Tip: Theorem of Scott)
Excercise 31 On what terms is the unit group of a division algebra simple?
(Tip: center and theorem of Wedderburn)
Excercise 32 On what terms is the unit group of a division algebra cyclic?
(Tip: exercise 6)
Excercise 33 Let D be a division algebra. What do we know about finite
normal subgroups of E(D)?
Excercise 34 Let D be a division algebra. What do we know about finite
subnormal subgroups of E(D)?
Excercise 35 Do a research in literature and prove that a nilpotent and
finitegenerated torsion group is finite. For what reason do we need this result
in this chapter? (Tip: Use the main theorem on finitegenerated abelian
groups and apply it on the factor groups for the nilpotent groups with respect
to a central chain of finite length!)
Excercise 36 By using the tip in exercise 35 extend the results if possible
to solvable groups. Is this statement also valid for nonsolvable groups?
Chapter 4
Nilradicals of Lie algebras
associated to associative
algebras
For every associative algebra the socalled associated Lie algebra can be de
rived in a natural way. In this chapter we focus on the determination of
the nilradical the greatest nilpotent ideal of the associated Lie algebra.
We describe it by the structure of the associative algebra: it is the sum the
center and the nilradical of the associative algebra. In particular, the nil
radical of the associated Lie algebra is an unital associative subalgebra. We
assume uneven characteristic of the ground field and the separability of the
factor algebra by the nilradical of the associative algebra. By assuming this
we can use the wellknown theorem of WedderburnMalcev for the existence
and conjugacy of radical complements in the associative algebra.
Our strategy is to start with solvable associative algebras. For this, the gen
eralized Jordan decomposition is the main tool for our analysis. We calculate
the nilradical for our standard examples (Solomon algebras, SolomonTits
algebras, group algebras and triangular matrices). In a next step we transfer
results of Herstein
1
about simple rings and its associated Lie ring to simple
1
Israel Nathan Herstein (born March 28, 1923, died February 9, 1988) was a math
ematician, appointed as professor at the University of Chicago in 1951. He worked on
a variety of areas of algebra, including ring theory, with over 100 research papers and
over a dozen books. Herstein was born in Lublin, Poland, in 1923. His family emigrated
to Canada in 1926, and he grew up in a harsh and underprivileged environment where,
according to him, you either became a gangster or a college professor. During his school
years he played football, ice hockey, golf, tennis, and pool. He also worked as a steeplejack
and as a barker at a fair. He received his B.S. degree from the University of Manitoba and
his M.A. from the University of Toronto. He received his Ph.D from Indiana University
in 1948. His advisor was Max Zorn. He held positions at the University of Kansas, Ohio
State University, University of Pennsylvania, and Cornell University before permanently
settling at the University of Chicago in 1962. He was a Guggenheim Fellow for the aca
demic year 1960/1961. He is known for his lucid style of writing, as exemplified by the
33
34
and semisimple algebras. The general case is studied in a last step using the
results for solvable and semisimple associative algebras.
We finish this chapter by applying the results to tensor products, adjunction
of a unit, matrix algebras, subalgebras etc. For this, we focus on describing
the nilradical by the underlying structure, e.g. by the factors of the tensor
product.
We use the term Lie nilpotent for the associated Lie algebra to pronounce
that the associated Lie algebra is nilpotent. In a similar matter we use the
term Lie solvable, Lie nilradical, Lie subalgebra, Lie ideal etc.
4.1
Solvable algebras
In this section we focus on solvable associative algebras and begin with the
definition of the most important tool for our analysis: the generalized Jordan
decomposition.
4.1.1
Generalized Jordan decomposition
Definitions and remarks 1 Let L be a KLie algebra and l L. We
define the multiplication with l also called adjoint representation by
ad(l) : L  L, x xl. The nilradical of L is the greatest nilpotent ideal
of L (if it is existing). The associated Lie algebra of an associative algebra
A with respect to the operation a b := ab  ba for all a, b A will be
symbolized by A
. If S, T are subsets of A
, then we denote by S T the
Klinear span of the set {s t  s S, t T }. Nilpotency and solvability is
used for groups and Lie algebras as usually.
By resp. we denote the left resp. rightregular representation of an
associative algebra A. Based on definition 5.2.1 in [74] we call a polynomial
fully separable if it is squarefree and separable. If A is an associative unitary
Kalgebra, then an element a A is fully separable if and only if its minimal
polynomial min
a,K
is fully separable. This is equivalent to the statement
that min
a,K
has no multiple roots in K[t] for every splitting field F of K.
Another description of this property is that min
a,K
and its formal deriva
tion min
a,K
have no nontrivial common divisor: gcd(min
a,K
, min
a,K
) = 1.
classic and widely influential Topics in Algebra, an undergraduate introduction to abstract
algebra that was published in 1964, which dominated the field for 20 years. A more ad
vanced classic text is his Noncommutative Rings in the Carus Mathematical Monographs
series. His primary interest was in noncommutative ring theory, but he also wrote papers
on finite groups, linear algebra, and mathematical economics. He had 30 Ph.D. students,
traveled and lectured widely, and spoke Italian, Hebrew, Polish, and Portuguese. He died
from cancer in Chicago, Illinois, in 1988. His doctoral students include Miriam Cohen,
Susan Montgomery, Karen Parshall, Claudio Procesi, Lance Small, and Murray Schacher.
35
It is straightforward to prove that a, a and a have the same minimal
polynomial. Thus a is nilpotent resp. fully separable if and only if a or a
has this property. Based on definition 5.1.4.1 in [74] a pair (r; s) A × A is
called a generalized Jordan decomposition of a A if a = r + s is valid, r
and s commute, r is nilpotent and s is fully separable. The nilradical of A
is denoted by rad(A). A is solvable if A/rad(A) the factor algebra by its
nilradical is commutative. If rad(A) possesses an complement T which is
a subalgebra, then we call T a radical complement: A = rad(A) T .
Lemma 1 Let K be a field, A an associative unitary finitedimensional K
algebra and a, r, s A. If (r; s) is a generalized Jordan decomposition of a,
then (ad(r); ad(s)) is one of ad(a).
Proof.
Step 1: It is straightforward to prove that a, (a) and (a) possess
the same minimal polynomial and ad(a) = ad(r + s) = ad(r) + ad(s) is valid.
Step 2: We prove now that ad(r) and ad(s) commute: A is associative.
We conclude that for all x, y A the functions (x) and (y) commute.
Because of rs = sr the functions (r) and (s) as well as (r) and (s)
commute. We conclude:
ad(r)ad(s)
=
((r)  (r))((s)  (s))
=
(r)(s)  (r)(s)  (r)(s) + (r)(s)
=
(s)(r)  (s)(r)  (s)(r) + (s)(r)
=
((s)  (s))((r)  (r))
=
ad(s)ad(r).
Step 3: By usage of Step 1 we derive that r and hence also (r) and (r)
are nilpotent. A is associative and therefor (r) (r) = (r) (r). As a con
sequence the subalgebra generated by {(r), (r)} is commutative and by
proposition 5 in [74] even nilpotent. Thus ad(r) = (r)  (r) is nilpotent,
too.
Step 4: By usage of Step 1 we derive that r and hence also (r) and (r)
are fully separable. A is associative and therefor (r) (r) = (r) (r). As a
consequence the subalgebra generated by {(r), (r)} is commutative and by
proposition 5 in [74] separable. Thus ad(r) = (r)  (r) is fully separable,
too.
4.1.2
The nilradical
In this section we determine the nilradical of a Lie algebra associated to
a solvable associative algebra. For this we will use our results about the
Jordan decomposition.
36
Definitions and remarks 2 An associative finitedimensional unitary com
mutative algebra with separable factor algebra by the nilradical possesses
exactly one radical complement: the set of all fully separable elements. If
the algebra is not commutative but solvable and T is a radical complement,
then T Z(A) is a radical complement of the center of A. This set coincides
with the intersection of all radical complements of rad(A) in A (see e.g. [74],
chapter 5).
Let L be a Lie algebra. A subalgebra M of L is called maximal nilpotent if
M is not contained in a proper nilpotent subalgebra of L.
Theorem 10 Let K be a field, A an associative unitary finitedimensional
solvable Kalgebra with separable factor algebra by the nilradical and Z the
radical complement of the center of A. The nilradical of A
is rad(A) Z =
rad(A) + Z(A). In particular, the nilradical of A
is an unital associative
subalgebra of A and the only maximal nilpotent subalgebra of A
containing
rad(A).
Proof.
Both ideals rad(A) and Z of A
are nilpotent, Hence by using
a theorem of Fitting their sum is nilpotent, too. Thus the nilpotent ideal
rad(A)Z is contained in the nilradical of A
. Let N be the nilradical of A
and n N . By the theorem of WedderburnMalcev a radical complement T
exists. Let r rad(A) and t T such that n = r + t is valid. As rad(A) is
contained in N we derive t N . N is a nilpotent Lie algebra and we conclude
that ad(t) is a nilpotent endomorphism of N . In particular, ad(t)
rad(A)
is a
nilpotent endomorphism of rad(A). A/rad(A) is separable and commutative
and by usage of theorem 5.3.1 in [74] every element of T is fully separable.
Lemma 1 implies that ad(t) is fully separable, too. In particular, ad(t)
rad(A)
is fully separable. We derive that ad(t)
rad(A)
is nilpotent and fully separable.
Lemma 1 implies ad(t)
rad(A)
= 0.
2
We conclude that t centralizes rad(A).
T is commutative and thus we get t T Z(A). This subalgebra is exactly
Z which is proven in 5.1.4 in [74].
The identity Z(A) = rad(Z(A)) + Z is valid. For an associative solvable
algebra the nilradical is exactly the set of all nilpotent elements (see e.g. [4]
or the section about reduced algebras in this volume). Therefor rad(A)+Z =
rad(A) + rad(Z(A)) + Z = rad(A) + Z(A) is valid
The addon is a consequence of the observation that every subalgebra of A
containing rad(A) is because of A A rad(A) an ideal of A
and the
nilradical coincide with the sum of an ideal and a subalgebra of A.
2
Every element possesses at most one Jordan decomposition (see e.g. chapter 5 in [74]).
37
4.2
Standard examples of solvable associative al
gebras
4.2.1
Solvable group algebras
In this section we determine the nilradical of the associated Lie algebra
(KG)
for a modular solvable group algebra KG with respect to dihedral
and quaternion groups. We need the following general insight about solvable
group algebras:
Preliminary remark 1 Let K be a field and G a finite group. By using
3.2.20 in [74] the group algebra KG is solvable if and only if G is abelian or
char(K) = p is valid and G the derived subgroup of G is a pgroup (p a
prime number). If G is abelian, then (KG)
is nilpotent. Let char(K) = p
and G a pgroup. Then there exists a pSylow subgroup P containing the
derived subgroup. Hence P is a normal pSylow subgroup of G, and therefor
its the only pSylow subgroup by using Sylow's theorem. The theorem of
SchurZassenhaus assured a complement H of P in G. If is the lineariza
tion of the natural epimorphism from G onto the factor group G/P , then
the kernel of is wellknown: Kern = KGAug(KP ) = Aug(KP )KG,
and Aug(KP ) is the augmentation ideal of KP . By a theorem of Wallace
Aug(KP ) is nilpotent, and thus Kern is nilpotent, too. The factor alge
bra KG modulo Kern is isomorphic to K(G/P ) which is isomorphic to
KH. KH is using Maschke's theorem semisimple, and we conclude by
1.9.4 in [74] that it is even separable. We derive rad(KG) = KG Aug(KP ),
and KH is a separable radical complement in KG. In particular, the di
mension of the nilradical is  G    H . By using results in [74], chapter
3, the intersection of all radical complements is the radical complement of
the center of KG. In particular, the radical complement of the center of
KG is contained in KH. The nilradical of (KG)
is determined by using
theorem 10 as the direct sum of the radical complement of the center of
KG and the nilradical (=KGAug(KP )) of KG. By using the same theorem
it is also the sum of the radical of KG with the center of KG, and the lat
ter subalgebra is Kspanned by the conjugacy class sums of G. The radical
complement of the center of KG can be constructed as follows: calculate the
generalized Jordan decomposition for a basis of Z(KG). The Klinear span
of the calculated fully separable parts is the desired radical complement of
the center of KG. This calculation is done within remark 3.
Let G be a group, K a field and T a finite subset of G. By T we denote the
sum of all elements of T in KG. C(G) symbolizes the set of all conjugacy
classes of G and k(G) the number of these classes. If U is a subgroup of G,
then the core of U in G is the greatest normal subgroup of G contained in
U . Its exactly the intersection of all conjugates of H in G and is symbolized
38
by core
G
(U ) =
gG
U
g
. By using the preliminary remark we prove now:
Theorem 11 (Lie nilradical of solvable group algebras) Let G be a
finite group and K a field with char(K) = p > 0 such that KG is solvable.
If H is a p Hall subgroup and P is the normal pSylow subgroup of G, then
(Aug(KP )KG) K(Z(G) H) is the nilradical of (KG)
. In particular,
the dimension of the Lie nilradical is  G    H  +  Z(G) H .
Proof.
We use some results of theorem 16 and the preliminary remark 1.
KH is a radical complement and Z(KG) KH is the radical complement of
Z(KG). We have to prove that this radical complement is exactly K(Z(G)
H). Then the proof is finished by using theorem 10. Surely, K(Z(G) H)
is contained in the other substructure. Let x :=
hH
k
h
h an element of
Z(KG)KH. The center of KG is spanned by the conjugacy class sums as a
Kspace, and we present x as
CC(G)
k
C
C. Let h H with k
h
= 0, then there
exists a conjugacy class C of G and an element c C with h = c and k
C
= 0.
All conjugates of c which is the set C are contained in H: the coefficient
of all elements of C is exactly k
C
and the conjugacy classes partition G.
Therefor all conjugates of c satisfy the identity
hH
k
h
h =
CC(G)
k
C
C and
are contained in H. We have proven h
G
= C H. By using h
G
H we
derive h
gG
H
g
. This set is the core of H in G. This normal subgroup has
trivial intersection with P , and thus every element of the core commutates
with all elements of P . H is abelian and G = P H is valid. This implies
that the core of H in G is central in G. We have proven h Z(G) H as
desired.
Corollary 1 Let p be a prime number, K a field with char(K) = p 3,
n N and G a group of order 2 · p
n
. The following conditions are valid:
(i) If a 2Sylow subgroup is central, then (KG)
is nilpotent.
(ii) If a 2Sylow subgroup is not central, then rad(KG) K · 1
G
is the
nilradical of (KG)
of dimension 2p
n
 1.
Proof.
The pSylow subgroup is a normal subgroup of order p
n
and of
index 2. The complement H in the preliminary remark 1 is of order 2. Thus
the radical complement KH is of dimension 2, and the radical complement
of the center of KG is KH or K · 1
G
(using again the preliminary remark
1). Now the proof is finished using again the preliminary remark 1.
For the dihedral groups we conclude:
39
Corollary 2 Let p be a prime number, K a field with char(K) = p 3,
n N and G = D
2p
n
. rad(KG) K · 1
G
is the nilradical of (KG)
of
dimension 2p
n
 1.
For the quaternion groups we derive:
Corollary 3 Let p be a prime number, K a field with char(K) = p 3,
n N
2
and G = Q
4p
n
. rad(KG) KZ(G) is the nilradical of (KG)
of
dimension 4p
n
 2.
Proof.
Let a, b G with o(a) = 2p
n
, o(b) = 4, z = b
2
= a
p
n
such that G is
generated by a, b, the center of G is generated by z and a
b
= a
1
z is valid.
We use the results of the preliminary remark 1. The derived subgroup of G
is generated by a
2
and is thus a pgroup of order p
n
. One of its complement
H is generated by b and is of order 4. 1 and z are fully separable because
p = 2 is valid, and H is noncentral. Therefor the radical complement of the
center of KG is contained in KH and possesses the dimension 2 or 3. By
usage of theorem 11 its exactly K(Z(G) H) and we conclude that it is of
dimension 2.
At the end of this section we determine (as announced earlier in this sec
tion) the generalized Jordan decomposition of the conjugacy classes within
the center Z(KG) of KG for solvable KG:
Remark 3 (Generalized Jordan decomposition of the conjugacy
class sums) Let K be a field with char(K) = p = 0, G a finite group with
normal pSylow subgroup P and abelian Hall complement H. Z(KG) =
(rad(KG) Z(KG)) K(Z(G) H) is valid using theorem 11. rad(KG) is
Klinear spanned by the elements (p  1)h with p = 1, p P, h H (using
the preliminary remark 1).
Let g
G
be a conjugacy class of G for an element g which is not central
in G. Thus the length of g
G
is at least 2.
We begin to focus on the case g = p P . Let C(p) be a set of conjugators
of p in G which is one of the minimal sets M in G with p
G
= p
M
. The
following equation is valid:
p
G
=
xC(p)
p
x
=
xC(p)
((p
x
 1) + 1)
=
(
xC(p)
(p
x
 1))+  p
G
 ·1
KG
.
40
The second summand which is a multiple of 1 is surely central. Thus the first
summand which lies in the nilradical of KG is central, too. As a consequence
we have found the decomposition. The fully separable part is zero (the class
sum consequently nilpotent) if and only if the length of the conjugacy class
is divided by p. For example, this is the case if P is abelian.
Let g = h H with h
G
H. Following the proof of theorem 11 the element
h is central in G. h
G
is no subset of H. Let C(h) be a set of conjugators of
h in G. Then  C(h)  =  h
G
 is valid. For every g C(h) elements h
g
H
and p
g
P exist with g = h
g
p
g
. As H is abelian, we derive h
g
= h
p
g
.
This element can be rewritten to (p
1
g
 1)hp
g
+ h(p
g
 1) + h. Thus there
exists an element j rad(KG) with h
G
= j+  h
G

K
·h. H is abelian, and
therefor H is contained in C
G
(h). We conclude that  h
G
 is a power of p
or h ist central. By our assumption h is not central, and we derive that h
G
is nilpotent (because of char(K) = p).
Let g = ph with p = 1 = h. We begin our analysis by assuming that p is
central. Thus h is not central and (ph)
G
= h
G
· p is valid. By our analysis so
far we deduct the nilpotency of h
G
and the one of (hp)
G
. Now let h central
and p not central. We calculate (ph)
G
= p
G
· h. The conjugates of ph in
G are closely connected to the ones of p in G, because h is central. There
exist elements j rad(KG) and k K with h
G
= j + k · 1
G
. We conclude
(ph)
G
= h · j + k · h. Because h Z(G) H is valid, the element h · j is
central in KG and contained in rad(KG). In this case we have determined
the Jordan decomposition. Now let p and h be not central. We proceed as
in the case for h
G
. Let C(ph) be a set of conjugators of ph in G. For every
x C(ph) let p
x
P and h
x
H such that x = h
x
p
x
is valid. We derive:
(ph)
G
=
xC(ph)
(p
x
 1)h
x
+
xC(ph)
h
x
.
The first sum is contained in rad(KG). For the second sum we calculate
further:
xC(ph)
h
x
=
xC(ph)
h
p
x
=
xC(ph)
(p
1
x
) + h(p
x
 1) +
xC(ph)
h.
Thus there exists an element j rad(KG) such that
xC(ph)
(ph)
x
= j+  C(ph) 
K
·h
41
is valid. This decomposition is unique as direct sum of elements of rad(KG)
and KH.
xC(ph)
(hp)
x
decomposes in Z(KG) as a sum of elements of
Z(KG) rad(KG) and of Z(KG) KH. This sum is Kdirect. There
for both decompositions are identically. h is not central, and thus  C(ph) 
must be divided by p and is zero. In particular, the whole conjugacy class
sum is nilpotent.
Only the case g Z(G) is remaining now. For this we prove that P Z(G)
is the normal pSylow subgroup with abelian complement Z(G)H in Z(G).
If g Z(G), then there exist p Z(G) P and h Z(G) H such that
g = ph = (p  1)h + h is valid. This is the desired decomposition. The prime
divisors of  Z(G)  are ones of  G , too. As an abelian group Z(G) is the
direct product of its Sylow subgroups. The theorem of Sylow implies that
we can conjugate them into the corresponding ones of G. But the center is
central and the conjugation on the center is the identity. As a consequence
they are contained in every corresponding Sylow subgroup of G.
As another application we determine within the next sections the Lie nil
radical for our standard examples: the SolomonTits algebras, the Solomon
algebras and the algebras of upper and lower triangular matrices.
4.2.2
Triangular matrices
Let K be a field and n N. The algebra of lower triangular matrices are
examples of solvable associative Kalgebras with separable factor algebra by
its nilradical. The nilradical of the algebra of lower triangular matrices of
K
n×n
 denoted by
u,n
 is the subalgebra of strict lower triangular matrices
 symbolized by s
u,n
. The nilradical has the dimension
n1
i=1
i =
1
2
(n  1)n.
u,n
is central, its center is of dimension 1. By theorem 10 we derive:
Corollary 4 For the Lie nilradical of the algebra of lower triangular ma
trices the following identities are valid:
(i) nil(
u,n
) = s
u,n
K · 1
(ii) dim
K
(nil(
u,n
)) = 1 +
1
2
(n  1)n.
Let K be a field and n N. The algebra of upper triangular matrices are
examples of solvable associative Kalgebras with separable factor algebra by
its nilradical. The nilradical of the algebra of upper triangular matrices of
K
n×n
 denoted by
o,n
 is the subalgebra of strict upper triangular matrices
 symbolized by s
o,n
. The nilradical has the dimension
n1
i=1
i =
1
2
(n  1)n.
o,n
is central, its center is of dimension 1. By theorem 10 we derive:
42
Corollary 5 For the Lie nilradical of the algebra of upper triangular ma
trices the following identities are valid:
(i) nil(
o,n
) = s
u,n
K · 1
(ii) dim
K
(nil(
o,n
)) = 1 +
1
2
(n  1)n.
4.2.3
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 the Klinear span of all class sums of socalled defect
classes in KS
n
: if S
n
, then D() := {i  i > (i + 1)}. The Solomon
algebra is the Klinear span of {
D()=D
 D n  1
}. The surprising
insight of Luis Solomon was that products of two defect class sums is a
linear combination of defect class sums. Thus the linear span of the defect
class sums is indeed an associative algebra. The Solomon algebra possesses
the dimension 2
n1
, and its factor algebra by the nilradical is of dimension
p(n)  the number of partitions of n. D
n
is a solvable associative algebra with
separable factor algebra by its nilradical. Its center is semisimple. Thus the
intersection of the center with the nilradical is trivial. For even resp. uneven
n the center is of dimension 3 resp. 2. For details the reader may study
the dissertation of Thorsten Bauer [4], in particular chapter 3, in which the
descriptions of the radical is contained, too. By theorem 10 we derive:
Corollary 6 For the Lie nilradical of the Solomon algebra in characteristic
zero the following identities are valid:
(i) nil(D
n
) = rad(D
n
) Z(D
n
)
(ii) dim
K
(nil(D
n
)) = 2
n1
 p(n) + 3, n even.
(iii) dim
K
(nil(D
n
)) = 2
n1
 p(n) + 2, n uneven.
4.2.4
SolomonTits algebras
Let K be a field and n N. By S(n, k) we denote the socalled Stirling
numbers with resp. to k n
0
. Its the number of unordered set partitions of
n consisting of exactly k subsets. In [76] the SolomonTits algebra K
n
is
analyzed in details for several questions and problems based on a paper by
Manfred Schocker. The algebra is defined as the monoid algebra K
n
with
resp. to the monoid
n
. This monoid consists of all ordered set partitions
of n. If (P
1
, · · · , P
l
) and (Q
1
, · · · Q
k
) are two elements of this monoid, then
their product 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
)
.
43
The symbol
signalizes that empty sets are deleted from this tuple. K
n
is again an example for a solvable associative algebra with separable factor
algebra by its nilradical.
The nilradical of K
n
is described in several ways which are not presented
here (see e.g. chapter 2 in [76]), and its dimension is dim
K
(rad(K
n
)) =
n
k=0
(k!  1) S(n, k) (see corollary 8 in [76]).
The center of K
n
is one
dimensional because K
n
is central (see theorem 13 in [76]). Therefor we
can apply theorem 10 and conclude:
Corollary 7 For the Lie nilradical of the SolomonTits algebra the follow
ing identities are valid:
(i) nil(K
n
) = rad(K
n
) K · 1
(ii) dim
K
(nil(K
n
)) = 1 +
n
k=0
(k!  1) S(n, k).
Details
 Pages
 Type of Edition
 Erstausgabe
 Year
 2016
 ISBN (PDF)
 9783960676034
 ISBN (Softcover)
 9783960671039
 File size
 10.6 MB
 Language
 English
 Publication date
 2016 (December)
 Keywords
 Cartan subalgebra Nilradical Lie algebra Associative algebra Maximal nilpotent Group algebra Exercise Lie nilpotency Salvatore Siciliano