Inter. J. Math. Math. Sc.
Vol. No. 2 (1980) 24-25
247
RINGS WITH INVOLUTION
WHOSE SYMMETRIC ELEMENTS ARE CENTRAL
TAW PIN LIM
Department of Actuarial Mathematics
University of Manitoba
Winnipeg, Manitoba R3T 2N2
Canada
(Received Hay 30, 1978 and in revised form February 6, 1979)
ABSTRAC______T. In
a ring
R with involution whose symmetric elements
the skew-symmetric elements
K
S
are centrl,
form a Lie algebra over the commutative ring
S.
The classification of such rings which are 2-torsion free is equivalent to the
classification of Lie algebras
K
over
S
that is symmetric, invariant and satisfies
If
S
is a field of char
Lie algebra if and only if
K’
equipped with a bilinear form
[[x,y],z]
2, f # 0 and dim K > i
f
is nondegenerate.
is either the pure quaternions over
orthogonal abelian Lie ideals of
S
then
f(y,z)x
K
f
f(z,x)y.
is a semisimple
Moreover, the derived algebra
or a direct sum of mutually
dim <_ 2.
/
KEF WORDS AN PHRASES. Ring h involation, ric and kemmetc leme,
L ebr, ’ymmi and invaiant 6iline form, Cn’ Criterion of
miimplicit of Li eb, p quernions, mutually orthogonal abtian
Lie idea.
1980 flhemZic Sbject Classification Cod:
16A68, 16A28, 17B05.
248
T.P. LIM
INTRODUCTION and MAIN RESULTS.
1.
Let
R
for all
(a + b)*
(ab)*
b’a*
R[a*
{a e R la*
a}, K
xy
R
2R
1
R
C
is called a
x e R,
For all
S
2x
+ K.
R
a CS-ring then for
x
R
If
implies that
R are respectively
-a}.
x,y e R
and the symbol
Z
R.
R
If the symmetric elements of
thus
such that
a.
denotes the commutator of
yx
denotes the center of
abbreviation,
a**
and
and .s.ke..W-symmetric elements of
_ymetr!.c
[x,y]
As usual,
+ b*,
a*
{a e
S
R
/
R
a,b
The sets of
R
i.e., a map
*,
be a ring with an involution
+
S,
then for
_.
cs-!in
x*
+
x* ,,.,ith
x
x
is 2-torsion free then
S
is a group direct sum
a
Z,
S
are central, i.e.,
x e K,
ax
S
e S,
0
x* e K
x
K
K.
If, additionally,
K
and
and hence
S
-(ax)*
xa
a Lie algebra over the commutative ring
+ x*
and therefore
R
K
is
is
with respect to commutation.
We have the following converse:
THEOREM i. If
over
S
S
f:K K
and
/
S
is an S-bilinear map such that
f(y,x)
(i)
f (x,y)
(2)
f(x,[y,z])
(3)
[[x,y],z]
f([x,y],z)
R
S
K
(4)
(a + x)(b + y)
(5)
(a+x)*
because
is
(f
is invariant)
can be made into a CS-ring by defining the
multiplication and the involution, for all
Let
symmetric)
(f
f(y,z)x- f(z,x)y
then the group direct sum
PROOF.
is a 2-torsion free Lie algebra
K
is a commutative ring,
ab
+ f(x,y) + ay +
a,b
bx
K,
as follows:
+ [x,y]
a- x.
a,b,c e S
and
x,y,z
K.
Multiplication in
R
is associative
249
RINGS WITH INVOLUTION
((a + x)(b + y))(c + z)
(a + x)((b + y)(c + z))
f (x,y) z+f ([x,y], z)+[ [x,y] ,z]-f (y, z)x-f (x, [y,z])-[x, [y, z]
f(x,y) z- f(y,z)x- [[z,x],y]
by
(2)
0
by
(3).
and Jacobl identity
From (I), (4) and (5),
((a + x)(b + y))*
Hence,
*
Since
(b + y)*(a + x)*.
is an involution in
K
is 2-torslon
free,
skew-symmetrlc elements of
A
CS-ring will, of
2(xy + yx)
f(x,y)
Note that if
R.
R
S
and
K
and therefore
are precisely the symmetric and
R
is a
CS-rlng.
course, satisfy identities (i)
(3) if we put
for all x,y e K.
K has an
S-basls
and
f
is the dot product then
(2) is the
triple dot product and (3) is the "triple cross product" with opposite sign,
that is,
[[x,y],z]
(x y).
z
We must, however, recall that the cross product
of vectors is valid only for dimension
that
_<
and it can also be ([3,p.61] or [5])
3
satisfy the standard polynomial of degree 4.
CS-rings
An example of a
commutative ring
is a ring of quaternlons
CS-ring
S, where
admits an
S-basis
over a 2-torslon free
l,i,j,ij
such that given
a,b e S
and
i2
a,
j2
b, ij =-ji
i*
i,
i*
-i, j *
The skew-symmetric part
K
-j.
is a Lie algebra (with respect to commutation)
of
of pure quaternions.
Henceforth, we shall tacitly assum__e that K
of char
# 2 and that K
identities (I)
is equipped with an
(3) such that
R
FK
is a Lie algebra over a field
F-billnear
is a
CS-rlng
and the involution defined according to (4) and (5).
frm
f
F
satisfying
with the multiplication
250
T.P. LIH
As usual, the derived algebra and the radical of
K’
[I,K] C I.
with
by
x,y e K
x
A Lie ideal I
<x,y>
shall denote the
dim K
If
We may assume
PROOF.
# 3 then K’
dim K > 3
Prom (3), we have
dimension < 3.
x,y,z,w
0
[[x,y],[z,w + vii
PROPOSITION 2.
0
K
E
R.
Since
If
z
f # 0
E
Hence,
K
Putting
for all
K
x,y,z
in
K.
then
+
w
<x,y,z>.
v
Consequently,
[K’,K’]
0
for arbitrary
is an ideal of
R
contained in
0. f
If
f
K
then by (2),
0
[[x,y],[z,w]]
then
K"
K’
K"
f(x,[y,z])
Hence,
K
[[z,y],y]
f(y,y)z
f([x,y],z)
K’
K’
K’
COROLLARY i.
and
is a Lie ideal of
y
in
and thus
K
with
z
0
for all
K
and an ideal of
f(y,y)
0.
If
f(y,y)-l[[z,y],y]
e
K’.
dim K/K’ > i
K
and thus
there is a nonzero vector
then by (3),
Since
<x,y>
are linearly independent then we choose a
<x,y,z>
v
[K,z]
z e K
dent modulo
E
I.
or
and thus
If
generated
[[x,y],[z,w]].
x,y,z,w in K and thus
x,y
K
of
O.
Continuing this argument, we obtain
PROOF.
K
contained in
is abellan.
[[x,y],z]
x,y,z
where
K such that
in
dim K/K’
F-subspace
are linearly independent vectors of
w e <x,y,z>
v
K
since the prop. is trivially true for
[[x,y], [z,w]] e <x,y> f <z,w>
vector
K
is a subspace of
R.
then
PROPOSITION i.
If
K
of
y.
and
Thus, if
0}
are respectively
It can be verified that a Lie ideal of
is a proper ideal of
If
El (x, K)
{x e
[K,K], Z
K
then let
By (3),
we have
x,y
[[x,y],K]
x e
K’
0,
we have
If
K
be vectors in
<x,y>
K’
a contradiction
is abellan and dim K > i
K’.
.
K which are linearly indepen0
which forces
Hence,
then
f
dim
K/K’
0
and
x E K
0
xy
or
i
0
for
251
RINGS WITH INVOLUTION
K.
x,y
all
COROLLARY 2.
0
then
dim
K/K’
> i
-
The second equivalence follows from (3).
PROOF.
dim
K’ #
If
f
=
0
f
If
@
[K’,K] -0.
then by prop. 2,
0
K/K’ < i.
K’.
x,y
and thus
0
f
Conversely, if
F.
K’
dim
K/K’ > i.
ax]
[x,y
0
and thus
ax e
Indeed, if
-
ax_
[x,y],
[x,y] # 0
K with
be vectors in
,
If suffices to show that the images
linearly independent over
y
x,y
then let
K/K’
in
a e F
for some
are
then
Hence,
a contradiction.
For the Lie algebras that we are considering there is a simple proof of
Cartan’s
criterion for semisimpllclty.
THEOREM 2.
f
If
@
ideals if and only if
PROOF.
in
If
K
"
0
# 0 then by prop. 2, K
K
Conversely, if
linearly independent.
By
THEOREM 3.
over
F
f
is a
nonzero Lie ideal contained
is abellan.
K
be any vector of
x
Hence,
O.
# 0
then
K
f(y,z)x- f(z,x)y
(3),
f(z,x)
If
has no nonzero abellan Lie
has a nonzero abelian Lie ideal
and
f(y,z)
and hence
0
nonzero vectors in I
and thus
K
then
is nondegenerate.
f
[K’,K ]
K’. By (3),
I
dim K >
and
K’
f(z,K)
0
I
y,z be
then let
such that
and
x
are
y
[[x,y],z] e [I,I]
0
and
is either a Lie algebra of
pure quaternlons
or a direct sum of mutually orthogonal abellan Lie ideals of
K with
dim < 2.
of
0
for otherwise, prop. 2 would imply that
We may assume
dim i.
We have only to consider the two cases,
Suppose K’
K
K’
PROOF.
K.
Since
KI V
# 0 then let K
which implies that
[K’,K’]
K’
[V,V]
where
is of
[K,K] # 0,
dim V < 2
dim <
I,
dim
K/K’
dim K
and by (3),
3
K
is
0, i.
by prop. i.
0
contradictory to
[K’,Kl]
K’
If
[K,Kl]
K.
252
T.P. LIM
Hence,
K
=
0.
Since the bilinear form
x,,z. As K’
has an orthogonal basis
also form a basis of
is orthogonal to
x
a,b e F.
where
K.
K’
x e
0
[K’, Kz]
0 # y
f # 0,
Let
d(x)
f(e,e)x
reducible as a module for
NOTE:
<_
2
and
[x,e]
d2
for
and
K
-
and
x
x
K’.
x e
K’
K’C
K
-a
F.
z,
[[x,y],z] e K’
K’. Moreover,
e e K/K’
By (3),
I
is in the
[2,p.87] that K’
Lx
by
-b, f(y,y)
To show
f(e,e)l where
Clearly, each
0
x e K
for all
it follows from
f(Lx,Ly)
f(x,x)
[z,x]
there exists a nonzero vector
and hence
d.
[x,y]
and hence
ax,
f(y,z)x- f(z,x)y
Hence,
Since every nonzero vector
<x,d(x)>,
0
z, [y,z]
by prop. 2.
By (3),
Since
K’.
dim
K’.
[K’, K’].
map of
with
[x,y]
K K’
We have
i.
0.
[Ix,el,el
Lx
f(y,[x,y])
is a Lie algebra of pure quaternions over
f(K,x)
d2(x)
subspace
K
K
the commutators Ix,y], [y,z], [z,x]
f(x,[x,y])
and thus
f(e,e) # 0.
with
K,
Consequently,
K’
Hence,
and choose
z e K
for all
y.
and
Suppose dim K/K’
let
By (2),
is symmetric and nondegenerate,
We can now easily derive from (3) that
f(z,z}= -ab.
and
f
is the identity
d-invariant
is completely
is an abelian Lie ideal of
K
# y. J
This paper constitutes a part of the author’s Ph.D. thesis written
under the supervision of Professor D. Z.
DJokovic
at the University
of Waterloo, Ontario, Canada.
PRESENT ADDRESS:
491 Parkside Drive
CANADA
Toronto, Ontario
REFERENCES
i.
Bourbaki, N., Elments de Mathmatlque Groupes et Algebres de Lie,
Chap. i, Hermann, Paris 1971.
2.
Curtis C. W. and Reiner, I., Representation Theory of Finite Groups and
Associative Algebras, Intersclence Publlshers, New York 1962.
3.
Herstein, I. N., Rings wtih Involution, Univ. of Chicago Press, Chicago 1976.
RINGS WITH INVOLUTION
253
REFERENCES
4.
Lim, T. P., Determinants over a class of
Univ. of Waterloo, Canada, 1976.
5.
Lim, T. P., Some Classes of Rings with Involution satisfying the Standard
Polynomial of Degree 4, submitted to the Pacific Journal of Mathematics.
rins
with involution, Ph. D. thesis,