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Lorentz type algebras
Introduction to the Lorentz algebra
Cándido Martı́n González
University of Málaga
[email protected]
October 22, 2015
Lorentz type algebras
An action of a group G on a set M is a map
G × M → M, (g , m) 7→ g · m such that
g (g 0 m) = (gg 0 )m
1m = m
for any g , g 0 ∈ G , m ∈ M. The orbit of an element
m ∈ M is denoted Gm := {gm : g ∈ G } and M is
the disjoint union of its (different) orbits. The
action is said to be transitive if there is only one
orbit. We say that a set M is a G -set (for a group
G ) if there is an action G × M → M.
Lorentz type algebras
Example. Let G be a group and H ⊂ G a
subgroup. Define in G the relation g1 ∼ g2 if and
only if g1−1 g2 ∈ H. This relation turns out to be an
equivalence relation and the set of all equivalence
classes is denoted G /H.
G /H := {gH : g ∈ G }.
This is not a group in general since H is not
necessarily a normal subgroup. However G /H is a
G -set for the action g1 · g2 H := (g1 g2 )H.
Lorentz type algebras
If we restrict an action G × M → M to a given
orbit Gm the restricted action is of course transitive.
Assume that G × M → M is a transitive action and
fix an element m ∈ M. Thus M = Gm and we have
a map G → Gm such that g 7→ gm which is
surjective. If we denote by Gm the isotropy group of
m, that is
Gm := {g ∈ G : gm = m},
then, it is easy to show that G /Gm ∼
= M (as
G -sets).
Lorentz type algebras
The relevance of this fact is that once that an
action has been given G × M → M, then the set M
is not anything new, since we may look for it in the
“quotients”of the group G . So we have many
information of the set in the acting group.
Also the maximum expression of this fact is
achieved if G is a Lie group or an algebraic group or
a finite group and M is the space of a representation
of G . Because the representations of simple Lie
groups are well known (recently completed since the
last case to describe was the representation ring of
the exceptional Lie algebra E8 ).
Lorentz type algebras
Some computations of Lie algebras
G ⊂ GLn (R) a Lie group,
Lie(G ) := {x ∈ gln (R) : exp(λx) ∈ G , ∀λ ∈ R}.
SLn (R) := {m ∈ GLn (R) : det(m) = 1}, then
Lie(SLn (R)) = {m ∈ gln (R) : exp(λm) ∈
SLn (R), ∀λ}. But we know that
det(exp(m)) = e tr(m) and so
Lie(SLn (R)) = {m ∈ gln (R) : det(exp(λm)) =
1, ∀λ}
Lie(SLn (R)) = {m ∈ gln (R) : exp(λtr(m))) = 1, ∀λ}
for λ = 1 we get
exp(tr(m)) = 1 ⇒ tr(m) = 0.
Lorentz type algebras
Reciprocally if tr(m) = 0 it is easy to prove that
det(exp(λm)) = 1 for any λ. Thus
Lie(SLn (R)) = sln (R) = {m ∈ gln (R) : tr(m) = 0}.
Prof. Dolores proved yesterday that for a fixed
matrix m, if we take the Lie group
G = {x : xmx t = m}, then
Lie(G ) := {x : xm + mx t = 0}.
Lorentz type algebras
If we take m = 1n then G = On (R) and
Lie(On (R)) := {m : m+mt = 0} = skewsymmetric matrices
and this algebra is denoted
Lie(On (R)) := on (R) orthogonal Lie algebra.
In a similar way can we define the symplectic Lie
algebra.
Lorentz type algebras
Let G = GLn (R), then
Lie(G ) := {x ∈ gln (R) : exp(λx) ∈ GLn (R)}. But
exp(m) is invertible for any matrix m and
exp(m)−1 = exp(−m). Thus
Lie(GLn (R)) = gln (R).
Lorentz type algebras
Heisenberg group


1 x z
H = {0 1 y  : x, y , z ∈ R},
0 0 1
we compute the algebra of those matrices
m ∈ gl3 (R) such that exp(λm) ∈ H. Thus


1 x z
exp(λm) = 0 1 y 
0 0 1
Lorentz type algebras


1 x z
exp(λm) = 0 1 y  ,
0 0 1

 

1 x z
0 x0 z0
d 
d
exp(λm) =
0 1 y  = 0 0 y 0  ,
dλ
dλ
0 0 1
0 0 0
d
exp(λm) = m exp(λm),
dλ
so that for λ = 0,


0 x 0 (0) z 0 (0)
m = 0 0 y 0 (0)
0 0
0
Lorentz type algebras

0 x 0 (0) z 0 (0)
m = 0 0 y 0 (0)
0 0
0

b
c  : a, b, c ∈ R}.
0

So if m ∈ Lie(H) we have
or

0 a

Lie(H) ⊂ { 0 0
0 0
And it is not really difficult to prove that the above
relation is really an equality.
Lorentz type algebras
The relevance of the Heisenberg group in Physics comes from the
fact that those processes governed by the principles of Quantum
Mechanics can be modeled by representations of the Heisenberg
group.
Antonio Jesús Calderón Martı́n,
Cristina Draper Fontanals,
Cándido Martı́n González,
José Marı́a Sánchez Delgado.
Linear Algebra and its Applications
Volume 458, 1 October 2014, Pages 463502
Gradings and symmetries on Heisenberg type algebras
Lorentz type algebras
Summarizing: for any Lie group G there is a Lie
algebra denoted Lie(G ) or g and both structures are
related by the exponential map exp : Lie(G ) → G .
This map is locally a diffeomorphism, that is, there
are neighborhoods 0 ∈ U ⊂ Lie(G ) and 1 ∈ V ⊂ G
such that we can restrict exp : U → V and this
restriction is a diffeomorphism. Thus the Lie group
and its Lie algebra are very similar if we look at
them “infinitesimally”.
Lorentz type algebras
If G and H are Lie groups and f : G → H a Lie
group homomorphism (a continuous map which is a
group homomorphism), then there is a unique Lie
algebra homomorphism f ∗ : Lie(G ) → Lie(H) such
that the following diagram commutes:
GO
f
exp
/H
O
Lie(G )
exp
f∗
/
Lie(H)
Lorentz type algebras
Moreover, if 1G is the identity in the Lie group G ,
then 1∗G = 1Lie(G ) and:
If f : G1 → G2 and g : G2 → G3 are Lie group
homomorphisms, then the composition g ◦ f is a Lie
group homomorphism such that
(g ◦ f )∗ = g ∗ ◦ f ∗ .
In other words, we have a functor (the so called Lie
functor) from the category of (real) Lie groups to
the category of (real) Lie algebras
Lie : LieGrpR → lieAlgR
Lorentz type algebras
The Lie algebra is a good substitution for the Lie
group for many interesting tasks. For instance
representation theory: any representation
r : G → GLn (R) of the Lie group G induces a
representation r ∗ : Lie(G ) → gln (R) of its
associated Lie algebra and
GO
r /
GLnO (R)
exp
Lie(G )
commutes.
exp
r∗
/ gl
n (R)
Lorentz type algebras
So representation theory of Lie algebras is a
fundamental tool in representation theory of Lie
groups. It is usually a convenient tool since the Lie
group is usually a curved object while its Lie algebra
is a vector space.
x y
SL2 (R) := {
: xt − yz = 1} is the algebraic
z t
variety xt −yz =1 in R4 . However its Lie algebra is
x y
sl2 (R) := {
: x + t = 0} which is a vector
z t
space of dimension 3. By the way, this proves
dim SL2 (R) = 3.
Lorentz type algebras
Introduction to gradings
If A is an algebra over a field K and G a group, a
decomposition
A = ⊕g ∈G Ag
is called a G -grading or a grading by the group G if
∀g , h ∈ G , one has Ag Ah ⊂ Agh . For instance a
Z2 -grading on A is a decomposition
A = A1 ⊕ A−1
such that A1 A1 ⊂ A1 , A−1 A−1 ⊂ A1 , A1 A−1 ⊂ A−1 ,
and A−1 A1 ⊂ A−1 .
Lorentz type algebras
grading of M2 (C)
Let A = M2 (C), then, up to isomorphism this
algebra has 3 possible gradings:
1) The Z
⊕ A−1 such that
2 -grading
A= A1 1 0
1 0
A1 = C
⊕C
and
0 0
0 0
0 1
0 0
A−1 = C
⊕C
.
0 0
1 0
2) The Z-grading
A=
⊕n∈Z
An such that
1 0
1 0
A0 = C
⊕C
,
0 0
0 0
0 1
0 0
A1 = C
, A−1 = C
, others An = 0.
0 0
1 0
Lorentz type algebras
3) The Z2 × Z2 -grading A = A00 ⊕ A01 ⊕ A10 ⊕ A11 .
Given by
0 1
A00 = C12 ,
A01 = C
,
1 0
1 0
0 1
A10 = C
,
A11 = C
.
0 −1
−1 0
Lorentz type algebras
A Z2 -grading A = A1 ⊕ A−1 produces an
automorphism f : A → A such that
f (a1 + a−1 ) = a1 − a−1 for any ai ∈ Ai (i = ±1).
Thus, f is an automorphism and f 2 = 1A .
Reciprocally, any automorphism f of any algebra A
such that f 2 = 1A produces a Z2 -grading
A = A1 ⊕ A−1 , where A1 = {x ∈ A : f (x) = x} and
A−1 = {x ∈ A : f (x) = −x}. The automorphism f
producing the grading is called the grading
automorphism of A.
Lorentz type algebras
Thus, sometimes it may be convenient to denote a
grading, not by the decomposition A = A1 ⊕ A−1 ,
but by a couple (A, f ) where f is the grading
automorphism of order 2, whose +1-eigenspace is
A1 and −1-eigenspace is A−1 .
Lorentz type algebras
Let A = X1 ⊕ X−1 and A = Y1 ⊕ Y−1 be two
Z2 -gradings on the same algebra A. We will say
that these gradings are isomorphic if there is an
automorphism f : A → A such that f (X1 ) = Y1 and
f (X−1 ) = Y−1 .
It is not difficult to prove (Exercise!!) that if (A, f )
and (A, g ) are isomorphic gradings on A, then f and
g are conjugated elements in the group Aut(A)
(that is g = pfp −1 for some p ∈ Aut(A)).
Reciprocally, if f and g are order two elements in
the group Aut(A) and they are conjugated, then
the gradings (A, f ) and (A, g ) are isomorphic.
Lorentz type algebras
Thus, the problem of determining all the posible
isomorphy classes of Z2 -gradings on an algebra A is
equivalent to that of describing the conjugacy
classes of order two elements in Aut(A). This
requires a knowledge of the group of automorphism
of the algebra A.
Lorentz type algebras
Similar ideas apply if we take any finitely-generated
abelian group G and consider all the isomorphy
classes of G -gradings on an algebra A (instead of
considering Z2 -gradings).
In any case, a fine study of Aut(A) is a good
starting position for the problem of describing all
the gradings on A.
Lorentz type algebras
Why studying gradings?
If we take for instance A = M4 (K ), being K a field,
there are only two Z2 -gradings (up to isomorphy) on
A. One is A = X1 ⊕ X−1 where




∗ 0 0 0
0 ∗ ∗ ∗
0 ∗ ∗ ∗
∗ 0 0 0 
,


X1 = 
X
=
−1
0 ∗ ∗ ∗
∗ 0 0 0  .
0 ∗ ∗ ∗
∗ 0 0 0
Lorentz type algebras
The other is A = Y1 ⊕ Y−1 , where



∗ ∗ 0 0
0
∗ ∗ 0 0 
0
,

Y1 = 
Y
=
−1
 0 0 ∗ ∗
∗
0 0 ∗ ∗
∗
0
0
∗
∗
∗
∗
0
0

∗
∗
.
0
0
So, the same algebraic object has arisen two
different graded algebras (even graded by the same
group Z2 ). And, of course, we can consider many
other gradings on M4 (K ) with different grading
groups.
Lorentz type algebras
Thus, while the underlying algebra is the same, we
have many possible ways to give a grading on A. In
a graded sense all these objects are nonisomorphic.
The study of gradings on an algebra plays also a
role in some Physical developments related to
graded contractions.
Lorentz type algebras
So, our original motivation for studying the
automorphism group of the Lorentz algebras over
different fields comes from the long term project of
computing the isomorphy classes of gradings on the
Lorentz algebra. And in order to study
automorphisms, we must take a look at the ideal
structure of the algebra (if any!!).
Lorentz type algebras
It is known that the Lorentz algebra over the reals is
simple, however the Lorentz algebra over the
complex field is a direct sum of two 3-dimensional
ideals isomorphic to the Lie algebra sl2 (C):
LC
= I ⊕ J,
I ∼
=J∼
= sl2 (C),
hence any automorphism of
or swaps them.
LC
either fixes I and J
Lorentz type algebras
This talk is part of a research jointly with:
Pablo Alberca (U. of Málaga)
Dolores Martı́n (U. of Málaga)
Daouda Ndoye (U. of Dakar)
The master document can be found in:
http://arxiv.org/abs/1508.01634
Lorentz type algebras
Precedents
E. G. Beltrametti and A. A. Blasi, Dirac spinors, covariant currents
and the Lorentz group over a finite field. Nuovo Cimento A,
http://dx.doi.org/10.1007/BF02759228 55, 301 (1968).
E. G. Beltrametti and A. A. Blasi, Rotation and Lorentz groups in
a finite geometry. J. Math. Phys.
http://dx.doi.org/10.1063/1.1664670 9, 1027 (1968).
H. R. Coish, Elementary particles in a finite world geometry. Phys.
Rev. http://dx.doi.org/10.1103/PhysRev.114.383 114, 383 (1959).
L. E. Dickson, Determination of the structure of all linear
homogeneous groups in a Galois field which are defined by a
quadratic invariant. Am. J. Math.
http://dx.doi.org/10.2307/2369602 21, 193 (1899).
Stephan Foldes. The Lorentz group and its finite field analogs:
Local isomorphism and approximation. J. Math. Phys. 49, 093512
(2008); http://dx.doi.org/10.1063/1.2982519.
Lorentz type algebras
From an algebraic view point the more interesting
work above is the one of Dickson who, as early as in
1899, makes a detailed study of groups defined by
quadratic forms in prime characteristic. As a by
product of his research, he identifies the Icosahedral
group as a subgroup of a Lorentz group in
characteristic 2.
Some recent developments on applications of finite
Lorentz groups to Signal and Image Processing,
seem to be under research.
Lorentz type algebras
Abstract
We start with the Lorentz algebra L = oR (1, 3) over the reals
and find a suitable basis B relative to which the structure
constants are integers. Thus we consider the Z-algebra LZ
which is free as a Z-module and its Z-basis is B. This allows
us to define the Lorentz type algebra LK := LZ ⊗Z K over any
field K . In this talk we study the ideal structure of Lorentz
type algebras over different fields. It turns out that Lorentz
type algebras are simple if and only if the ground field has no
square root of −1. Thus, they are simple over the reals but
not over the complex. Also, if the ground field is of
characteristic 2 then Lorentz type algebras are neither simple
nor semisimple. We extend the study of simplicity of the
Lorentz algebra to the case of a ring of scalars where we have
to use the notion of m-simplicity (relative to a maximal ideal
m of the ground ring of scalars).
Lorentz type algebras
Over finite fields....
The Lorentz type algebras over a finite field Fq
where q = p n and p is odd, are simple if and only if
n is odd and p of the form p = 4k + 3. In case
p = 2 then the Lorentz type algebras are not simple.
The ideal structure of the algebras, provides some
information of their automorphism groups.
Lorentz type algebras
Classical Lorentz algebra
The Lorentz algebra over the reals, denoted by o(1, 3), is the Lie
algebra of the orthogonal Lie group O(1, 3):
o(1, 3) = Lie(O(1, 3)) = {M ∈ gl4 (R) : MI13 + I13 M t = 0},
where M t denotes matrix transposition of M and
I13 = diag(−1, 1, 1, 1) (some authors take I13 = diag(1, 1, 1, −1)
which is equivalent). A straightforward computation reveals that a
generic element of o(1, 3) is of the form


0 x1
x2 x3
 x1
0
x4 x5 


 x2 −x4
0 x6 
x3 −x5 −x6 0
Lorentz type algebras
and then denoting by eij the elementary matrix with 1 in the entry
(i, j) and 0 elsewhere we have a basis of o(1, 3) given by
B = {s12 , s13 , s14 , a23 , a24 , a34 } where sij := eij + eji and
aij = eij − eji .
[, ]
s12
s13
s14
a23
a24
a34
s12
0
−a23
−a24
−s13
−s14
0
s13
a2,3
0
−a34
s12
0
−s14
s14
a2,4
a34
0
0
s12
s13
a23
s1,3
−s12
0
0
a34
−a24
a24
s1,4
0
−s12
−a34
0
a23
Figure : Multiplication table of o(1, 3).
a34
0
s14
−s13
a24
−a23
0
Lorentz type algebras
A nice basis

s12
0
1
=
0
0
a23



0 0
0 0 1 0


0 0
 , s13 = 0 0 0 0 , s14
1 0 0 0
0 0
0 0
0 0 0 0



0 0 0 0
0 0
0 0 1 0
0 0


=
0 −1 0 0 , a24 = 0 0
0 0 0 0
0 −1


0 0 0 0
0 0 0 0

a34 = 
0 0 0 1 .
0 0 −1 0
1
0
0
0

0 0
0 0
=
0 0
1 0

0 0
0 1
,
0 0
0 0
Relative to this basis the structure constants are 0, 1 or −1.
0
0
0
0

1
0
,
0
0
Lorentz type algebras
Lorentz algebra over rings
Thus we can construct the Z-algebra
LZ := Zs12 ⊕ Zs13 ⊕ Zs14 ⊕ Za23 ⊕ Za24 ⊕ Za34 whose
multiplication table is given in Figure 1. Fix now an
associative, commutative and unital ring Φ and consider the
category algΦ of associative commutative and unital
Φ-algebras.
Then for any object R in algΦ we may define the Lorentz type
algebra LR := LZ ⊗Z R. This is nothing but the free
R-module with basis s12 , s13 , s14 , a23 , a24 and a34 , enriched
with an R-algebra structure by the multiplication table as in
Figure 1. As a free R-module we have
dim LR = 6.
Lorentz type algebras
Lorentz functor
Of course if we take R = R then LR ∼
= o(1, 3), the
Lorentz algebra. If R = C then LR is the
complexified Lorentz algebra. If R and S are objects
in algΦ and f : R → S a Φ-algebras homomorphism,
then we may define a Lie Φ-algebras homomorphism
Lf : LR → LS in an obvious way. Thus, if LieΦ
denotes the category of Lie algebras over Φ, we
have defined a covariant functor
L:
algΦ −→ LieΦ
R 7−→ LR
f 7−→ Lf
Lorentz type algebras
Orthogonal Lie algebra? Not exactly!
Let O(n) be the orthogonal Lie group over the reals: the
group of all matrices M in GLn (R) such that MM t = 1n .
Then, its Lie algebra o(n) consists of all matrices M in gln (R)
such that M + M t = 0. This is generated (as a vector space)
by the matrices eij − eji where i < j with i, j ∈ {1, . . . , n} and
the structure constants relative to the basis of these elements
are again 0 or ±1. Thus, we can consider as before the
Z-algebra o(n; Z) := ⊕i<j Z(eij − eji ). Fix as before a ring Φ
and then, for any algebra R in algΦ , it is tempting to define
the scalar extension o(n; R) := o(n; Z) ⊗Z R. So, this is the
Lie R-algebra with basis eij − eji as before and multiplication
table as the one for o(n) in the corresponding basis.
Lorentz type algebras
However in characteristic two, the orthogonal Lie
algebra is different from this and in order to be
coherent with the classical definition we should not
use the notation o(n; R). Instead we will use but
Γn (R) for the algebraic group of matrices in GLn (R)
such that MM t = 1n and γn (R) for its Lie algebra
γn (R) := o(n; Z) ⊗ R.
We will also use the notation γ(n; R) meaning
γn (R), and if the ground ring of scalars is free of
2-torsion, we will write o(n; R) or on (R) instead of
γn (R).
Lorentz type algebras
Summarizing γn (R) = γ(n; R) is the algebra of
skew-symmetric n × n matrices with entries in R. It
is a free R-module with basis the set of all matrices
eij − eji with i < j.
Thus we have dimR (γ(n; R)) = n(n − 1)/2 and we
have again a functor
γ(n) : algΦ → LieΦ
such that R 7→ γ(n; R). If f : R → S is a
homomorphism of algebras in algΦ then we will
denote by γ(n; f ) : γ(n; R) → γ(n; S) the
homomorphism of Lie algebras γ(n; f ) := 1 ⊗ f .
Lorentz type algebras
Remark
If Φ is a ring agreeing with its 2-torsion, that is,
1 + 1 = 0, then for any Φ-algebra R in algΦ , the
Lie algebra γ(4; R) agrees with the Lorentz type
Lie algebra LR . In particular this is the case for a
field K of characteristic two: LK = γ(4; K).
Lorentz type algebras
Lorentz Type algebra over a field of char=2
Definition
√
Consider next the full subcategory −1√
Φ of algΦ whose
objects are the Φ-algebras √
R such that −1 ∈ R. Denote by
I the inclusion functor I : −1Φ → algΦ .
This allows a more general result:
Lemma
√
For any Φ, the functors L ◦ I and γ(4) ◦ I : −1Φ → LieΦ
are isomorphic. More precisely (i) for any algebra R in algΦ
such that the equation x 2 + 1 = 0 has a solution in R, there is
an isomorphism ηR : LR ∼
= γ(4; R);√(ii) If f : R → S is a
homomorphism of Φ-algebras and −1 ∈ R, the following
diagram commutes:
Lorentz type algebras
LR
Lf
LS
ηR
/
γ(4; R)
γ(4;f )
/ γ(4; S).
ηS
Proof.
Take i ∈ R such that i2 = −1. Starting from the standard
basis B of LR , we define a new basis
0
C = {aij0 : i, j ∈ {1, 2, 3, 4}, i < j} where a12
:= is12 ,
0
0
0
a13 := is13 , a14 := is14 and aij := aij for the remaining
elements. Then the isomorphism LR → γ(4; R) is the induced
by aij0 7→ eij − eji for i < j. On the other hand, the
commutativity of the square above is straightforward.
Lorentz type algebras
Simplicity
We would like to study under what conditions the
Lorentz functor
L:
algΦ → LieΦ
produces simple Lie algebras. A second goal would
be to describe the algebraic group (in the sense of
affine group schemes):
Aut(LΦ ) : algΦ −→ Grp
R 7−→ AutR (LR )
Lorentz type algebras
To shorten the notations, we write b1 := s12 , b2 := s13 ,
b3 := s14 , b4 := a23 , b5 := a24 , b6 := a34 so that the basis B
of LR is now B = {bi }61 and has the multiplication table:
[, ]
b1
b2
b3
b4
b5
b6
b1
0
−b4
−b5
−b2
−b3
0
b2
b4
0
−b6
b1
0
−b3
b3
b5
b6
0
0
b1
b2
b4
b2
−b1
0
0
b6
−b5
b5
b3
0
−b1
−b6
0
b4
b6
0
b3
−b2
b5
−b4
0
Figure : Second version of table in Figure 1.
Lorentz type algebras
Simplicity
As an primary goal we investigate the simplicity of
LΦ when Φ is a field. It is known that LR is simple
while LC is not. In fact LC ∼
= sl2 (C) ⊕ sl2 (C). How
can we investigate the simplicity of an algebra (just
from its multiplication table)?
Lorentz type algebras
Investigating simplicity
If we start with an algebra A over Φ
(not necessarily satisfying any
particular identity) such that A2 6= 0,
the simplicity of A is equivalent to
the fact that the ideal generated by
any nonzero element is the whole
algebra A.
Lorentz type algebras
Multiplication algebra
For any a ∈ A denote La : A → A the left
multiplication operator La (x) := ax and Ra : A → A
the right multiplication operator such that
Ra (x) = xa. Denote by EndΦ (A) the algebra of all
linear maps A → A. Denote by M(A) the
subalgebra of EndΦ (A) generated by all the left and
right multiplication operators. The elements of
M(A) are linear combinations of compositions of
left and right multiplication operators.
Lorentz type algebras
A preorder relation in A
If x, y ∈ A we say x ` y if there is some T ∈ M(A)
such that y = T (x). Roughly speaking: if y is
obtained form x after a finite number of
multiplications we have x ` y .
The relation ` is reflexive and transitive (a preorden
in A).
Observe that x ` y is equivalent to the assertion
that the ideal hy i generated by y is contained in the
ideal hxi generated by x.
Lorentz type algebras
A simplicity condition
Theorem
Let A be an algebra A such that A2 6= 0 and {bi }i a
basis of the algebra. Then a sufficient condition for
the simplicity of A is:
1
bi ` bj for any i and j.
2
∀x ∈ A \ {0}, ∃i : x ` bi .
Proof. Let 0 6= I / A and 0 6= x ∈ I . Since ∃i such
that x ` bi , and bi ` bj for any j, we have x ` bj for
any index j. So hbj i ⊂ hxi for each j. Consequently
A ⊂ hxi or hxi = A. hence A is simple.
Lorentz type algebras
Ground field with no
√
−1
√
By a field Φ with −1 6∈ Φ we mean that there is no element
x ∈ Φ such that x 2 = −1. This condition implies that Φ is of
characteristic other than 2. Also, this condition is equivalent
to the assertion that:
∀x, y ∈ Φ, (x 2 + y 2 = 0 implies x = y = 0).
If x 2 + y 2 = 0 and (say) x 6= 0, then 1 + ( yx )2 = 0 so that
√
−1 exists.
Lorentz type algebras
Ground field with no
Theorem
If Φ is a field and
√
−1
√
−1 6∈ Φ, then LΦ is simple.
Sketch of proof.
[, ]
b1
b2
b3
b4
b5
b6
b1
0
−b4
−b5
−b2
−b3
0
b2
b4
0
−b6
b1
0
−b3
b3
b5
b6
0
0
b1
b2
b4
b2
−b1
0
0
b6
−b5
b5
b3
0
−b1
−b6
0
b4
b6
0
b3
−b2
b5
−b4
0
Lorentz type algebras
bi ` bj
By looking at the multiplication table
we see that bi ` bj . For instance
b1 `1 {b1, b2, b3, b4, b5} and b2 `1 b6
hence b1 ` bi for any i. The same
applies to the others so that bi ` bj
for any i, j.
Lorentz type algebras
0 6= x ` bi for some i
P
Take 0 6= x = λi bi , if we take
T = λ2 Rb2 Rb1 + λ5 Rb6 Rb2 , this is an element of
M(LΦ ) and T (x) = −(λ22 + λ25 )b1 hence
x ` (λ22 + λ25 )b1
and if λ2 √
or λ5 6= 0 then λ22 + λ25 6= 0 (on the
contrary −1 ∈ Φ). Thus, if λ2 or λ5 6= 0 x ` b1
and we are done.
So assume λ2 = λ5 = 0.
Lorentz type algebras
0 6= x ` bi for some i
0 6= x = λ1 b1 + λ3 b3 + λ4 b4 + λ6 b6 . Things are
getting easier because x is “shorter”. Take
S := λ4 Rb1 + λ3 Rb6 ∈ M(LΦ ), then it is easy to see
that S(x) = −(λ23 + λ24 )b2 .
Hence if λ3 or λ4 6= 0 we have as before x ` b2 and
we are done again.
In the worst case λ3 = λ4 = 0.
Lorentz type algebras
0 6= x ` bi for some i
So far we have λ2 = λ3 = λ4 = λ5 = 0. Thus
x = λ1 b1 + λ6 b6 and λ1 or λ6 6= 0. Thus
λ21 + λ26 6= 0. The operator
R = λ1 Rb2 + λ6 Rb5 ∈ M(LΦ ) satisfies
R(x) = (λ21 + λ26 )b4 hence
x ` b4
which finishes the proof.
Lorentz type algebras
Representation theory language
For any Φ-algebra A, the multiplication algebra
M(A) is associative and unital. Furthermore A is
an M(A)-module and the irreducible
M(A)-submodules of A are precisely the ideals of A.
Thus the simplicity of LΦ proved above is just the
assertion that the M(LΦ )-module LΦ is irreducible.
Lorentz type algebras
√
When Φ is a field such that −1 ∈ Φ the same
relation ` helps to realize that LΦ is not simple. Of
course there are elements x such that x ` LΦ , but
one realizes immediately that there are other
elements x such that x ` I for a certain
three-dimensional ideal I of Lφ . Thus we detect
proper nonzero ideals inside LΦ . Moreover there are
only two nonzero proper ideals I and J and
LΦ = I ⊕ J being I ∼
= sl2 (Φ) ∼
= J.
Lorentz type algebras
Let now Φ be any commutative unital ring and R any algebra
in the category algΦ . Let Max(R) denote the maximal
spectrum of R (the set of maximal ideals) and m ∈ Max(R).
Let V be any R-algebra (not necessarily associative) with
V 2 6= 0 and I / V an ideal. Of course the R-submodule mV is
an ideal of V , but we would like to exclude these ideals from
our study. Also, we want to exclude, those ideals which
complement mV . Thus
Definition
We say that I / V is m-null if I ⊂ mV . Also I is said to be
m-total if V = I + mV . The algebra V is said to be m-simple
if its unique ideals are the m-null and the m-total ones.
Lorentz type algebras
Definition
We say that a field K is 2-formally real if for any
x, y ∈ K, the equality x 2√
+ y 2 = 0 implies
x = y = 0 (equivalently −1 ∈
/ K).
More generally for an ideal i of a Φ-algebra R we
say that R is i-2-formally real if for any x, y ∈ R,
the fact x 2 + y 2 ∈ i implies x, y ∈ i.
Lorentz type algebras
Theorem
Let m ∈ Max(R) be a maximal ideal. Then the Lorentz type
algebra LR is m-simple if and only if R is m-2-formally real. In
particular, the Lorentz type algebra LK over a field K is simple if
and only if K is 2-formally real (equivalently, if and only if
√
−1 ∈
/ K).
Theorem
Let R be an algebra in algΦ with Jacobson radical rad(R) = 0.
ThenQ
LR is m-simple for any m ∈ Max(R) if and only if
R ⊂ i∈I Ki is a subdirect product of 2-formally real field {Ki }i∈I
and for anyQm ∈ Max(R) there is some j ∈ I such that πj (m) = 0,
being πj : i∈I Ki → Kj the canonical projection onto the field Kj .