Download Introduction to the Lorentz algebra

Lorentz type algebras
Introduction to the Lorentz algebra
Cándido Martı́n González
University of Málaga
[email protected]
October 13, 2015
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 .