16667347

doi:10.1006rjabr.2000.8660, available online at http:rrwww.idealibrary.com on

On Infinite Root Systems

Jose Antonio Cuenca Mira1

´

´

Departamento de Algebra, Geometrıa y Topologıa, Facultad de Ciencias,´ ´

Uni¨ersidad de Malaga, 29071 Malaga, Spain´ ´

Communicated by Georgia Benkart

Received April 12, 2000

The structure theory of finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic 0 is obtained by using the

w x

classification of root systems in the Bourbaki sense 5 . The analogous

w x

theory in prime characteristic is very hard 2, 4, 21᎐24 , being first focused

w x

in the determination of the classical Lie algebras 17᎐19 . The last heavily depends on the study of the associated sets of roots. In this case the nonzero characteristic introduces new complications yielding to

computa-Ž w x.

tions which can be very long see 19 . In this note we introduce ab-root

w x

systems. They were previously introduced by Winter 25 for the finite case

w x

in a slightly different way. The main result in 25 asserts that every finite

ab-root system is isomorphic to a root system in the Bourbaki sense with zero added. Since sets of roots appearing in the study of the classical Lie algebras in prime characteristic are finite ab-root systems, then the men-tioned Winter theorem avoids the difficulties arising in these characteris-tics.

On the other hand, infinite sets of roots were studied by Kaplansky and

w x

Kibler 9, 11 and play a fundamental role in the structure of some classes of Lie algebras. This is so in the determination of the topologically simple

U _{w x}

Lie H -algebras 7, 12, 15, 16 . From the necessity to consider sets of roots of an arbitrary cardinal arises the following question: Is the Winter result true for infinite ab-root systems? This note is devoted to this problem. As an application of our main result we give a theorem of classification of

ab-root systems.

1

Ž . Ž .

This work was partially supported by DGYCIT PB97-1497 , AECI, and PAI FQM-0125 . 669

0021-8693r01 $35.00 Copyright䊚2001 by Academic Press All rights of reproduction in any form reserved.

1. BASIC CONCEPTS AND NOTATIONS

Concepts and notations given in this section closely follow those of

w x

Winter 8, 25, 26 . A subset ⌫ of an additive abelian groupG is said to be a set of sumin G when 0g⌫. Let GX be an additive abelian group and ⌫X a set of sum in GX. A map f:⌫ª⌫X is called a homomorphismof sets of

Ž . Ž . X

sum when for all ␣,␤g⌫ with ␣q␤g⌫ we have f ␣ qf ␤ g⌫

Ž . Ž . Ž .

and f ␣q␤ sf ␣ qf ␤ . Composition of homomorphisms of sets of sum are homomorphisms. If A is an additive abelian group, ⌫ a set of sum, and f,g:⌫ªA homomorphisms of sets of sum then the map

Ž . Ž .

fqg:⌫ªA given by fqg:␣¬f ␣ qg ␣ for all ␣g⌫ is a

homo-Ž .

morphism of sets of sum. So we have an abelian group Hom⌫,A whose elements are the homomorphisms of sets of sum with domain ⌫ and

U _{Ž .} X

codomain A. For every set of sum ⌫ we put⌫ sHom ⌫,⺪. If f:⌫ª⌫ is a homomorphism of sets of sum, then the map fU :⌫XUª⌫U given by

fU :␸¬␸(f is a homomorphism between both abelian groups. In fact * is a contravariant functor from the category of the sets of sum to that of

UU _Ž U _.

the abelian groups. For every set of sum ⌫ we put ⌫ sHom ⌫ ,⺪. In a similar way for any homomorphism f between sets of sum we write fUU

Ž U.U w x

for f . As it was pointed by Winter 25 , for all ␣g⌫ the map

U _{Ž .} UU

_ˆ

␣

ˆ

:⌫ ª⺪ given by␣

ˆ

:␸¬␸ ␣ belongs to ⌫ , being the subset ⌫ of

UU

_ˆ

these elements a set of sum in ⌫ and the map H_⌫ :⌫ª⌫ given by

H⌫ :␣¬␣

ˆ

a homomorphism of sets of sum. It is easy to show that if

X UU_Ž

_ˆ

_.

_ˆ

X

f:⌫ª⌫ is a homomorphism of sets of sum then f ⌫ ;⌫. Indeed,

ˆ

X

for all ␣g⌫ and ␺g⌫ we have

$

UU U

f

Ž

␣ ␺

ˆ

. Ž

.

s

Ž

␣

ˆ

(f

. Ž

␺

.

s␣ ␺

ˆ

Ž

(f

.

s␺

Ž

f

Ž

␣

.

.

sf

Ž

␣ ␺

. Ž

.

.

$ _X

UU_{Ž .}

_ˆ

So f ␣

ˆ

sf

Ž

␣

.

g⌫.

As a consequence of the above for every homomorphism f:⌫ª⌫X

ˆ ˆ

ˆ

X

between sets of sum there exists a unique homomorphism f:⌫ª⌫ making commutative the diagram

H⌫6

ˆ

⌫ ⌫

6 fˆ

f ₆

X

H⌫

X 6

_ˆ

X

⌫ ⌫

ˆ

UU

Moreover, f can be obtained by restriction of f .

If f:⌫ª⌫X is a bijective homomorphism of sets of sum, f is said to be an isomorphismwhen fy1 _{is also a homomorphism. Isomorphisms from ⌫} to⌫are called automorphismsof ⌫. For all ␣ in the set of sum ⌫ we have a binary relation ;_␣ in ⌫ writing ␤;_␣ ␥ when␥y␤s␣. We denote

Ž .

by ⌫_␤ ␣ the corresponding class of ␤ in the equivalence relation

gener-Ž .

ated by ;_␣ . We say that ⌫_␤ ␣ is the ␣-string of ␤. If there exist two

Ž . Ž .

integers q₁,q₂G0 such that ␤y q₂q1 ␣f⌫ and ␤q q₁q1 ␣f⌫

Ž .

then the ␣-chain ⌫_␤ ␣ is said to be bounded. In this case there are two

Ž . Ž .

integers q,

ll

G0 such that ␤y

ll

q1 ␣f⌫, ␤q qq1 ␣f⌫ with

Ž .

⌫_␤ ␣ being the set of the different elements ␤y

ll

␣, . . . ,␤, . . . ,␤qq␣.

Ž .

The integer c_␣␤s y qy

ll

is called the Cartan number of the bounded

Ž . w x

chain ⌫_␤ ␣ . In a similar way to 25, p. 242 we can prove the following criterion for bijective homomorphisms to be isomorphisms.

LEMMA1.1. Let f:⌫ª⌫X be a bijecti¨e homomorphism between sets of

Ž . Ž . XXŽ X.

sum. Assume that all the chain ⌫_␤ ␣ ␣,␤g⌫, ␣/0 and ⌫_␤ ␣ Ž␣X,␤Xg⌫X, ␣X/0. are bounded. Then f is an isomorphism if and only if cfŽ␣.fŽ␤.sc␣␤ for any␣,␤g⌫ with ␣/0.

Let ⌫ be a set of sum, ␣g⌫, ␣/0. The automorphism s:⌫ª⌫ is said to be a symmetry of ⌫ at the point ␣ when the following conditions hold:

䢇 s_Ž␣_.s y␣_;

䢇 sŽ⌫_␤Ž␣..s⌫_␤Ž␣..

Obviously a symmetry at a nonzero point ␣ is an involutive map. As it

w x

was pointed in 25, p. 239 , if ⌫ is a set of sum and ␣ is a nonzero element

Ž .

in ⌫ such that ⌫_␤ ␣ is bounded for all ␤g⌫, then there is at most a symmetry of ⌫ at the point ␣. Moreover, if s_␣ is the symmetry of ⌫ at ␣ then s_␣ is given by s_␣:␤¬␤yc_␣␤␣ for all ␤g⌫. Elementary but useful is the following lemma.

LEMMA 1.2. Let ⌫ be a set of sum, s:⌫ª⌫ a map, and ␣ a nonzero element in ⌫.Then we ha_¨e:

Ž .1 The map s is a symmetry at ␣ iff it is an in¨olute homomorphism

Ž . Ž . Ž .

satisfying s ␣ s y␣ and s ␤ g⌫_␤ ␣ for all ␤g⌫.

Ž .2 If s is a symmetry at␣ and␤ is an element in⌫such that⌫_␤Ž␣.is unbounded, then ␤qn␣g⌫ for all ng⺪. Moreo¨er, if Ann␣s ng

4 X Ž .

⺪:n␣s0 s0 then for all integers n)0 there exists some ␤ g⌫_␤ ␣

Ž X. X 4

such that s ␤ y␤ does no belong to i␣:yn-i-n.

Ž .

Proof. 1 It suffices to show that the given conditions imply that sis a symmetry at ␣. The homomorphism s is an automorphism, since s2_sId.

Ž Ž .. Ž Ž ..

By the homomorphic character of s we have s ⌫_␤ ␣ ;⌫_sŽ␤. s ␣ . Thus

Ž Ž .. Ž . Ž . Ž . 2Ž Ž .. Ž Ž ..

s ⌫_␤ ␣ ;⌫_sŽ␤. ␣ s⌫_␤ ␣ . Moreover, ⌫_␤ ␣ ss ⌫_␤ ␣ ;s⌫_␤ ␣ . Therefore s is a symmetry at ␣.

Ž .2 Assume the existence of an integerqG0 such that ␤, . . . ,␤qq␣

Ž . Ž . Ž .

are in ⌫ but ␤q qq1␣f⌫. Since s ␤qq␣ g⌫_␤ ␣ , there exists an

X

Ž . X

integer

ll

Fq such that s ␤qq␣ s␤q

ll

␣. From this we obtain

Ž . Ž X. Ž X . Ž .

s ␤ s␤q qq

ll

␣. Taking into account that ␤q

ll

y1 ␣g⌫_␤ ␣ ,

Ž Ž X . . Ž . Ž X . Ž .

then we have s ␤q

ll

y1␣ ss ␤ y

ll

y1␣s␤q qq1 ␣.

Ž .

But this contradicts that sis a symmetry, since␤q qq1 ␣f⌫. We can argue in a similar way when there exists an integer

ll

G0 such that

Ž .

␤y

ll

␣, . . . ,␤ are in ⌫ but ␤y

ll

q1␣f⌫.

Ž .

If Ann ␣s0 there exists a unique i0g⺪ such that s ␤ s␤qi0␣. Choose an integer j such that 2jF ynqi0. Put ␤Xs␤qj␣. Then we

Ž X. X Ž . X Ž .

have s ␤ y␤ ss ␤ yj␣y␤ s i0y2j ␣, being i0y2jGn.

Ž X. X 4

Therefore s ␤ y␤ does not belong to i␣:yn-i-n.

The set of sum ⌫ is said to be an ab-root system if the following conditions hold:

Ž .1 ⌫_␤Ž␣.is bounded for all ␤g⌫ and every nonzero ␣g⌫.

Ž .2 For all ␣g⌫, ␣/0, there exists a homomorphism of sets

ⴢ ⴢ_{Ž .} ⴢ_{Ž .}

of sum ␣ :⌫ª⺪ such that ␣ ␣ s2 and satisfying ␣ ␤ sc_␣␤ for any ␤g⌫.

Obviously if ⌫ is an ab-root system and ␣g⌫, ␣/0, then y␣g⌫ and

ⴢ_{Ž .}

the map s_␣:⌫ª⺪given by s_␣:␤¬␤y␣ ␤ ␣ is a symmetry at ␣. LetV be a real prehilbert space. A subset ⌫#of V such that 0f⌫#is said to be a root system when ⌫# generates V and for all ␣g⌫# there exists a linear form ␣k:Vª⺢satisfying the following conditions:

䢇 ␣ ␣kŽ .s2.

䢇 ␣k_Ž␤_.g⺪_{for all} ␤g⌫#_.

䢇 The linear map r_␣:VªV given by r_␣:¨¬¨y␣kŽ .¨ ␣ is an

Ž .

isometry such that r_␣ ⌫# ;⌫#.

Obviously in the finite case our definition of root systems agrees with

w x

the one given in 5, p. 144; 20, p. 27 .

Let ⌫#be a root system in the real prehilbert spaceV. For all ␣g⌫#

Ž .

we havey␣sr_␣ ␣ g⌫#. Moreover it is easy to show that r_␣ is the only isometry of V yielding ␣ to y␣ and that fixes all the points of the

␣¬¨

Ž .

Ž .

hyperplane orthogonal to ⺢␣. Therefore r_{␣ ¨} s_¨y2_Ž␣¬␣.␣ for all

¨gV. So

␣¬¨

Ž

.

k

␣

Ž .

¨ s2 .

Ž .

1

␣¬␣

Ž

.

w x Ž .

If ␣,␤g⌫# then as in 5 one can prove the following assertions: i if

y1 1 _{k k}

Ž . Ž . Ž .

Ž .

with equality if and only if ␣ and ␤ are proportional; iii the integers

k_{Ž .} k_{Ž . Ž .}

␣ ␤ and ␤ ␣ have the same sign; iv if ␣ and ␤ are

nonpropor-k_{Ž .} k_{Ž . Ž}

tional and nonorthogonal and ␣ ␤ and ␤ ␣ are both positive resp.

. Ž . Ž .

negative then at least one of these numbers is 1 resp. y1 ; v if

k_{Ž . Ž .} k_{Ž .}

␣ ␤ F y1 then ␤q␣g⌫#; vi if ␣ ␤ G1 then ␤y␣g⌫#. If ⌫#is a root system in the real prehilbert spaceVandW is the vector

4

subspace generated by a finite subset ␥1, . . . ,␥n of ⌫#, then ⌫#lW is

Ž . Ž . Ž . kŽ .

also a finite set. Indeed, by ii , iii , and Eq. 1 we havey4F␥i ␣ F4 for all ␣g⌫#and is1, . . . ,n and we can define the map

ntimes

4

4

h:⌫#lWª ig⺪:y4FiF4 = ⭈⭈⭈ = ig⺪:y4FiF4

Ž kŽ .. Ž .

given by h:␤¬ ␥_i ␤ . If for ␤₁,␤₂g⌫#lW we have h ␤₁ s

Ž . Ž . Ž . Ž .

h ␤₂ , then 1 gives ␥i¬␤1 s ␥i¬␤2 for is1, . . . ,n. Thus ␤1y␤2 is

Ž .

orthogonal to W. In particular, ␤₁y␤₂¬␤₁y␤₂ s0 and so ␤₁s␤₂. This proves the injectivity of hand so the finite character of ⌫#lW.

If ␣ and ␤ are nonproportional elements in the root system ⌫# then

4

the above gives that Is ig⺪:␤qi␣g⌫# is a finite set. Let qG0 be the greatest element in this set and

ll

G0 such that y

ll

is the lowest

w x

integer in I. In a similar way to 5, p. 149 we can prove that Is ig

4 Ž .

⺪:y

ll

FiFq. Now it is easy to show that r_␣ ␤y

ll

␣ s␤qq␣. Taking

Ž . kŽ . kŽ .

into account that r_␣ ␤y

ll

␣ s␤q

ll

␣y␣ ␤ ␣, this gives ␣ ␤ s

Ž .

y qy

ll

. So the given definition of root systems of arbitrary cardinal is

w x

equivalent to that of Neher 12᎐14 .

By the previous considerations all root systems with zero added are

ab-root systems. The main result of this paper asserts that every ab-root

Ž .

system is isomorphic as set of sum to some root system with zero added.

2. ab-ROOT SYSTEMS

This section is devoted to prove some properties of the ab-root systems.

LEMMA 2.1. Let ⌫ be an ab-root system and ␣ a nonzero element in ⌫.

Then the following assertions hold:

Ž .1 If 2␣g⌫ then4␣f⌫.

Ž .2 If for some integer iG1 we ha¨e ␣, . . . ,i␣g⌫, then is1 or 2.

Ž .3 If 2␣g⌫ then3␣f⌫.

Ž .

Proof. 1 By contradiction we assume 4␣g⌫. Since 2␣g⌫ then

Ž .ⴢŽ . Ž .ⴢŽ . Ž .ⴢŽ . Ž .ⴢŽ .

4␣/0. So 2s 4␣ 4␣ s2 4␣ 2␣ s4 4␣ ␣ . Therefore 4␣ ␣

Ž .2 We have 2sŽi␣.ⴢŽi␣.si iŽ ␣ ␣.ⴢŽ .. Since i/0, then Ži␣ ␣.ⴢŽ .

2 2

s _i. Taking into account that _ig⺪, we obtain is1 or 2.

Ž .3 It is an immediate consequence from 2 .Ž .

PROPOSITION 2.2. Let ⌫ be an ab-root system ␣,␤g⌫ with ␣/0.

Then the following assertions hold:

Ž .1 If ␤/0 and2␣f⌫ then not all of

␤y2␣,␤y␣,␤,␤q␣,␤q2␣

belong to ⌫.

Ž .2 Not all the elements ␤y2␣,␤y␣,␤,␤q␣,␤q2␣,␤q3␣

belong to ⌫.

Ž .3 There are at most 5 different elements in ⌫_␤Ž␣.. Moreo¨er, when Ž .

␤/0,"␣ and2␣f⌫the␣-string ⌫_␤ ␣ has at most4 different elements.

Ž .4 If ␤/0 and ␣ ␤ⴢŽ .s0 then ␤ ␣ⴢŽ .s0.

Ž .5 If ␤/␣ and ␣ ␤ⴢŽ .s2 then ␤ ␣ⴢŽ .s1. Moreo¨er, 2␤f⌫.

Ž .6 If ␤/0 and ␣ⴢs␤ⴢ then ␣s␤.

Ž .

Proof. 1 Suppose, contrary to our claim, that ␤y2␣,␤y␣,␤,␤

Ž . ⴢŽ

q␣,␤q2␣ are all in ⌫. Since ␤y2␣ y␤s y2␣f⌫, then ␤ ␤

. Ž . ⴢŽ .

y2␣ F0. Since ␤q2␣ y␤s2␣f⌫, then ␤ ␤q2␣ F0. From

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

these inequalities, 2y2␤ ␣ F0 and 2q2␤ ␣ F0. So 1F␤ ␣

ⴢ_{Ž .}

and ␤ ␣ F y1, which is a contradiction.

Ž .2 By Lemma 2.1 we assume without loss of generality that ␤/0. First we prove that ⌫ cannot contain ␤y3␣,␤y2␣,␤y␣,␤,␤q␣,

Ž .

␤q2␣,␤q3␣. Suppose all these elements are contained in ⌫. By 1 ,

ⴢ_{Ž .}

2␣g⌫. By Lemma 2.1, 3␣f⌫andy3␣f⌫. Therefore ␤ ␤y3␣ F0 2

ⴢ_{Ž .} ⴢ_{Ž .}

and ␤ ␤q3␣ F0. This is equivalent to the inequalities 3F␤ ␣ and 2

ⴢ_{Ž .}

␤ ␣ F y3, which is the required contradiction. Thus ⌫ cannot contain

␤y3␣,␤y2␣,␤y␣,␤,␤q␣,␤q2␣,␤q3␣. Now we argue by

Ž .

contradiction to prove 2 . Assume ␤y2␣,␤y␣,␤,␤q␣,␤q2␣,␤

Ž .

q3␣ belong to ⌫. By 1 , 2␣g⌫ and so Lemma 2.1 gives 3␣f⌫. As 2

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

above ␤ ␤q3␣ F0 and so ␤ ␣ F y₃. Since ␤ ␣ is an integer,

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

␤ ␣ F y1. Put ␥s2␣. We have ␤ ␥ s2␤ ␣ F y2. Thus␥y␤,

Ž .

␥,␥q␤,␥q2␤,␥q3␤ are elements in ⌫. By 1 , 2␤g⌫. By Lemma 2

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

2.1, 3␤f⌫. Thus ␥ ␥q3␤ F0. So, ␥ ␤ F y3. With ␥ ␤ being

ⴢ_{Ž .}

an integer, ␥ ␤ F y1. Since ␤y␥s␤y2␣g⌫, we have ␤q␥,

␤q2␥g⌫. Therefore ␤y2␣,␤y␣,␤,␤q␣,␤q2␣,␤q3␣,␤q 4␣ are elements in ⌫. This contradicts that which was previously proved.

Ž .4 Taking into account 3 and thatŽ . ␣ ␤ⴢŽ .s0, we have only the

Ž . Ž . 4 Ž .

following possibilities for ⌫_␤ ␣ :⌫_␤ ␣ s ␤ , or ⌫_␤ ␣ s ␤y␣,␤,␤

4 Ž . 4

q␣ , or ⌫_␤ ␣ s ␤y2␣,␤y␣,␤,␤q␣,␤q2␣ . We will see

inde-Ž . 4

pendently each one of these cases. First we suppose⌫_␤ ␣ s ␤ . Since in

Ž . 4 ⴢŽ .

this case ␤y␣f⌫ and ␤q␣f⌫, we have ⌫_␣ ␤ s ␤ . Thus ␤ ␣

Ž . 4 Ž .

s0. Now we assume⌫_␤ ␣ s ␤y␣,␤,␤q␣ . In this case⌫_␣ ␤ > ␣

4

y␤,␣,␣q␤ . If this inclusion were strict then ␣q2␤g⌫ or ␣y2␤

ⴢ_{Ž .}

g⌫. Suppose ␣q2␤g⌫. Since ␣ ␣q2␤ s2, the elements 2␤ and

Ž .

2␤y␣ belong to ⌫. Thus ␣y2␤g⌫. So taking into account 3 ,

Ž . 4 ⴢŽ .

⌫_␣ ␤ s ␣y2␤,␣y␤,␣,␣q␤,␣q2␤ and ␤ ␣ s0. In a similar

ⴢ_{Ž .}

way it is obtained ␤ ␣ s0 when ␣y2␤g⌫.

Ž . 4

Finally we suppose ⌫_␤ ␣ s ␤y2␣,␤y␣,␤,␤q␣,␤q2␣ . By

Ž .1 ,␥s2␣ belongs to ⌫. By 1 and taking into account that 2Ž . ␥s4␣f⌫, we have that not all the elements ␤y2␥,␤y␥,␤,␤q␥,␤q2␥

be-Ž . Ž .

long to ⌫. So we have only three possibilities for ⌫_␤ ␥ :⌫_␤ ␥ s ␤y␥,

4 Ž . 4 Ž .

␤,␤q␥ , or ⌫_␤ ␥ s ␤y␥,␤,␤q␥,␤q2␥ , or ⌫_␤ ␥ s ␤y2␥,

4 Ž .

␤y␥,␤,␤q␥ . Suppose ␤y2␥g⌫. Since ␤y4␣ q␣s␤y3␣

ⴢ_{Ž .}

f⌫ we obtain ␣ ␤y4␣ G0. So

0F␣ⴢ

Ž

␤y4␣

.

s␣ⴢ

Ž

␤y2␣

.

y␣ⴢ

Ž

2␣

.

s␣ⴢ

Ž

␤

.

y8.

ⴢ_{Ž .}

Thus, ␣ ␤ G8, which is a contradiction. So it must be ruled out the

Ž . 4

possibility ⌫_␤ ␥ s ␤y2␥,␤y␥,␤,␤q␥ . In a similar way it is ruled

Ž . 4 Ž .

out the case ⌫_␤ ␥ s ␤y␥,␤,␤q␥,␤q2␥ . Therefore ⌫_␤ ␥ s ␤

4 Ž . ⴢŽ .

y␥,␤,␤q␥ . As in the first part of the proof of 4 , ␤ ␥ s0 and 1

ⴢ_{Ž .} ⴢ_{Ž .}

␤ ␣ s2␤ ␥ s0.

Ž .5 First we prove 2␤f⌫. Otherwise ␦s2␤g⌫ and ␣ ␦ⴢŽ .s ⴢ_{Ž .}

2␣ ␤ s4. So the elements ␣y4␦,␣y3␦,␣y2␦,␣y␦,␣ are in⌫.

Ž .

By 3 , 4␤s2␦g⌫, but this contradicts Lemma 2.1. Therefore 2␤f⌫. Now we will consider in a separate way the cases where ␣y2␤f⌫ or

␣y2␤g⌫. First we assume ␣y2␤f⌫. Since ␣g⌫ in this case we

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

have ␤ ␣y␤ F y1 and so ␤ ␣ F1. If it were ␤ ␣ -0 then

ⴢ_{Ž .}

␣q␤,␣q2␤g⌫, since ␣y␤g⌫. Hence ␣ ␣q2␤ s6, which

Ž . ⴢŽ . Ž .

contradicts 3 . Thus, 0F␤ ␣ F1. By 4 can be ruled out the

possibil-ⴢ_{Ž .} ⴢ_{Ž .}

ity ␤ ␣ s0. Therefore, ␤ ␣ s1. Now we assume ␣y2␤g⌫. Since

ⴢ_{Ž . Ž . Ž .}

␣ ␣y2␤ s y2, then the elements ␣y2␤ q␣s2 ␣y␤ and 3␣

Ž . Ž .

y2␤ belong to ⌫. Since ␣/␤ and 2 ␣y␤ g⌫, then 3 ␣y␤ f⌫

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

and so 0G␤ 3␣y2␤ s2␤ ␣y␤ q␤ ␣ s3␤ ␣ y2␤ ␤ s 4

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

3␤ ␣ y4. Thus ␤ ␣ F3 and so ␤ ␣ F1. From this we obtain

ⴢ_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

␤ ␣y2␤ s␤ ␣ y2␤ ␤ F y3. Taking into account that 2␤f⌫,

Ž . ⴢŽ . ⴢŽ .

Ž .6 Assume ␣/␤. Since ␣ⴢs␤ⴢ then 2s␤ ␣ⴢŽ .. By Ž .5 , 1s

ⴢ_{Ž .} ⴢ_{Ž .}

␣ ␤ s␤ ␤ s2, which is a contradiction.

Let ⌫ be an ab-root system and␥ and ␦ nonzero elements in ⌫. As in the classical case of the root systems, we say that␥ is connected to␦ in ⌫ if there exists a finite sequence ␣1,␣2, . . . ,␣n of nonzero elements in ⌫

ⴢ_{Ž .}

such that␥s␣1,␦s␣n, and satisfying ␣ ␣i iq1 /0 for all ig 1, . . . ,n 4

y1 . The elements ␣1, . . . ,␣n are said to be a pathfrom␥ to␦. If S is a subset of ⌫and there exists a path from␥ to ␦ such that every␣i belongs to S, then we say that ␥ and ␦ are connected in S. Connectness is an

4 Ž

equivalence relation in ⌫y 0 the symmetry property is a consequence

Ž . .

of part 4 of Proposition 2.2 . The union of zero with one equivalence class is a subset of ⌫, which is said to be a connected componentof ⌫. The

4

ab-root system ⌫ is said to be irreduciblewhen ⌫/ 0 and it has only one connected component.

If ⌫X is a subset of the ab-root system ⌫ which contains 0 and where

␣y␤g⌫X for all ␣,␤g⌫X such that ␣y␤g⌫, then we say that ⌫X is an ab-root subsystemof ⌫. Every ab-root subsystem ⌫X of ⌫ is an ab-root

ⴢ _Ž X _4.

system relative to the restriction of the homomorphism ␣ ␣g⌫ y 0 .

4

Let⌫ be an ab-root system and ⌫i igAA the connected components of ⌫. The following assertions hold:

Ž .1 If ␣g⌫i, ␤g⌫j with i/j and ␣/0/␤, then ␣q␤f⌫ and ␣y␤f⌫.

Ž .2 Every ⌫i is an ab-root subsystem.

Ž .

Indeed, to prove 1 we argue by contradiction assuming ␣q␤g⌫. So

ⴢ_{Ž .} ⴢ_{Ž .}

␣ ␣q␤ s2 and ␤ ␣q␤ s2. Transitivity of the connectness rela-tion gives ␣ connected to ␤, which is a contradiction. In a similar way it

Ž .

can be shown that ␣y␤f⌫. To see 2 we consider ␣,␤g⌫_i such that

␣y␤g⌫. We can assume without loss of generality that ␣/0/␤. Put

␥s␣y␤. If it were ␥f⌫_i, then ␥g⌫_j for some j/i. But now

Ž .

␥q␤s␣g⌫ contradicts 1 .

3. THE MAIN THEOREM

LetG be an abelian group and⌫;Gan ab-root system. We say that ⌫ is nicely embedded in G when for all nonzero ␣g⌫ there exists a homomorphism of abelian group ␣⬚:Gª⺪ extending ␣ⴢ.

If ⌫ is an ab-root system in the abelian group of a real vector space V

such that for any non-zero ␣g⌫ there exists a linear form ␣k:Vª⺢

4

is a root system in the real prehilbert spaceV then ⌫#j 0 is an ab-root system very nicely embedded inV.

Ž

Obviously if ⌫ is an ab-root system nicely embedded resp. very nicely

. Ž .

embedded in an abelian group resp. a real vector space then every

Ž .

ab-root subsystem of ⌫ is nicely embedded resp. very nicely embedded in

Ž .

the subgroup resp. vector subspace which it generates. So every

con-Ž

nected component of an ab-root system nicely embedded resp. very nicely

. Ž .

embedded in an abelian group resp. real vector space is nicely

embed-Ž . Ž .

ded resp. very nicely embedded in the subgroup resp. vector subspace

Ž .

which it generates see the end of Section 2 .

4

Let ⌫#be a subset of a real vector spaceV. Let ⌫#i igAA be a family of subsets of ⌫# such that ⌫#sDi⌫#i. Suppose the following conditions:

Ž .1 every ⌫#i generates a vector subspace Wi which is a prehilbert space

Ž . Ž . Ž .

relative to an inner product ¬ i; 2 the set⌫#iis a root system inWi; 3

Ž .

Wil Ýj/iWj s0 for all igAA. Then ⌫# is a root system in the

pre-Ž . Ž . Ž .

hilbert spaceWs

[

Wi of inner product ¬ given by ¬ sÝi ¬ i.

LEMMA 3.1. Let ⌫ be an ab-root system ¨ery nicely embedded in a real 4

¨ector space V and ⌫i igAA the family of the connected components of ⌫. If

k 44

the set of linear forms ␣ :␣g⌫y 0 separates points in V and e¨ery ⌫i 4 generates a¨ector subspace W which is a prehilbert space such thati ⌫iy 0 is

4

a root system in Wi, then ⌫y 0 is a root system in the space W that it generates.

Proof. By the previous comments it suffices to show that W_il ŽÝj/iWj.s0 for all igAA. To see this consider xgWilŽÝj/iWj.. Since

k_{Ž .}

xgÝj/iWj, for all ␣/0 in ⌫i we have ␣ x s0. Since xgWi, for all

Ž . kŽ .

␤/0 in ⌫_j j/i we have ␤ x s0. So the separating hypothesis for

␣k:␣g⌫y 440 yields to xs0.

Ž .

THEOREM 3.2 Main Theorem . For the set of sum ⌫ the following assertions are equi¨alent:

Ž .1 ⌫ is an ab-root system.

Ž .2 ⌫ is isomorphic to an ab-root system nicely embedded in a torsion

-free abelian group.

Ž .3 ⌫ is isomorphic to an ab-root system_¨ery nicely embedded in a real

¨ector space.

Ž .4 ⌫ is isomorphic to a root system with zero added.

Ž . Ž .

Proof. 1 « 2 . For any ␣g⌫, ␣/0, let s_␣ be the symmetry at ␣. $

$ $

ˆ

ˆ

Ž

ˆ

.

There exists a unique homomorphism_$ s_␣:⌫ª⌫such thats_␣ ␤ ss_␣

Ž

␤

.

UU UU

$ _U involutive automorphism, then s_␣ is so. Moreover, for ␤g⌫, ␸g⌫ s

Ž .

Hom⌫,⺪ we have $ $

ˆ

s␣

Ž

␤ ␸

.

Ž

.

ss␣

Ž

␤ ␸

. Ž

.

s␸(s␣

Ž

␤

.

ⴢ

_ˆ

ⴢ

s␸ ␤

Ž

.

y␣ ␤ ␸ ␣

Ž

.

Ž

.

s ␤y␣ ␤ ␣ ␸

Ž

.

ˆ

Ž

.

.

Thus

$ _ⴢ

ˆ

ˆ

s_␣

_Ž

␤

_.

s␤y␣ ␤ ␣

Ž

.

ˆ

.

Ž .

2

ⴢ_{Ž .}

_ˆ

ⴢ_{Ž .}

Sincen is a homomorphism and_$ ␣ ␤ sc␣␤, we have_$ ␤y␣ ␤ ␣

ˆ

g

ˆ

Ž . Ž . Ž .

⌫␤ˆ ␣

_ˆ

. Equation 2 givess␣ ␣

_ˆ

s y␣

_ˆ

. By Lemma 1.2,s␣is a symmetry of

ˆ

ⴢ_{Ž .}

⌫ at ␣

ˆ

. By Proposition 2.2, y4F␣ ␤ F4. Taking into account that

UU

_ˆ

_{Ž .}

⌫ is torsion-free, from Lemma 1.2 it is obtained that_$ ⌫_␤ˆ ␣

ˆ

is bounded

UU

ˆ

ˆ

Ž

ˆ

.

ˆ

ˆ

ˆ

for all ␤g⌫. Therefore, s␣ ␤ s␤yc␣␤ˆˆ␣

ˆ

for all ␤g⌫. Since ⌫ is

Ž . ⴢŽ . w x

torsion-free, by 2 we havec␣␤s␣ ␤ sc␣␤ˆˆ. As in 25, pp. 240᎐241 we

ⴢ 44

can prove that the set of homomorphisms ␣ :␣g⌫y 0 separates

ˆ

points in ⌫. So the mapH_⌫ :⌫ª⌫ is a bijective homomorphism. Lemma

ˆ

1.1 gives that H_⌫ is an isomorphism. Now we will prove that ⌫ is an

ab-root system nicely embedded in the abelian group⌫UU. For any nonzero

Ž ⴢ.UU UU

␣g⌫ we put ␣⬚

ˆ

s␧( ␣ for ␧:⺪ ª⺪ the homomorphism given

Ž . UU Ž UU .

by␧:␺¬␺ Id⺪ for any␺g⺪ . Obviously ␣⬚

ˆ

gHom⌫ ,⺪. More-over

$

ⴢ ⴢ ⴢ

ˆ

ˆ

␣⬚ ␤

ˆ

_Ž

_.

s␧(

Ž

␣

.

**

_Ž

␤

_.

s␧ ␣ ␤

Ž

Ž

.

.

s␣ ␤

Ž

.

sc␣␤sc␣␤ˆˆ.

ˆ

Therefore, ⌫ is an ab-root system nicely embedded in the torsion-free abelian group⌫UU.

Ž .2 «Ž .3 . Without loss of generality we will assume that ⌫ is an ab-root system nicely embedded in a torsion-free abelian group G. The tensor product Gm_⺪ ⺢is a real vector space in a natural way. Moreover, for all

xgG such that xm1s0 we have xs0. Indeed, in this case there exists a finitely generated subgroup H of G such that xm1s0 in Hm_⺪ ⺢. Since G is torsion-free, H is a free abelian group. From this it is easily

4

obtained that xs0. In particular, ␣m1 :␣g⌫ is a set of sum which has bounded every␣m1-string for all ␣m1/0. Moreover the map from

4

⌫ to the set ␣m1 :␣g⌫ given by ␣¬␣m1 is a bijective homomor-phism between sets of sum which is an isomorhomomor-phism. For all ␣g⌫, ␣/0

Ž .k Ž .

we put ␣m1 s␪( ␣⬚mId_⺢ where ␪:⺪m_⺪ ⺢ª⺢is the canonical

Ž .k

isomorphism. Obviously ␣m1 is a ⺢-linear form. For all ␤g⌫ we

Ž .kŽ . Ž Ž . . Ž .

have ␣ m1 ␤m 1 s ␪ ␣⬚ ␤ m 1 s ␣⬚ ␤ sc_␣␤ sc_␣m1,␤m1.

4 Ž .k Ž .

Therefore ␣m1 :␣g⌫ with the linear maps ␣m1 ␣/0 is an

Ž .3 «Ž .4 . Let ⌫ be an ab-root system very nicely embedded in a real vector spaceV. PutWsVrN for N the subspace of the xgV such that

k_{Ž . 4}

␣ x s0 for all ␣g⌫y 0 . Let ᎐be the projection map of V onto W

and ⌫ the image of ⌫ under ᎐. First we will prove that ⌫ is an ab-root system and ᎐ induces an isomorphism between ⌫ and ⌫. If ␣,␥g⌫ and

ⴢ_{Ž .} ⴢ_{Ž .}

␣s␥, then ␤ ␣ s␤ ␥ for all ␤/0 in ⌫. As it was pointed out in

Ž . Ž . ⴢ 44

the proof of 1 « 2 , the set ␤ :␤g⌫y 0 separates points in ⌫

Žsee 25, pp. 240w ᎐241 . Therefore,x. ␣s␥ and so ᎐ induces a bijection between ⌫and⌫. We consider next ␣₁,␣₂g⌫such that␣₁q␣₂g⌫and we will show that ␣₁q␣₂g⌫. We have the existence of ␣₃g⌫ such that ␣1q␣2s␣3. Without loss of generality we assume ␣3/0. So

k_{Ž .} k_{Ž .} k_{Ž .} k_{Ž .} k_{Ž .}

␣ ␣3 1 q␣ ␣3 2 s␣ ␣3 3 s2. This gives␣ ␣3 1 G1 or␣ ␣3 2 G1.

k_{Ž .}

If it were ␣ ␣_{3 1} G1 then ␦s␣₁y␣₃ belongs to ⌫. Thus ␦s y␣₂, and by that which was previously proved,␦s y␣2. Therefore,␣1q␣2s

k_{Ž .}

␣₁y␦s␣₃g⌫. We can argue in a similar way in the case that ␣ ␣_{3 2}

G1. So we obtain that ␣1q␣2g⌫ always implies ␣1q␣2g⌫, which gives that ᎐ induces an isomorphism between the sets of sum ⌫ and ⌫. Considering for every nonzero element ␣ in ⌫ the linear form induced by

k

␣ we see that ⌫ is an ab-root system very nicely embedded in the vector

Ž .

space W also in the vector subspace generated by ⌫ . Moreover, these induced maps separate points inW.

By Lemma 3.1 and the above we can assume without loss of generality that ⌫ is an irreducible ab-root system very nicely embedded in a real

4

vector space W, with W generated by ⌫. To show that ⌫y 0 is a root system inW we will prove first that if F0 is a nonempty finite subset of ⌫ and FF is the family of all finite subsets of ⌫ containing F0, then there

4

exists a family ⌫_{F FgFF} of ab-root subsystems of ⌫satisfying the following conditions:

Ž .i Every ⌫F is finite irreducible and contains F.

Ž .ii ⌫sD_FgFF⌫_F.

Žiii. ⌫_F;⌫_FX for any F,FXgFF such that F;FX.

First for all nonzero ␣,␤g⌫ we choose a set C_␣␤ whose elements are a

4

path from ␣ to ␤. To see the existence of the family ⌫F FgFF, for all

FgFF we denote by WF the vector subspace of W generated by the set

4

SFsD␣,␤gFy04C␣␤. Putting ⌫Fs⌫lWF, we obtain a family ⌫F FgFF

Ž . Ž .

of ab-root subsystems of ⌫, satisfying ii , iii , and with the property that every ⌫_F contains F. Let F be arbitrary in FF. Let␥₁, . . . ,␥_n be the whole of the different elements of the finite set SF. By Proposition 2.2, y4F

ⴢ_{Ž .}

␣ ␥_i F4 for all ␣g⌫_F, ␣/0. So we can define a map

ntimes

4

4

h_F: ⌫_Fy 0 lWª ig⺪:y4FiF4 = ⭈⭈⭈

4

= ig⺪:y4FiF4

Ž ⴢŽ . ⴢŽ .. 4

given by h_F:␣¬ ␣ ␥₁ , . . . ,␣ ␥_n for all ␣g⌫_Fy 0 . If ␣,␤g⌫_F,

Ž . Ž . kŽ . kŽ .

␣/0/␤ and h_F ␣ sh_F ␤ , then ␣ ␥_i s␤ ␥_i for all is

k_{Ž .} k_{Ž .} k_{Ž .} k_{Ž .}

1, . . . ,n. Therefore, ␣ x s␤ x for all xgWF. So ␣ ␥ s␤ ␥ for all␥g⌫_F. The last part of Proposition 2.2 applied to ⌫_F gives ␣s␤. Now injectivity of hF yields to the finite character of ⌫F. Let ␦1,␦2 be

ⴢ_{Ž .}

arbitrary elements in ⌫F. If it were ␦ ␥1 i s0 for all ␥igSF, then

k_{Ž .} ⴢ_{Ž .} ⴢ_{Ž .}

0s␦₁ ␦₁ s␦ ␦_{1 1} s2, a contradiction. Thus, ␦ ␥₁ i₁ /0 for some ␥i₁ ⴢ_{Ž .}

gSF. In a similar way, ␦ ␥2 i2 /0 for some␥i2gSF. The way to define

4 Ž 4.

SF yields to the existence of ␣1gFy 0 resp. ␣2gFy 0 such that

Ž . Ž .

␣₁ resp. ␣₂ is connected in ⌫_F to ␥_i resp. ␥_i . Since C_{␣ ␣} ;⌫_F,

1 2 1 2

transitivity of the connectness relation gives that ␦1 and ␦2 are connected in ⌫F.

Ž .X

Now we consider an inner product ¬ making W a prehilbert space

Ž .

and we will prove the existence of a new inner product ¬ inW such that

4 kŽ .

for all ␣g⌫y 0 the map r_␣:WªW given by r_␣:¨¬¨y␣ ¨ ␣ is

an isometry. If FgFFand WWF is the subgroup generated by r␣:␣g⌫Fy 440 then as in 20, p. 27 we can define another inner productw x Ž ¬ .X_F in the vector subspaceWF given by

X X

x¬y s t x ¬t y

Ž

.

F

Ý

Ž

Ž .

Ž .

.

tgWWF

4

for any x,ygWF. For all ␣g⌫Fy 0 the linear map r␣ induces an

Ž Ž .X . 4

isometry of WF, ¬ F . So every subset ⌫Fy 0 is a root system in WF.

Ž .

By 1 it is easy to show that the inner product of the embedding real prehilbert space of an irreducible root system ⌫#is completely determined

k 4

up to a positive factor by the set ␣ :␣g⌫# . Thus for all FgFF we

Ž .X Ž

can assure the existence of a real numberkF)0 such that x¬y FskF x

.X Ž . Ž .Ž .X

¬y _F for any x,ygW_F. Setting ¬ _Fs 1rk_F ¬ _F we obtain an inner

0 0

Ž . Ž .

product inWF which extends ¬ F0. Now an inner product ¬ inW can

Ž . 4

be defined extending every ¬ F. Moreover, ⌫y 0 is a root system in

Ž Ž ..

the real prehilbert space W, ¬ .

4. ON THE CLASSIFICATION OF ab-ROOT SYSTEMS

An ab-root system ⌫ is called reduced when for all nonzero ␣g⌫ we have 2␣f⌫.

Let V be a real prehilbert space which have an orthonormal base

⑀i i4gII. As in the classical case 5 the setw x ⑀iy⑀j:i,jgII,i/j4 is a root system in the prehilbert vector space which it generates and each one of the following sets

"⑀i:igII

4

j

"⑀i"⑀j:i,jgII,i/j

4

"2⑀i:igII

4

j

"⑀i"⑀j:i,jgII,i/j

4

"⑀ "⑀ :i,jgII,i/j

i j

4

w x

is a root system in V. As in 5 these different classes of root systems are called respectively types A, B, C, D.

The key result in the classification of the ab-root systems is given in the following theorem.

THEOREM4.1. Let ⌫ be a reduced irreducible ab-root system.Then ⌫ is 4

isomorphic to ⌫#j 0 ,where ⌫# is either a finite root system or an infinite root system of one of the types A, B,C, or D.

Proof. Let ⌫ be an infinite reduced irreducible ab-root system. By the main theorem there exists a root system ⌫# in a real prehilbert space W

4

such that ⌫ and ⌫#j 0 are isomorphic as sets of sum. Let F0/⭋be a finite subset of ⌫# and FF the family of all the finite subsets of ⌫#

Ž . Ž .

containing F₀. As in the proof of 3 « 4 of Theorem 3.2 for all FgFF

there exists a vector subspace WF of W and a finite subset ⌫#F of ⌫# containing F and satisfying the following conditions:

Ž .i Every ⌫#_F is a reduced irreducible root system inW_F. Ž .ii ⌫#Fs⌫#lWF for all FgFF.

Žiii. ⌫#_F;⌫#_FX for any F,FXgFF such that F;FX.

4

Let ␣,␤ be nonzero arbitrary elements in ⌫#. We choose F₀s ␣,␤ and F>F₀ such that dim_⺢WFG3. Taking into account that dim⺢WFG3, then the well-known classification of the finite reduced irreducible root

w x

systems and 5, Chap. vi, Sect. 1, Proposition 12 applied to the finite root

5 5␣ 1

system ⌫#F give _{5 5}␤ s1, 2, . Moreover, the norms of the elements in2 ⌫#

w x

take at most two different values 5, Chap. vi, Sect. 1, Proposition 12 . By multiplying the inner product of W by a real positive factor if it were necessary we can assume without loss of generality that ⌫# is either an

w x

infinite character of ⌫# and the classification of H-systems and J-systems

w x

given in 9, 11 allow us to assert that ⌫# is a root system of one of the types A, B, C, or D.

The classification of ab-root systems plays a fundamental role in the

w x

study of C-algebras 6 . These are a wide class of Lie algebras closely

w x w x

related with classical Lie algebras 17, 18 ,V-algebras 10 , and graded Lie

w x

algebras 1, 3, 13 .

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