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arXiv:1209

.4380v2

[math.RT]26Aug2013

THE EXTENDED AFFINE LIE ALGEBRA ASSOCIATED WITH A CONNECTED

NON-NEGATIVE UNIT FORM

GUSTAVO JASSO

Abstract. Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a

Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalizedSerre relations by M. Barot, D. Kussin and H. Lenzing. This is done in an analogous way to the construction

of the simply-laced affine Kac-Moody algebras. Thus, we obtain a family of extended affine Lie algebras ofsimply-laced Dynkin type and arbitrary nullity. Furthermore, there is a one-to-one correspondence b etweenthese Lie algebras and the equivalence classes of connected non-negative unit forms.

1. Introduction

To each unit form q : Zn Z, M. Barot, D. Kussin and H. Lenzing associate in [2] a complex Lie algebra

G(q) by means of generalized Serre relations. We are interested in the case when the form q is connected andnon-negative, see Section 2 for definitions. When the rank of the radical of q is 0 or 1, the algebra G(q) is asimply-laced finite dimensional simple Lie algebra or a simply-laced affine Kac-Moody algebra respectively.Also, the algebra G(q) admits a root space decomposition with respect to a finite dimensional abelian Cartan

subalgebra H with the roots of q, i.e. the set R(q) := q1(0) q1(1), as root system. Moreover, R(q) is aso-called extended affine root system, see Definition 2.2 and Proposition 2.4. Extended affine root systemsare a certain generalization of affine root systems; one important difference is that, in general, isotropic rootsgenerate a subspace (of an euclidean space, say) of dimension higher than 1. Extended affine root systemsare preciseley the root systems associated with extended affine Lie algebras. Roughly speaking, extend affineLie algebras are characterized by the existence of a non-degenerate symmetric invariant bilinear form whichinduces a root-space decomposition with respect to a finite dimensional abelian subalgebra; moreover, theadjoint representations of homogeneous elements of non-isotropic degree are locally nilpotent, see Definition

2.1. Examples of extended affine Lie algebras are both the simple finite-dimensional Lie algebras and theaffine Kac-Moody algebras. We refer the reader to [1] for an introductory treatment of the theory of extendedaffine Lie algebras and motivation for their study.

Our aim is to study the connection between extended affine Lie algebras and the Lie algebra G(q) described

above. In fact, the Lie algebra G(q) is very close to being an extended affine Lie algebra, but in general itlacks an invariant non-degenerate symmetric bilinear form (the non-degeneracy being the main obstruction).

We can obtain such a form by passing to a natural quotient of G(q). Thus, this article can be regarded as asimple extension of [2] towards the theory of extended affine Lie algebras.

Let us explain the contents of this article. In Section 2, we recall the definitions of an extended affine Liealgebra and of an extended affine root system. We also show how to associate an extended affine root systemto a given connected non-negative unit form. In Section 3, given a connected non-negative unit form q, weconstruct a Lie algebra E(q) with root system R(q) and with nullity the corank of q. This construction is aslight modification of the construction of [2, Sec. 2]. The modification concerns both the Cartan subalgebra

and the root spaces associated to non-isotropic roots. Our main result is the following:

Theorem 1.1 (see Theorems 3.7 and 4.5). Let q : Zn Z be a connected non-negative unit form withassociated root system R(q). Then the Lie algebra E(q) is an extended affine Lie algebra with root systemR(q). Furthermore, if q is a connected non-negative unit form which is equivalent to q, then there is anisomorphism of graded algebras between E(q) and E(q).

2010 Mathematics Subject Classification. 17B67.The author wishes to acknowledge the support of M. Barot, who directed the authors undergraduate thesis from which

this article is derived. This acknowledgemnt is extended to L. Demonet for his helpful comments and suggestions on an earlierversion of this article which greatly improved its readability.

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In Section 4, we recall from [2] the construction of the Lie algebra G(q) by means of generalized Serre

relations. We show that E(q) can also be obtained from G(q) in a way that imitates the construction of theaffine Kac-Moody algebras, cf. [5, Chapter 14]. More precisely, we have the following result:

Theorem 1.2. Let q : Zn Z be a connected non-negative unit form. Then there is an isomorphism ofgraded Lie algebras between G(q)/I and E(q) where I is the unique maximal ideal ofG(q) with I H = {0}.

2. Preliminaries

We recall the definitions of and extended affine Lie algebra and of an extended affine root system. Wefollow closely the exposition of [1]. Afterwards, we recall basic concepts regarding unit forms and present astraightforward way to associate an extended affine root system with a connected non-negative unit form,see Proposition 2.4.

2.1. Extended affine Lie algebras and their root systems. Let L be a Lie algebra over the field C ofcomplex numbers having the following properties:

(EA1) L has a non-degenerate symmetric bilinear form

(, ) : L L C

which is invariant in the sense that

([x, y], z) = (x, [y, z]) for all x,y,z L.

(EA2) L has a non-trivial finite dimensional abelian subalgebra H which equals its own centralizer in Land such that adL h is diagonalizable for all h H.

It follows from (EA2) that L admits a root space decomposition

L =H

L,

where, for each H,

L := {x L | [h, x] = (h)x for all h H}.

We call

R = { H | L = {0}}

the root system of L with respect to H (note that (L, L) whenever + = 0). Since the form (, ) isinvariant and adL h is diagonalizable for all h H, the restriction of the form to H is non-degenerate. Thuswe may transfer the bilinear form (, ) to H via the induced isomorphism

H H

h (h, ).

Then, we let

R0 = { R | (, ) = 0} and R = R \ R0.

We call R0 the set of isotropic roots and R the set ofnon-isotropic roots.

(EA3) If R is a non-isotropic root and x L, then adL x acts locally nilpotently on L. That is,for each y L there exists n > 0 such that

(adL x)ny = 0.

(EA4) R is a discrete subset of H.(EA5) L is irreducible, i.e. R cannot be decomposed as a disjoint union R1R2 where R1, R2 are non-empty

subsets ofR satisfying (R1, R2) = {0}.

Definition 2.1. [1, Ch. I] Let L be a Lie algebra over C such that it satisfies the five properties (EA1)-(EA5).Then, the triple (L,H, (, )) is called an extended affine Lie algebra.

The root systems of extended affine Lie algebras can be axiomatically described. They belong to the classof extended affine root systems.

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Definition 2.2. [1, Ch. II, Def. 2.1] Assume that V is a finite dimensional real vector space with anon-negative symmetric bilinear form (, ), and that R is a subset of V. Let

R0 = { R | (, ) = 0} and R = { R | (, ) = 0}.

We call R an extended affine root system in V if R has the following properties:

(R1) 0 R.

(R2) R = R.(R3) R spans V.(R4) If R then 2 / R.(R5) R is discrete in V.(R6) If R and R then there exist d, u Z0 such that

{+ n : n Z} R = { d , . . . , + u} and d u = 2(, )

(, ).

(R7) R cannot be decomposed as a disjoint union R1 R2 where R1, R2 are non-empty subsets of R

satisfying (R1, R2) = {0}.(R8) For any R0 there exists R such that + R.

2.2. Unit forms. A unit form is a quadratic form q : Zn Z of the form

q(x) =ni=1

x2i +i 0 (resp.q(x) 0) for all x Zn \ {0}. Recall that any unit form has an associated symmetric matrix C = Cq Znn

defined by

Cij := q(ci + cj) q(ci) q(cj),

where {c1, . . . , cn} is the standard basis ofZn. Also, recall that

q(x, y) := xCy, for x, y Zn

is the symmetric bilinear form associated with q and that we have 2q(x) = q(x, x) for each x Zn. Theradical of q is the subgroup ofZn defined by

rad q := {x Zn

| q(x, ) = 0}.Further, we set corank q := rank(rad q). Observe that rad q q1(0). If q is non-negative, then rad q =q1(0). The associated bigraph of q is the graph whith vertices 1, . . . , n and |qij | solid edges between verticesi and j if qij < 0 and |qij| dotted edges between i and j if qij > 0. We say that a unit form is connected ifits associated bigraph is connected. We say that two non-negative unit forms q and q are equivalent if thereexist an automorphism T ofZn such that q = q T. Connected non-negative unit forms can be classified asfollows:

Theorem 2.3. [3, Thm. (Dynkin-types)] Let q : Zn Z be a non-negative unit form. Then there exist aZ-invertible transformation T : Zn Zn such that

(q T)(x1, . . . , xn) = q(x1, . . . , xn)

where = corank q and is a Dyknin diagram with n vertices uniquely determined by q. We call the

Dynkin type of q.Note that Theorem 2.3 implies that two connected non-negative unit forms are equivalent if and only if

they have the same corank and the same Dynkin type.There is a straightforward way to associate an extended affine root system to a given connected non-

negative unit form.

Definition-Proposition 2.4. Let q : Zn Z be a connected non-negative unit form. Then

R(q) := q1(0) q1(1) Zn.

is an extended affine root system in Rn with respect to the symmetric bilinear form of q.

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Proof. All but properties (R6) and (R7) are easily verified (for propery (R3) observe that the standard basisofZn belongs to R(q) since q(ci) = 1 for all i {1, . . . , n}). We first show that R(q) verifies axiom (R6). Let q1(1), R(q), and n Z. We need to determine the values of n for which q(+ n) {0, 1}. Notethat we have

q(+ n) = 12 q(+ n,+ n)

= 12

(q(, ) + 2nq(, ) + n2q(, ))

=q() + nq(, ) + n2.

First, suppose that q() = 0. Since q is non-negative we have that rad q and thus q(, ) = 0. Then wehave that + n R(q) if and only if n {1, 0, 1}. Thus, taking d = u = 1 gives the result in this case.Now suppose otherwise that q() = 1. A simple calculation shows that q(+ n) = 1 if and only if n = 0 orn = q(, ). On the other hand, q(+ n) = 0 if and only if

n =q(, )

q(, )2 4

2

Since q(+ n) 0, by putting x = 12 q(, ) and evaluating q(+ x) we obtain that 2 q(, ) 2.

Hence q(+ n) = 0 has integer solution if and only if q(, ) {2, 2}, which is given by n = 12 q(, ).Thus our analysis shows that

q(+ n) R(q) if and only if

n {0, q(, )} if q(, ) {1, 1}n {0, 12 q(, ), q(, )} if q(, ) {2, 2}.

Since q(, ) = 2q(,)q(,) the claim follows in this case also.

We now show that R(q) satisfies axiom (R7). Let R(q) = R1 R2. Without loss of generality, we canassume using the connectedness of q that {c1, . . . , cn} R1. On the other hand, for any R2 there existi Z such that =

ni=1 ici. Thus we have

q(, ) =

ni=0

iq(ci, ) for all R2 R(q).

Hence if (R1, R2) = {0}, then R2 must be empty. This shows that R(q) is irreducible. Thus R(q) satisfiesproperties (R1)-(R8) and is thus an extended affine root system.

3. Main result

To a connected non-negative unit form q, we associate an extended affine Lie algebra E by modifying theconstruction of [2, Sec. 2]. The modification concerns the root spaces attached to non-zero isotropic roots.We give the proof of Theorem 1.1 in the second part of this section.

3.1. Construction. We fix a connected non-negative unit form q : Zn Z with symmetric matrix C Znn. For simplicity, we let

R := R(q), R := R(q) = q1(1) and R0 := R(q)0 = q1(0).

Recall that since q is non-negative we have that R0 = q1(0) = rad q. Let radC q := C Z rad q and definecomplex vector spaces N = N(q) for R by

N :=Ce if R,

Cn

/C

if R0

\ {0},Cn (radC q) if = 0,

where (radC q) denotes the C-dual of radC q and define

N = N(q) :=R

N

as a vector space. We can endow N with a Lie algebra structure as follows: Let : Cn N be the canonicalprojection for each non-zero R0; by convention, 0(v) = v for all v Cn. Choose a non-symmetric bilinearform B : Zn Zn Z such that q(x) = B(x, x) for all x Zn and set (, ) = (1)B(,). Finally, choosea projection : Cn radC q and define the following bracket rules that depend on the choice of B and .

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Let , R, , R0, v, w Cn and , (radC q):

[(v), (w)] = (, )q(v, w)+()(B1)

[(v), e] = [e, (v)] = (, )q(v, )e+ .(B2)

[e, e] =

(, )e+ if + R,

(, )+() if + R0,

0 otherwise.

(B3)

[, e] = [e, ] = (())e .(B4)

[, (w)] = [(w), ] = (())(w).(B5)

[, ] = 0.(B6)

Remark 3.1. If , R are such that + R0 = rad q, then +() = +(). Hence the abovebracket is anticommutative.

Proposition 3.2. The rules (B1)-(B3) define a Lie algebra structure on N which is independent of thechoice of B and .

Proof. The fact that N is a Lie algebra and that the Lie bracket defined above is independent of the choiceof B is shown in [2, Prop. 2.2]. We only need to verify Jacobis identity

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

for homogeneous elements x,y,z N when at least one of them belong to (radC q). Let , (radC q) andlet x N and y N for some non-zero R. We have two cases:

[, [, x]] + [, [x, ]] + [x, [, ]] = (())[, x] (())[, x]

= (())(())x (())(())x = 0,

and since [x, y] N+ , we have that

[, [x, y]] + [x, [y, ]] + [y, [, x]] = (( + ))[x, y] (())[x, y] + (())[y, x] = 0.

Finally, that the Lie algebra structure of N is independent of the choice of can be shown exactly as inthe proof of [2, Prop. 5.3(i)].

The Lie algebra N is close to being an extended affine Lie algebra, but it lacks a non-degenerate invariant

bilinear form. To achieve this we must consider the following quotient of N.

Definition 3.3. Let E = E(q) be the factor Lie algebra of N induced by the canonical epimorphismsN Cn/ radC q for R

0 \ {0}. Thus the root spaces of E are given by

E =

Ce if R,

Cn/ radC q if R0 \ {0},

Cn (radC q) if = 0.

Remark 3.4. With some abuse of notation we also denote by the composition Cn N radC q. Thus

rule (B1) induces the following rule on E:

[(v), (w)] = (, )q(v, w)+() =

q(v, w)0() if + = 0,

0 otherwise

where , R0 and v, w Cn.

3.2. The invariant bilinear form on E(q). We define a bilinear form on E by the following rules:

(e, e) = 1 for R,

((v), (w)) = q(v, w) for R0 and v, w Cn,

(, 0(w)) = (0(w), ) = ((w)) for w Cn,

and zero in other cases. We need the following properties of (, ) which follow directly from the definition.

Lemma 3.5. Let, R and , R0. Then

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(i) (, ) = 1 and (, ) = 1.(ii) (, ) = (, ) and (, ) = (, ).

(iii) ( + , + ) = (, )(, )(, )(, ).(iv) If + R then (, ) = (, ).(v) If + R0 then (, ) = (, ).

Proposition 3.6. The symmetric bilinear form (, ) : E E C is invariant and non-degenerate.

Proof. We show first that (, ) is non-degenerate. Let x E be non-zero. If R then x = e for

some non-zero C. Hence (x, e) = = 0. Now suppose that R0 is non-zero and x = (v) for

some v / radC q. Then there exists some non-zero w Cn such that

(x, (w)) = ((v), (w)) = q(v, w) = 0.

Finally, suppose that = 0 and that x radC q (resp. x (radC q)). Then it is immediate that there existsome y (radC q) (resp. y radC q) such that (x, y) = 0. This shows that the form (, ) is non-degenerate.

Now we prove that the form is invariant. Observe that it is sufficient to prove that

(x, [y, z]) = ([x, y], z)

for homogeneous elements x,y,z E such that deg x + deg y + deg z = 0, where deg x = d for x Ed as theform vanishes in other cases. In order to prove this, we distinguish various cases. First, we treat the case

where x,y,z / (radC q)

. Let ,, R

, ,, R0

and u,v,w Cn

.(i) Since q(, ) = q(, ) = 0 we have

((u), [(v), (w)]) =((u), (, )q(v, w)+()) = (, )q(v, w)q(u, ) = 0 and

([(u), (v)], (w)) =((, )q(u, v)+(), (w)) = (, )q(u, v)q(, w) = 0.

(ii) We have

((u), [e, e]) = ((u), (, )+()) = (, )q(u, ).

On the other hand, since + + = 0, we have

([(u), e], e) =((, )q(u, )e+, e) = (, )q(u, ) = ((+ ), )q(u, )

= (, )(, )q(u, ) = (, )q(u, ).

Since + R0, by Lemma 3.5(v) we have (, ) = (, ). Thus we have the required equality in this

case.(iii) We have

(e, [(v), e]) = (e, (, )q(v, )e+tau) = (, )q(v, ).

On the other hand, since + + , we have

([e, (v)], e) =((, )q(v, )e+, e) = (, )q(v, ) = (, ( + ))q(v, )

=(, )(, )q(v, ) = (, )q(v, )

as we also have q(u, + ) = q(u, ) = 0 and thus q(u, ) = q(u, ).(iv) We have

(e, [e, e]) = (, )q(, )(e, e) = (, )q(, ).

On the other hand, since + + = 0, we have

([e, e], e) =(, )q(, )(e, e) = (, )q(, ) = ((+ ), )q(, )

= (, )(, )q(, ) = (, )q(, ).

Since + R, by Lemma 3.5(iv) we have (, ) = (, ). Finally, we have

q(, ) q(, ) = q(, ) q(, ) = 0,

since = ( + ). Hence q(, ) = q(, ). Thus we have the required equality in this case.We now consider the case when at least one of the elements involved lies in (radC q). We must show that

(, [x, x]) = ([, x], x)

for (radC q), x E(q), R and = 1. There are two non-trivial cases:

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(i) Let (radC q), v , w Cn and let R0 be non-zero. Then

(, [(v), (w)]) = q(v, w)(, 0()) = q(v, w)() and

([, (v)], (w)) = ()((v), (w)) = ()q(v, w).

(ii) Let (radC q) and let R. Then

(, [e, e]) = (, )(, 0()) = () and

([, e], e) = ()(e, e) = ().

The following is the main result of this section.

Theorem 3.7. Let q : Zn Z be a connected non-negative unit form. Then the triple(E(q), (, ), H) isan extended affine Lie algebra with R(q) as root system.

Proof. It is shown in Proposition 3.6 that there exists a non-ndegenerate invariant symmentric bilinear form(, ) : E E C. Let H = E0. Then the triple (E, (, ), H) satisfies axioms (EA1), (EA2), (EA4)and (EA5) of extended affine Lie algebras, see Definition 2.1. It remains to show that E satisfies axiom(EA3). For this, let x E for some R and x E for some R. Then for each n > 0 we havethat (ad x)

n(x) E+n. But since R is an extended affine root system it satisfies property (R6), hence + n R for only finitely many values of n. It follows that E+n = {0} for only finitely many valuesof n, thus ad x acts locally nilpotently on E. This shows that the triple (E, (, ), H) satisfies axioms of

extended affine Lie algebras.

4. A Kac-Moody-like Construction of E(q)

We give an alternative construction of E = E(q) as a quotient of the Lie algebra G = G(q) defined in[2]. This is done by analogy to construction of the simply-laced affine Kac-Moody algebras. We recall the

construction ofG and some necessary results regarding the Lie algebra G. We give the proof of Theorem 1.2at the end of the section.

Let q : Zn Z be a connected non-negative unit form with symmetric matrix C Znn and let R := R(q)be the root system of q, see Definition-Proposition 2.4. We keep the notation of the previous section. Let Fbe the free Lie algebra on 3n generators {fi, hi, fi | 1 i n} which are homogeneous of degree ci, 0, cirespectively. Let I(q) be the ideal of F defined by the following generalized Serre relations:

(G1) [hi, hj] = 0 for i, j {1, . . . , n}.

(G2) [hi, fj ] = Cijfj for i, j {1, . . . , n} and = 1.(G3) [fi, fi] = hi for i, j {1, . . . , n} and = 1.

(G) [ftit , [ft1it1 , . . . , [f2i2 , f1i1 ]] = 0 whenevert

k=1 kcik / R for k = 1.

Let G = G(q) := F/I(q) and let H be the abelian subalgebra of G generated by h1, . . . , hn.A monomial in G is an element obtained from the generators using the bracket only. Thus every monomial

has a well defined degree. Let G be the space generated by the monomials of degree . In general, thealgebra G does not have a root space decomposition, see [2, Sec. 5]. In order to achieve such a decompositionwe need to extend F/I(q) by (radC q), the C-dual of radC q. Let : C

n radC q be any projection. Wedefine G := F/I(q) (radC q) and we let H = H (radC q). We define the following bracket rules whichdepend on the choice of:

[, x] = [x, ] = (())x for (rad q) and x G.

[, ] = 0 for , (radC q).

Proposition 4.1. [2, Lem. 5.2, Prop. 5.3(i)] Using the Lie algebra structure on G, the rules above define a

Lie algebra structure on G which is independent of the choice of .

For h =n

i=1 ihi H G, define r(h) =n

i=1 ici Cn. We define a bilinear form (, ) : H H

Cn by the rules

(h, h) = hCh for h, h H,

(, h) = ((r(h))) for (radC q)andh H.

We observe that this form is non-degenerate in the second variable. Indeed, if r(h) radC q is non-zero thenthere exist (radC q) such that (r(h)) = 0.

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Proposition 4.2. [2, Prop. 5.3 (ii)] The Lie algebra G admits a root space decomposition, that is G =R{0} G, where

G := {x G : [h, x] = (r(h), )x for all h H} for R.

Moreover, we have the following result.

Theorem 4.3. [2, Thm. 1.1] Letq and q

be two equivalent connected non-negative unit forms. Then thereexists an isomorphim of graded Lie algebras : G(q) G(q) which maps the generators of G(q) to the

generators of G(q).

We now proceed by analogy with the construction of the Kac-Moody algebras, cf. [5, Ch. 14]. The Lie

algebra G is an H-graded H-module, hence it contains a unique maximal ideal I such that I H = {0}.

The following proposition describes the relationship between G and E.

Proposition 4.4. Letq : Zn Z be a connected non-negative unit form and let R(q) be the extended affineroot system associated with q, see Proposition 2.4. Then there is an epimorphism of graded Lie algebrasG E which induces an isomorphism between H G and H E.

Proof. Let hi = [eci, eci] E for i {1, . . . , n}. It is readily verified that the elements eci, hi, eci of E

satisfy relations (G1)-(G) and that they generate Cn =0 E as a Lie algebra, see Definition 2.2 and(R6). Hence there is an epimorphism of graded Lie algebras G C

n

=0 E mapping hi to hi and fi

to eci for i {1, . . . , n} and = 1 with the required property. This morphism extends in a natural way toan epimorphism of graded Lie algebras G E. The last statement is immediate.

We are ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2. It suffices to show that E has no non-zero ideal I with I H = {0}, as in this casethe epimorphism from Proposition 4.4 induces an isomorphism of graded Lie algebras G/I E. Suppose Iis a non-zero ideal of E with I H = {0}. Then I is a graded ideal, that is

I =R

(E I).

Let x (E I) be a non-zero homogenous element of degree R. We distinguish two cases:

(i) If R

then we may assume that x = e. But 0 = [e, e] = 0() (H I) as I is an ideal,which contradicts I H = {0}.(ii) If R0 is non-zero then x = (v) for some v Cn \ radC q. Hence there exists some w Cn such

that q(v, w) = 0. Thus 0 = [(v), (w)] = q(v, w)0() (H I), which again contradicts I H = {0}.This concludes the proof.

Finally, we obtain the following result.

Theorem 4.5. Letq and q be two equivalent connected non-negative unit forms which are equivalent. Thenthere is an ismorphism of graded Lie algebras : E(q) E(q) which maps the generators of E(q) to thegenerators of E(q).

Proof. By Theorem 4.3 there exists an isomorphism of graded Lie algebras : G(q) G(q) which maps thegenerators ofG(q) to the generators ofG(q). By Proposition 4.4, the isomorphism induces an isomorphism

of graded Lie algebras : E(q) E(q

) which maps the generators of E(q) to the generators of E(q

).

References

[1] B. Allison, S.Azam, S. Berman, Y. Gao and A. Pianzola: Extended affine Lie algebras and their root systems, chap. I. Mem.

Amer. Math. Soc., 603 (1997).

[2] M. Barot, D. Kussin and H. Lenzing: The Lie algebra associated to a unit form. J. Algebra 296 (2006), 1-17.[3] M. Barot, J. A. de la Pena: The Dynkin type of a non-negative unit form. Expo. Math. 17 (1999), 339-348.

[4] R. Borcherds: Vertex algebras, Kac-Moody algebras and the monster. Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.[5] R. Carter: Lie Algebras of Finite and Affine Type. Cambridge University Press, Cambridge, (2005).

[6] G. Jasso: Algebras de Lie de tipo afn extendido y formas cuadraticas. Undergraduate Thesis, Universidad NacionalAutonoma de Mexico (2010).

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Instituto de Matematicas, Univerisdad Nacional Autotonoma de Mexico. Mexico City, Mexico

Current address: Graduate School of Mathematics, Nagoya University. Nagoya, JAPAN

E-mail address: jasso.ahuja.gustavo@b.mbox.nagoya-u.ac.jp