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On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

Kh. S. Nirov?

Max-Planck-Institut f¨ ur Gravitationsphysik – Albert-Einstein-Institut Am M¨ uhlenberg 1, D-14476 Golm b. Potsdam, Germany E-mail: nirov@aei.mpg.de

arXiv:math-ph/0504038v1 12 Apr 2005

A. V. Razumov

Institute for High Energy Physics 142280 Protvino, Moscow Region, Russia E-mail: Alexander.Razumov@ihep.ru

Abstract We de?ne the twisted loop Lie algebra of a ?nite dimensional Lie algebra g as the Fr? echet space of all twisted periodic smooth mappings from R to g. Here the Lie algebra operation is continuous. We call such Lie algebras Fr? echet Lie algebras. We introduce the notion of an integrable Z-gradation of a Fr? echet Lie algebra, and ?nd all inequivalent integrable Z-gradations with ?nite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.

1 Introduction

The theory of loop groups and loop Lie algebras has a lot of applications to mathematical and physical problems. In particular, it is a necessary tool for formulation of many integrable systems and construction of appropriate integration methods. Here one or another version of factorization problem for the underlying group arises (see, for example, [1]). For the so-called Toda systems associated with loop groups the required factorization is induced by a Z-gradation of the corresponding loop Lie algebra, and, at least from this point of view, the classi?cation of Z-gradations of loop Lie algebras is quite important. The de?nition and general integration procedure for the Toda systems can be found in [2, 3, 4]. The classi?cation of Z-gradations of complex semisimple ?nite dimensional Lie algebras is well known (see, for example, [5]). The corresponding classi?cation of the Toda systems associated with compex classical Lie groups was given in papers [6, 7]. There are two main de?nitions of the loop Lie algebras. In accordance with the ?rst definition used, for example, by Kac in his famous monograph [8], a loop Lie algebra is the set of ?nite Laurent polynomials with coef?cients in a ?nite dimensional Lie algebra. It is rather dif?cult to associate a Lie group with such a Lie algebra. Actually this is connected to the fact that the exponential of a ?nite polynomial is usually not a ?nite polynomial. However, it

On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia. E-mail: nirov@ms2.inr.ac.ru

?

1

should be noted that with this de?nition in the case when the underlying Lie algebra is complex and simple one can classify all Z-gradations of the loop Lie algebras with ?nite dimensional grading subspaces [8]1 . In accordance with the second de?nition, used in the monograph by Pressley and Segal [9], a loop Lie algebra is the set of smooth mappings from the circle S 1 to a ?nite dimensional Lie algebra. This set is endowed with the structure of a Fr? echet space. Here the Lie algebra operation de?ned pointwise is continuous. The de?nition given in [9] is more convenient for applications to the theory of integrable systems, because in this case we always have an appropriate Lie group. Therefore, it would be interesting and useful to obtain a classi?cation of Z-gradations for loop Lie algebras de?ned as in [9]. In the present paper we introduce the concept of an integrable Z-gradation and classify all integrable Z-gradations with ?nite dimensional grading subspaces of loop Lie algebras and twisted loop Lie algebras of ?nite dimensional complex simple Lie algebras. The result of the classi?cation is actually the same as for loop Lie algebras and twisted loop Lie algebras de?ned as in [8]. Namely, to classify all integrable Z-gradations with ?nite dimensional grading subspaces of the Lie algebras under consideration one has to classify all ZK -gradations of the underlying Lie algebras or, equivalently, all their automorphisms of ?nite order.

2 Loop Lie algebras and loop Lie groups

Consider the vector space C ∞ (S 1 , V ) of smooth mappings from the circle S 1 to a ?nite dimensional vector space V . It is convenient to treat the circle S 1 as the set of complex numbers of modulus one. There is a natural mapping from the set R of real numbers to S 1 which takes ? from R to V by σ ∈ R to eiσ ∈ S 1 . Given an element ξ ∈ C ∞ (S 1 , V ), one de?nes a mapping ξ the equality ?(σ ) = ξ (eiσ ). ξ ? is smooth and satis?es the relation ξ ?(σ + 2π ) = ξ ?(σ ). Conversely, any smooth The mapping ξ periodic mapping from R to V induces a smooth mapping from S 1 to V . Introduce the notation ?(k) = dk ξ/ ? dsk , ξ ?(0) = ξ ?. Given where s is the standard coordinate function on R. It is customary to assume that ξ ?(k) . an element ξ ∈ C ∞ (S 1 , V ), we denote by ξ (k) the element of C ∞ (S 1 , V ) induced by ξ ∞ 1 Endow C (S , V ) with the structure of a topological vector space in the following way. Let · be a norm on V . De?ne a countable collection of norms { · m }m∈N on C ∞ (S 1 , V ) by ξ m = max max ξ (k) (p) , 1

0 k<m p∈S

? : R → V by or via the corresponding mapping ξ ? ξ

m

= max

0 k<m σ∈[0,2π ]

max

?(k)(σ ) . ξ

m1

Note that for any ξ ∈ C ∞ (S 1 , V ), if m1 < m2 , then ξ Given a positive integer m, denote Um = {ξ ∈ C ∞ (S 1 , V ) | ξ

1

ξ

m2 .

m

< 1/m}.

Actually in [8] one can ?nd the classi?cation of Z-gradations of the af?ne Kac–Moody algebras. The classi?cation of Z-gradations of the corresponding loop Lie algebras immediately follows from that classi?cation.

2

The collection formed by the sets Um is a local base of a topology on C ∞ (S 1 , V ). As a base of the topology we can take the collection of subsets of the form Uξ,m = ξ + Um , ξ ∈ C ∞ (S 1 , V ).

A sequence (ξn ) in C ∞ (S 1 , V ) converges to ξ ∈ C ∞ (S 1 , V ) relative to this topology, if and k) ?( ?(k) . One only if for each nonnegative integer k the sequence ξ converges uniformly to ξ n can show that actually we have a Fr? echet space. We de?ne a Fr? echet space as a complete topological vector space whose topology is induced by a countable family of semi-norms.2 Let now g be a real or complex ?nite dimensional Lie algebra. Supply the Fr? echet space C ∞ (S 1 , g) with the Lie algebra structure de?ning the Lie algebra operation pointwise. The obtained Lie algebra is called the loop Lie algebra of g and denoted L(g). It is clear that constant mappings form a subalgebra of L(g) which is isomorphic to the initial Lie algebra g. Let again V be a ?nite dimensional vector space, and let a be an automorphism of V . Consider the quotient space E of the direct product R × V by the equivalence relation which identi?es (σ, v ) with (σ + 2π, a(v )). De?ne the projection π : E → S 1 by the relation π ([(σ, v )]) = eiσ . It is not dif?cult to show that in such a way one obtains a smooth vector bundle E → S 1 with ?ber V . ?(σ ) ∈ V Let ξ be a smooth section of E . For any σ ∈ R there exists a unique element ξ such that ?(σ ))] = ξ (eiσ ). [(σ, ξ ? from R to V which satis?es the relation This relation de?nes a smooth mapping ξ ?(σ + 2π ) = a(ξ ?(σ )), ξ ? : R → V which is twisted periodic, called twisted periodicity. Conversely, given a mapping ξ the equality ?(σ ))], ξ (p) = [(σ, ξ p = eiσ , de?nes a smooth section of E . One can make the space C ∞ (S 1 ← E ) of smooth sections of π E → S 1 a Fr? echet space in the same way as it was done above for the space C ∞ (S 1 , V ). Here it is natural and useful to assume that the corresponding norm on V is invariant with respect to the automorphism a. If the vector space V is a Lie algebra g and a is an automorphism of g, one can supply the π vector space C ∞ (S 1 ← E ), or equivalently the vector space of twisted periodic mappings from R to g, with the structure of a Lie algebra de?ning Lie algebra operation pointwise. We denote this Lie algebra by La (g) and call a twisted loop Lie algebra. The loop Lie algebra L(g) can be considered as the twisted loop Lie algebra La (g) with a = idg. Let G be a Lie group whose Lie algebra coincides with g and Ad be the adjoint representation of G in g. For any g ∈ G the linear operator Ad(g ) is an automorphism of g. Such automorphisms are called inner automorphisms. They form a normal subgroup Int g of the group Aut g of automorphisms of g. One can show that if the automorphisms a and b of g differ by an inner automorphism of g then the twisted loop Lie algebras La (g) and Lb (g) are naturally isomorphic. This means, in particular, that if the Lie algebra g is semisimple one can consider only the twisted loop Lie algebras La (g) with a belonging to the ?nite subgroup of

2

π

π

Sometimes a more general de?nition of a Fr? echet space is used (see, for example, [10]).

3

Aut g identi?ed with the automorphism group Aut Π of some simple root system Π of g. In particular, one can assume that aK = idg for some positive integer K . It is convenient for our purposes to assume that K does not necessarily coincide with the order of a. Let g be a semisimple Lie algebra. Consider an arbitrary element η of La (g) and the corre? de?ned as sponding mapping η ? from R to g. It is clear that the mapping ξ ?(σ ) = η ξ ?(Kσ ), is a periodic mapping from R to g. Therefore, it induces an element ξ of L(g). It is clear that in this way we obtain an injective homomorphism from La (g) to L(g). The image of this homomorphism is formed by the elements ξ satisfying the condition ξ (εK p) = a(ξ (p)), where εK = exp(2π i/K ) is the K th principal root of unity. We will denote this image as ? from R to g one has La,K (g). For the corresponding mapping ξ ?(σ + 2π/K ) = a(ξ ?(σ )). ξ Thus, when g is a semisimple Lie algebra, the twisted loop Lie algebra La (g) can be identi?ed with a subalgebra of the loop Lie algebra L(g). We call a Lie algebra G a Fr? echet Lie algebra if G is a Fr? echet space and the Lie algebra operation in G, considered as a mapping from G × G to G, is continuous. Actually one can consider a Fr? echet Lie algebra as a smooth manifold modelled on itself. Here the Lie algebra operation is a smooth mapping. To prove that La (g) is a Fr? echet Lie algebra we start with the following simple lemmas. Lemma 2.1 Let g be a ?nite dimensional Lie algebra and · be a norm on g. There exists a positive real number C such that [x, y ] for all x, y ∈ g. Proof. First prove the statement of the proposition for a special choice of the norm · . Let (ei ) be a basis of g. Expand an arbitrary element x of g over the basis (ei ), x = i ei xi , and de?ne x = max{|xi |}.

i

C x

y

In this case for any i one has |x | x, y ∈ g one has [x, y ] =

i

x . It is also evident that ei = 1. For arbitrary elements [ei xi , ej y j ] =

i,j i,j,k

ek ck ij xi y j ,

where ck ij are structure constants of the Lie algebra g, [ei , ej ] =

k

ek ck ij .

Therefore, [x, y ]

i,j,k

ek |ck ij ||xi ||y j |

i,j,k

|ck ij |

x y .

4

Thus, the statement of the proposition is valid for the norm chosen and for C=

i,j,k

|ck ij |.

Since in the ?nite dimensional case all norms are equivalent, the statement of the proposition is valid for an arbitrary norm · . Lemma 2.2 There are positive real numbers Cm , m = 1, 2, . . ., such that [ξ, η ] for all ξ, η ∈ La (g). Proof. For m = 1, using Lemma 2.1, one easily obtains [ξ, η ] For m = 2 one has [ξ, η ] It is clear that max [ξ, η ](1) (p) = max [ξ (1) , η ](p) + [ξ, η (1) ](p) 1 1

p∈S p∈S 2 1 m

Cm ξ

m

η

m

= max [ξ (p), η (p)] 1

p∈S

max C ξ (p ) 1

p∈S

η (p )

C ξ

1

η 1.

= max max [ξ (p), η (p)] , max [ξ, η ](1) (p) 1 1

p∈S p∈S

.

C ξ (1) Taking into account that ·

1

1

η

1

+C ξ

1

η (1)

1

2C ξ

2

η 2.

·

2,

we conclude that

2

[ξ, η ] Similarly one obtains [ξ, η ]

m

2C η

2

ξ 2. η

m.

2m?1 C ξ

m

Thus, the statement of the proposition is valid for Cm = 2m?1 C . Now we are able to prove the desired result. Proposition 2.1 The twisted loop Lie algebra La (g) is a Fr? echet Lie algebra. Proof. It suf?ces to show that for any ?xed elements ξ1 , ξ2 ∈ La (g) and any positive integer ′ ′ m, there are positive integers m1 and m2 such that for any ξ1 ∈ Uξ1 ,m1 and ξ2 ∈ Uξ2 ,m2 one has ′ ′ [ξ1 , ξ2 ] ∈ U[ξ1 ,ξ2 ],m . It is clear that one can assume that m1 m and m2 m. Let m be a ?xed positive integer, ξ1 and ξ2 be arbitrary elements of L(g), m1 , m2 be ′ ′ ∈ Uξ2 ,m2 write the arbitrary positive integers greater than m. For any ξ1 ∈ Uξ1 ,m1 and ξ2 equalities

′ ′ ′ ′ [ξ1 , ξ2 ] ? [ξ1 , ξ2 ] = [(ξ1 ? ξ 1 ) + ξ 1 , (ξ 2 ? ξ2 ) + ξ2 ] ? [ξ1 , ξ2 ] ′ ′ ′ ′ = [ξ1 ? ξ1 , ξ2 ? ξ2 ] + [ξ1 , ξ2 ? ξ2 ] + [ξ1 ? ξ1 , ξ2 ].

5

Using Lemma 2.2, we obtain

′ ′ [ξ1 , ξ2 ] ? [ξ1 , ξ2 ] m ′ Cm ( ξ 1 ? ξ1 m ′ ξ2 ? ξ2 m m

′ + ξ1 ? ξ1 ′ Cm ( ξ 1 ? ξ1 m1

ξ2

m

+ ξ1

m2

m

′ ξ2 ? ξ2

m)

′ ξ2 ? ξ2 m1

′ + ξ1 ? ξ1

ξ2

m

+ ξ1

m

′ ξ2 ? ξ2

m2 ).

Thus, we have

′ ′ [ξ1 , ξ2 ] ? [ξ1 , ξ2 ] m

< Cm

1 1 1 + ξ2 m1 m2 m1

m

+ ξ1

m

1 m2

.

It is clear that for suf?ciently large m1 and m2 one has

′ ′ [ξ1 , ξ2 ] ? [ξ1 , ξ2 ] ′ ′ that means that [ξ1 , ξ2 ] ∈ U[ξ1 ,ξ2 ],m. m

< 1/m,

Let G be a ?nite dimensional Lie group with the Lie algebra g. The loop group L(G) is de?ned as the set of all smooth mappings from the circle S 1 to G with the group law being pointwise composition in G. Here, as for the case of loop Lie algebras, for any element γ of L(G) one can de?ne a smooth mapping γ ? from R to G connected with γ by the equality γ ? (σ ) = γ (eiσ ), and satisfying the relation γ ? (σ + 2 π ) = γ ? (σ ). Conversly, any periodic smooth mapping from R to G induces an element of L(G). One can endow the loop group L(G) with the structure of an in?nite dimensional manifold and a Lie group in the following way. Recall that the exponential mapping exp : g → G is a local diffeomorphism near the ?e be an open neighbourhood of the identity of G diffeomorphic to some open identity. Let U neighbourhood of the zero element of g, and ? ? be the restriction of the inverse of the exponential ∞ 1 ? ? ?e )) by mapping to Ue . Denote Ue = C (S , Ue ) and de?ne a mapping ? : Ue → C ∞ (S 1 , ? ?(U ? (γ ) = ? ? ? γ. ?e )) is open in L(g) and we can consider the pair (Ue , ?) as a chart Note that the set C ∞ (S 1 , ? ? (U on L(G). For an arbitrary element γ ∈ L(G) denote Uγ = γ Ue , and de?ne the mapping ?γ : Uγ → ?e )) by C ∞ (S 1 , ? ?(U ? γ (γ ′ ) = ? ? ? (γ ? 1 γ ′ ). In this way we obtain an atlas which makes L(G) into a smooth manifold modelled on the Fr? echet space L(g). Actually in this way L(G) becomes a Lie group. The Lie algebra of L(G) can be naturally identi?ed with L(g). We say that the set U ? L(G) is open if for any γ ∈ L(G) the set ?γ (U ∩ Uγ ) is open. This de?nition supplies L(G) with the structure of a topological space. As any Lie group the loop Lie group L(G) is a Hausdorff topological space. Twisted loop groups are de?ned in full analogy with twisted loop Lie algebras. Let a be an automorphism of a Lie group G and E be the quotient space of the direct product R × G 6

by the equivalence relation which identi?es (σ, g ) with (σ + 2π, a(g )). De?ning the projection π : E → S 1 by the relation π ([(σ, g )]) = eiσ , we obtain a smooth ?ber bundle E → S 1 with ?ber G. Endow the space of smooth sections of this bundle with the structure of a group de?ning the group composition pointwise. This group is called the twisted loop group of G and denoted La (G). Similarly, as for the case of L(g), one endows La (G) with the structure of an in?nite dimensional manifold modelled on the Fr? echet space La (g), where we denote the automorphism of g induced by the automorphism of G by the same letter a. One can verify that in such a way La (G) becomes a Lie group with the Lie algebra La (g). Recall that for any g ∈ G the mapping Int(g ) : h ∈ G → ghg ?1 ∈ G is an automorphism of G. Such automorphisms are called inner and form a normal subgroup of the group Aut G. Similarly, as for the twisted loop Lie algebras, if the automorphisms a and b of G differ by an inner automorphism of G, then the twisted loop Lie groups La (G) and Lb (G) are naturally isomorphic. Therefore, for the case of a semisimple Lie group G one can consider only twisted loop groups La (G) where aK = idG for some positive integer K . One can show that there is a bijective correspondence between elements of La (G) and twisted periodic mappings from R to G. We denote by γ ? the twisted periodic mapping from R to G corresponding to the element γ ∈ La (G). Let G be a semisimple Lie group, and a be an automorphism of G such that aK = idG for some positive integer K . The transformation σ → Kσ induces an injective homomorphism from La (G) to L(G) whose image is formed by the elements γ satisfying the condition γ (εK p) = a(γ (p)), and will be denoted by La,K (G). For the corresponding mapping γ ? from R to G the above condition becomes γ ? (σ + 2π/K ) = a(? γ (σ )). Thus, when G is a semisimple Lie group the twisted loop group La (G) can be identi?ed with a subgroup of L(G).

π

3 Automorphisms of twisted loop Lie algebras

In this section g is always a complex simple Lie algebra and a is an automorphism of g. As was shown in Section 2, studying the twisted loop Lie algebra La (g), one can assume without any loss of generality that aK = idg for some positive integer K and consider instead of La (g) the corresponding subalgebra La,K (g) of L(g). A linear homeomorphism A from a Fr? echet Lie algebra G to itself is said to be an automorphism of G if A [ ξ, η ] = [ Aξ, Aη ] for any ξ, η ∈ G. Treating G as a smooth manifold, we see that since A is linear and continuous, it is smooth. There are two main classes of automorphisms of the Fr? echet Lie algebra La,K (g). The automorphisms of the ?rst class are generated by diffeomorphisms of S 1 . Let us recall that the group Di?(S 1 ) of smooth diffeomorphisms of the circle S 1 can be supplied with the structure of a smooth in?nite dimensional manifold in such a way that it becomes a Lie group. Necessary information on groups of diffeomorphisms of compact manifolds and 7

some relevant references are given in Appendix A. The Lie algebra of the Lie group Di?(S 1 ) is the Lie algebra Der C ∞ (S 1 ) of smooth vector ?elds on S 1 . Here the one-parameter subgroup associated with a vector ?eld X is actually the ?ow generated by X . Let f be a diffeomorphism of S 1 . Consider a linear continuous mapping Af : La,K (g) → L(g) de?ned by the equality Af ξ = ξ ? f ?1 . It is easy to see that if η = Af ξ , then η (f (εK f ?1 (p))) = a(η (p)). Hence, if f (ε K p ) = ε K f (p ) for any p ∈ S 1 , then Af can be considered as a mapping from La,K (g) to La,K (g). In this case Af is an automorphism of La,K (g). Conversely, if the mapping Af is a mapping from La,K (g) to La,K (g), then f satis?es the above condition and Af is an automorphism of La,K (g). One can show that the diffeomorphisms satisfying the condition f (εK p) = εK f (p) form a Lie subgroup of the Lie group Di?(S 1 ). We denote it by Di? K (S 1 ). The Lie algebra of Di? K (S 1 ) is the subalgebra of Der C ∞ (S 1 ) formed by the vector ?elds X such that (X (?))(εK p) = (X (?))(p) for any function ? ∈ C ∞ (S 1 ) satisfying the condition ? (ε K p ) = ? (p ). Denote this subalgebra by DerK C ∞ (S 1 ). It is clear that we have a left action of Di? K (S 1 ) on La,K (g) realised by automorphisms of La,K (g). Since this action is effective, we can say that the group of automorphisms Aut La,K (g) has a subgroup which can be identi?ed with the Lie group Di? K (S 1 ). The Lie group Di? K (S 1 ) can be identi?ed with a subgroup of the group Di?(R). To construct this identi?cation we start with consideration of general smooth mappings from S 1 to S 1. ? ∈ C ∞ (R, R), connected with f For any f ∈ C ∞ (S 1 , S 1 ) one can ?nd a smooth mapping f by the equality ? f (eiσ ) = eif (σ) . ? satis?es the relation The function f ?(σ + 2π ) ? f ?(σ ) = 2πk, f where k is an integer, called the degree of f . From the other hand, any smooth mapping ? ∈ C ∞ (R, R) which satis?es the above relation induces a smooth mapping from S 1 to S 1 . It f is evident that two functions differing by a multiple of 2π induce the same mapping. If f is a diffeomorphism, then its degree is 1 for an orientation preserving mapping, and it ? is ?1 for an orientation reversing mapping. Note that in this case the corresponding function f is strictly monotonic, and that any smooth strictly monotonic function satisfying the relation ?(σ + 2π ) ? f ?(σ ) = ±2π, f induces a diffeomorphism of S 1 . 8

If f ∈ Di? K (S 1 ) one obtains that ?(σ + 2π/K ) = f ?(σ ) + 2π/K, f for K ≥ 2 and that ?(σ + π ) = f ?(σ ) ± π f

for K = 2. Note that if ξ is an element of La,K (g) and f ∈ Di? K (S 1 ) then ?? f ??1 , Af ξ = ξ where Af is the automorphism of La,K (g) induced by f . The second interesting class of automorphisms of La,K (g) is formed by automorphisms generated by automorphisms of g acting on the elements of La,K (g) pointwise. Let α be an element of the Lie group L(Aut g). Consider a linear mapping from La,K (g) to L(g) de?ned by the equality Aα ξ = αξ, where (αξ )(p) = α(p)(ξ (p)). It is clear that Aα is a homomorphism from La,K (g) to L(g). Moreover, if α satis?es the relation α(εK p) = aα(p)a?1 , then the mapping Aα is an automorphism of La,K (g). In other words, any element of the Lie group LInt(a),K (Aut g) induces an automorphism of the Lie algebra La,K (g), and we have a left action of LInt(a),K (Aut g) on La,K (g) realised by automorphisms of La,K (g). This action is again effective and, therefore, Aut La,K (g) has a subgroup which can be identi?ed with the Lie group LInt(a),K (Aut g). Actually, if for f ∈ Di? K (S 1 ) and α ∈ LInt(a),K (Aut g) we de?ne the automorphism A(f,α) of La,K (g) by A(f,α) ξ = α(ξ ? f ?1 ), we obtain a left effective action of the semidirect product Di? K (S 1 ) ? LInt(a),K (Aut g) on La,K (g) realised by automorphisms of La,K (g). Here the group operations in Di? K (S 1 ) ? LInt(a),K (Aut g) are given by (f1 , α1 )(f2 , α2 ) = (f, α), where f = f1 ? f2 , and (f, α)?1 = (f ?1 , α?1 ? f ?1 ). Thus, we see that Di? K (S 1 ) ? LInt(a),K (Aut g) can be identi?ed with a subgroup of the group Aut La,K (g). In fact, this subgroup exhausts the whole group Aut La,K (g). Theorem 3.1 The group of automorphisms of La,K (g) can be naturally identi?ed with the semidirect product Di? K (S 1 ) ? LInt(a),K (Aut g).

?1 α = α1 (α2 ? f1 ),

9

Proof. The main idea of the proof is borrowed from [9]. Let A be an automorphism of La,K (g). Fix a point p ∈ S 1 and consider the mapping Ap from La,K (g) to g de?ned by the equality Ap (ξ ) = (Aξ )(p). This mapping is linear and continuous. Some necessary information on such mappings are given in Appendix B. Certainly, Ap is a homomorphism from La,K (g) to g. 1 Let m be a nonnegative integer. Denote by χm p a smooth function on S such that

(m) χm (p ) = 1 , p m(k ) χp (p ) = 0 ,

k = m,

m and supp χm p ∩ εK supp χp = ?. Let x be an arbitrary element of g. It is not dif?cult to get convinced that for any nonnegative integer m the mapping K ?1 m ηp,x = l=0 ?l χm εl p a (x)

K

is an element of La,K (g) satisfying the conditions

m(m) ηp,x (p) = x, m(k ) ηp,x (p ) = 0 ,

k = m.

0 The linear mapping A is invertible by de?nition. Therefore, there is an element ξp,x ∈ 0 0 0 La,K (g) such that A(ξp,x) = ηp,x . This implies that Ap (ξp,x) = x. Thus the mapping Ap is surjective. For any open set U ? S 1 the set

LU a,K (g) = {ξ ∈ La,K (g) | supp ξ ? U } is an ideal of La,K (g). If an element x ∈ g belongs to the image of the restriction of Ap to U LU a,K (g), then there is an element ξ ∈ La,K (g) such that Ap (ξ ) = x. Since Ap is surjective, it follows that for any element y ∈ g one can ?nd an element η ∈ La,K (g) such that Ap (η ) = y . Since [ξ, η ] belongs to LU a,K (g), it follows that [x, y ] = Ap ([ξ, η ]) belongs to the image of is an ideal of g. As the Lie algebra g is . This means that the image of Ap |LU Ap |LU a,K (g) a,K (g) is either trivial or surjective. simple, the mapping Ap |LU a,K (g) The support of the mapping Ap is the union of sets of the form {q }, where q ∈ S 1 (see Appendix B). Suppose that supp Ap = {q } ∪ {q ′ } and {q } ∩ {q ′ } = ?. Let U and U ′ be disjoint neigbourhoods of {q } and {q ′ } respectively. Since S 1 is a normal topological space U′ such neighbourhoods do exist. It is clear that LU a,K (g) and La,K (g) are commuting ideals of and Ap |LU ′ (g) are commuting ideals of g. Hence, La,K (g). Therefore, the images Ap |LU a,K (g) a,K ′ is surjective, the other one is trivial. Thus, the and A | one of the mappings Ap |LU U p ( g ) (g) L a,K

a,K

support of Ap has the form

{f ′ (p)} for

some mapping f ′ : S 1 → S 1 , and we can write

M (m) cm (f ′ (p))), p (ξ m=0

Ap (ξ ) =

for some nonnegative integer M and endomorphisms cm p (see Appendix B). We assume that the endomorphisms cm are de?ned for all nonnegative m , but cm p p = 0 for m > M . It is clear that m m Ap (ηf ′ (p),x ) = cp (x). 10

Using the relations

m n (m+n) [ηp,x , ηp,y ] (p ) =

m+n [x, y ], m

m n (k ) [ηp,x , ηp,y ] (p ) = 0 ,

k = m + n,

we obtain

m n Ap ([ηf ′ (p),x , ηf ′ (p),y ]) =

m + n m+n cp ([x, y ]). m

Since Ap is a homomorphism, we have

m n m n Ap ([ηf ′ (p),x , ηf ′ (p),y ]) = [Ap (ηf ′ (p),x ), Ap (ηf ′ (p),y )],

therefore,

m n m n Ap ([ηf ′ (p),x , ηf ′ (p),y ]) = [cp (x), cp (y )].

Thus, one has the equalities m + n m+n n cp ([x, y ]) = [cm p (x), cp (y )]. m In particular, for m = n = 0 the equality

0 0 c0 p ([x, y ]) = [cp (x), cp (y )]

(? )

(??)

is valid. Since g is simple, the mapping c0 p is either trivial or surjective. Suppose that it is trivial. Putting in the equality (*) n = 0, we obtain

m 0 cm p ([x, y ]) = [cp (x), cp (y )].

Since the Lie algebra g is simple, then [g, g] = g. Therefore, for any m the mapping cm p is trivial. Hence, the mapping Ap is also trivial. This contradicts surjectivity of Ap . Thus, c0 p is surrjective, and the equality (??) says that it is an automorphism of the Lie algebra g. Putting in (?) m = 0 and n = 1, we obtain

0 1 c1 p ([x, y ]) = [cp (x), cp (y )]

for any x, y ∈ g. Rewrite this equality as

?1 1 0 ?1 1 (c0 p ) (cp ([x, y ])) = [x, (cp ) (cp (y ))].

Therefore,

?1 1 0 ?1 1 ((c0 p ) cp ) ad(x) = ad(x)((cp ) cp ) ?1 1 for any x ∈ g. Since the Lie algebra g is simple, the linear operator (c0 p ) cp is multiplication 1 0 by some scalar, denote it by ρ. Thus, we have cp = ρcp . Relation (?) for m = 1 and n = 1 takes the form 1 1 2 0 2 c2 p ([x, y ]) = [cp (x), cp (y )] = ρ cp ([x, y ]). 2 0 m m 0 Therefore, c2 p = (ρ /2)cp . In general case we have cp = (ρ /m!)cp for any positive m. From m the other hand, cp = 0 for m > M . It is possible only if ρ = 0. Hence, cm p = 0 for all m > 0. 1 De?ne a mapping α : S → Aut g by

α (p ) = c0 p, 11

then one can write Aξ = α(ξ ? f ′ ). Since for any ξ ∈ La,K (g) the mapping Aξ belongs to La,K (g), the mappings f ′ and α must be smooth. The mapping f ′ is actually an element of Di? K (S 1 ), and α belongs to LInt(a),K (Aut g). Hence, de?ning f = f ′?1 , we see that Aξ = α(ξ ? f ?1 ). Thus, an arbitrary automorphism of La,K (g) has the above form for some f ∈ Di? K (S 1 ) and some α ∈ LInt(a),K (Aut g). Note that in the case where g is a complex Lie algebra, the Lie group Aut g is a complex Lie group. In this case LInt(a),K (Aut g) is also a complex Lie group. From the other hand, any diffeomorphism from the identity component of the Lie group Di? K (S 1 ) to a complex Lie group is trivial (see, for example, [9]). This implies that Di? K (S 1 ) cannot be endowed with the structure of a complex Lie group. Therefore, even in the case where g is a complex Lie algebra we consider LInt(a),K (Aut g) as a real Lie group. Thus, the identi?cation described in Theorem 3.1 supplies the group Aut La,K (g) with the structure of a real Lie group. Here the action of the group Aut La,K (g) on La,K (g), where La,K (g) is treated as a real manifold, is smooth. The Lie algebra of the Lie group Aut g is the Lie algebra Der g of derivations of g. The situation is almost the same for the case of the Lie group Aut La,K (g). Actually, any element of the Lie algebra of the Lie group Aut La,K (g) induces a derivation of La,K (g), but in the case where g is a complex Lie algebra there are derivations of La,K (g) which cannot be obtained in such a way. To show this, let us consider ?rst the Lie algebra of Aut La,K (g). Using the identi?cation described in Theorem 3.1, we see that this Lie algebra can be identi?ed with the semidirect product of the Lie algebra of the Lie group Di? K (S 1 ) and the Lie algebra of the Lie group LInt(a),K (Aut g). As we already noted the Lie algebra of Di? K (S 1 ) is the subalgebra DerK C ∞ (S 1 ) of the Lie algebra Der C ∞ (S 1 ) of smooth vector ?elds on S 1 . The Lie algebra of the Lie group LInt(a),K (Aut g) is LAd(a),K (Der g). Thus, the Lie algebra of the group of automorphisms of La,K (g) can be naturally identi?ed with the Lie algebra DerK C ∞ (S 1 ) ? LAd(a),K (Der g). By a derivation of a Fr? echet Lie algebra G we mean a continuous linear mapping D from G to G which satis?es the relation D [ξ, η ] = [Dξ, η ] + [ξ, Dη ]. Note again that continuity and linearity imply smoothness. The derivation of La,K (g) corresponding to an element of DerK C ∞ (S 1 ) ? LAd(a),K (Der g) is constructed as follows. De?ne the action of a vector ?eld X ∈ DerK (S 1 ) on an element ξ ∈ La,K (g) in the usual way. Let (ei ) be a basis of g, then for any element ξ ∈ La,K (g) one can write ξ= ei ξ i ,

i

where ξ i are smooth functions on S 1 . Then one assumes that X (ξ ) =

i

ei X (ξ i ).

One can get convinced that this de?nition does not depend on the choice of a basis (ei ). Let (X, δ ) be an element of DerK C ∞ (S 1 ) ? LAd(a),K (Der g). Consider the corresponding oneparameter subgroup of the Lie group Di? K (S 1 ) ? LInt(a),K (Aut g). It is determined by two 12

mappings λ : R → Di? K (S 1 ) and θ : R → LInt(a),K (Aut g). For any ?xed element ξ ∈ La,K (g) one has a curve τ ∈ R → θ(τ )(ξ ? (λ(τ ))?1 ) in La,K (g). The tangent vector to this curve at zero can be treated as the action of a linear operator D on the element ξ . It is clear that Dξ = ?X (ξ ) + δ (ξ ), where (δ (ξ ))(p) = δ (p)(ξ (p)). One can verify that D is a derivation of the Lie algebra La,K (g). In can be shown also that in the case where g is a real Lie algebra the derivations of the above form exhaust all possible derivations of the Lie algebra La,K (g). In the case where g is a complex Lie algebra to exhaust all derivations one should assume that the vector ?eld X may be complex.

4 Z-gradations of twisted loop Lie algebras

In general, dealing with Z-gradations of in?nite dimensional Lie algebras we confront with necessity to work with in?nite series of their elements, or, in other words, with series in Fr? echet spaces. The relevant information on such series is given in Appendix C. Let G be a Fr? echet Lie algebra. Suppose that for any k ∈ Z there is given a closed subspace Gk of G such that (a) for any k, l ∈ Z one has [Gk , Gl ] ? Gk+l , (b) any element ξ of G can be uniquely represented as an absolutely convergent series ξ=

k ∈Z

ξk ,

where ξk ∈ Gk . In this case we say that the Fr? echet Lie algebra G is supplied with a Zgradation, and call the subspaces Gk the grading subspaces of G and the elements ξk the grading components of ξ . If F is an isomorphism from the Fr? echet Lie algebra G to a Fr? echet Lie algebra H, then taking the subspaces Hk = F (Gk ) of H as grading subspaces we endow H with a Z-gradation. In this case we say that the Z-gradations of G and H under consideration are conjugated by the isomorphism F . It is clear that if the grading components of an element ξ ∈ G are ξk , then the grading components F (ξ )k of the element F (ξ ) ∈ H are F (ξk ). As the simplest example, let us consider the so-called standard gradation of L(g). Denote by λ the standard coordinate function on C and its restriction to S 1 . The grading subspaces for the standard gradation are de?ned as L(g)k = {λk x | x ∈ g}, and the expansion of a general element ξ of L(g) over grading subspaces is the representation of ξ as a Fourier series: ξ= λk xk ,

k ∈Z

? has the usual form that in terms of the mapping ξ ?= ξ

k ∈Z

eiks xk ,

13

with xk =

1 2π

?ds. e?iks ξ

[0,2π ]

From the theory of Fourier series it follows that the Fourier series of any element ξ ∈ L(g) converges absolutely to ξ as a series in the Fr? echet space L(g). Hence, we really have a Z-gradation of L(g). The necessity to include the requirement of absolute convergence in the de?nition of Z-gradation is justi?ed by the following proposition. Proposition 4.1 Let a Fr? echet Lie algebra G be supplied with a Z-gradation. For any two elements of G, ηk , ξk , η= ξ=

k ∈Z k ∈Z

the grading components of [ξ, η ] are given by [ξ, η ]k =

l∈Z

[ξk?l , ηl ].

Here the series at the right hand side converges absolutely. Proof. First prove that the series (k,l)∈Z×Z [ξk , ηl ] converges absolutely. Let α be an element of D (Z × Z), ?x a positive integer m, and de?ne rα,m =

(k,l)∈α

[ξk , ηl ]

m.

There are elements β, γ ∈ D (Z) such that α ? β × γ . Using Lemma 2.2, we obtain rα,m

(k,l)∈β ×γ

[ξk , ηl ]

m

Cm

(k,l)∈β ×γ

ξk

m

ηl

m

= Cm

k ∈β

ξk

m l∈γ

ηl

m

Cm

k ∈Z

ξk

m l∈Z

ηl

m

.

It is clear that for any positive integer m the net (rα,m )α∈D(Z) is monotonically increasing, that means that rα,m ≥ rβ,m if α β . The above inequalities show that it is also bounded above. Similarly as it is for the case of sequences, such a net is convergent. Therefore, the series (k,l)∈Z×Z [ξk , ηl ] converges absolutely. As follows from Proposition C.2 one can write [ξk , ηl ] =

(k,l)∈Z×Z k ∈Z l∈Z

[ξk , ηl ] .

For a ?xed k the net l∈α [ξk , ηl ] α∈D (Z) converges absolutely. It is clear that this net coincides with the net [ξk , l∈α ηl ] α∈D(Z) . Since the net l∈α ηl α∈D (Z) converges to η and the Lie algebra operation in G is continuous, one has [ξk , ηl ] = [ξk , η ].

l∈Z

14

Similarly, one obtains [ξk , η ] = [ξ, η ].

k ∈Z

Using again Proposition C.2, we come to the equality [ξ, η ] =

k ∈Z l∈Z

[ξk?l , ηl ],

where for any k the series

l∈Z [ξk ?l , ηl ]

converges absolutely.

Suppose that a Fr? echet Lie algebra G is supplied with a Z-gradation such that for any element ξ = k∈Z ξk of G the series k∈Z k ξk converges unconditionally. In this case one can de?ne a linear operator Q in G, acting on an element ξ = k∈Z ξk as Qξ =

k ∈Z

k ξk .

Actually the elements k ξk are the grading components of the element Qξ , therefore, the series k ∈Z k ξk converges absolutely by the de?nition of a Z-gradation. It is clear that Gk = {ξ ∈ G | Qξ = kξ }. We call the linear operator Q the grading operator and say that the Z-gradation under consideration is generated by grading operator. If a Z-gradation of a Fr? echet Lie algebra G and a Z-gradation of a Fr? echet Lie algebra H are conjugated by an isomorphism F , and the Zgradation of G is generated by a grading operator Q, then the Z-gradation of H is generated by the grading operator F QF ?1 . The standard gradation of L(g) is generated by a grading operator Q such that ? ds. Qξ = ?idξ/ Here the operator Q is a derivation of L(g). In general we have the following statement. Proposition 4.2 Let a Z-gradation of a Fr? echet Lie algebra G be generated by a grading operator Q. The equality Q [ξ, η ] = [Qξ, η ] + [ξ, Qη ]. is valid for any ξ, η ∈ G. Proof. Using Proposition 4.1, one obtains Q [ξ, η ] =

k ∈Z

(Q [ξ, η ])k =

k ∈Z

k [ξ, η ]k =

k ∈Z l∈Z

k [ξk?l , ηl ] .

In a similar way one comes to the equalities [Qξ, η ] =

k ∈Z l∈Z

(k ? l)[ξk?l , ηl ] ,

[ξ, Qη ] =

k ∈Z l∈Z

l[ξk?l , ηl ] .

The three above equalities imply the validity of the statement of the proposition. 15

It follows from this lemma that if the grading operator Q generating a Z-gradation of a Fr? echet Lie algebra G is continuos, it is a derivation of G. We call a Z-gradation of a Fr? echet Lie algebra G integrable if the mapping Φ from R × G to G de?ned by the relation e?ikτ ξk Φ(τ, ξ ) =

k ∈Z

is smooth. Here as usually we denote by ξk the grading components of the element ξ with respect to the Z-gradation under consideration. For each ?xed ξ ∈ G the mapping Φ induces a smooth curve Φξ : R → G given by the equality Φξ (τ ) = Φ(τ, ξ ). Proposition 4.3 Any integrable Z-gradation of a Fr? echet Lie algebra G is generated by grading operator. The corresponding grading operator Q acts on an element ξ ∈ G as Qξ = i d dt Φξ ,

0

where we denote by t the standard coordinate function on R. Proof. Since the mapping Φ is smooth and linear in ξ , then Q is a continuous linear operator on G. Therefore, for any net (ξα )α∈D(Z) in G which converges to an element ξ ∈ G, the net (Qξα )α∈D(Z) converges to Qξ . The net (ξα )α∈D(Z) , where ξα =

k ∈α

ξk ,

where ξk are the grading components of ξ , converges to ξ . Since for any α ∈ D (Z) the element ξα is the sum of a ?nite number of grading components, one has Qξα = i This means that Qξ = consideration.

k ∈Z k ξk .

d dt

Φξα =

0 k ∈α

k ξk .

Thus, the linear operator Q generates the Z-gradation under

Proposition 4.4 Let a Fr? echet Lie algebra G be supplied with an integrable Z-gradation. Then for any ?xed τ ∈ R the mapping ξ ∈ G → Φ(τ, ξ ) ∈ G is an automorphism of G. The mapping Φ satis?es the relation Φ(τ1 , Φ(τ2 , ξ )) = Φ(τ1 + τ2 , ξ ). Proof. From Proposition 4.1 it follows that one can write [Φ(τ, ξ ), Φ(τ, η )] =

k ∈Z l∈Z

[(Φ(τ, ξ ))k?l, (Φ(τ, η ))l ].

It is clear that (Φ(τ, ξ ))k?l = e?i(k?l)τ ξk?l , Therefore, one has [Φ(τ, ξ ), Φ(τ, η )] =

k ∈Z

(Φ(τ, η ))l = e?ilτ ηl . e?ikτ [ξ, η ]k = Φ(τ, [ξ, η ]).

k ∈Z

e?ikτ

l∈Z

[ξk?l , ηl ] =

That proves the ?rst statement of the proposition. The second statement of the proposition is evident. 16

Let us return to consideration of twisted loop Lie algebras. Suppose that g is a complex simple Lie algebra, and a is an automorphism of g satisfying the relation aK = idg for some positive integer K . Assume that the twisted Lie algebra La,K (g) is endowed with an integrable Z-gradation. De?ne a mapping ? from R to the Lie group Aut La,K (g) by the equality (?(τ ))(ξ ) = Φ(τ, ξ ). It is a curve in the Lie group Aut La,K (g). Using the identi?cation of Aut La,K (g) with the Lie group Di? K (S 1 ) ? LInt(a),K (Aut g), for any τ ∈ R one can write ?(τ ) = (λ(τ ), θ(τ )), where λ is a mapping from R to the Lie group Di? K (S 1 ) and θ is a mapping from R to the Lie group LInt(a),K (Aut g). The mapping λ induces a mapping Λ from R × S 1 to S 1 given by Λ(τ, p) = (λ(τ ))(p), and the mapping θ induces a mapping Θ from R × S 1 to Aut g given by Θ (τ, p) = (θ(τ ))(p). Using the mappings Λ and Θ one can write (Φ(τ, ξ ))(p) = Θ (τ, p)(ξ (Λ?1(τ, p))), where the mapping Λ?1 : R × S 1 → S 1 is de?ned by the equality Λ?1 (τ, Λ(τ, p)) = p. Since the mapping Φ is smooth, also the mappings Λ and Θ are smooth. Therefore, by the exponential law (see Appendix A), the mappings λ and θ are also smooth. Thus, the curve ? is a smooth curve in the Lie group Aut La,K (g). Actually, as follows from Proposition 4.4, it is a one-parameter subgroup of Aut La,K (g). The tangent vector to the curve ? at zero is a derivation of La,K (g) which coincides with the linear operator ?i Q. Therefore, one has the equality Qξ = ?i X (ξ ) + i δ (ξ ). Here X ∈ DerK C ∞ (S 1 ) is the vector ?eld being the tangent vector at zero to the curve λ in Di? K (S 1 ), and δ is the tangent vector at zero to the curve θ in LInt(a),K (Aut g). Note that the mapping Λ corresponding to the mapping λ is a ?ow on S 1 , and X is the vector ?eld which generates this ?ow. Proposition 4.5 Either the vector ?eld X is zero vector ?eld, or it has no zeros. Proof. It is clear that Φ(τ + 2π, ξ ) = Φ(τ, ξ ) for any ξ ∈ L(g). It implies that Λ(τ + 2π, p) = Λ(τ, p) for any p ∈ S 1 . According to the mechanical interpretation of the ?ow, Λ(τ, p) is the position of a particle at time τ , if its position at zero time is p. Here the velocity of the particle at time τ is X (Λ(τ, p)). If p ∈ S 1 is a zero of X , then a particle placed at the point p at some instant of time will forever remain at that point. If X (p) = 0, a particle placed at the point p will instantly move in the same direction, and it cannot pass any zero of the vector ?eld X . If X has zeros, this contradicts the periodicity of Λ in the ?rst argument. There is no contradiction only if X is zero vector ?eld. 17

Recall that any derivation of a simple Lie algebra is an inner derivation. Therefore, if δ is an element of LInt(a),K (Der g), then there exists a unique element η of La,K (g) such that δ (ξ ) = [η, ξ ]. Thus, we come to the following proposition. Proposition 4.6 The grading operator Q generating an integrable Z-gradation of a twisted loop Lie algebra La,K (g) acts on an element ξ ∈ La,K (g) as Qξ = ?i X (ξ ) + i[η, ξ ]. where X ∈ DerK C ∞ (S 1 ), and η is an element of La,K (g). We will not consider Z-gradations with in?nite dimensional grading subspaces. Therefore, the vector ?eld X cannot be zero vector ?eld. Indeed, suppose that X = 0 and Qξ = i[η, ξ ] = kξ , then [η, ξ ] 1 = |k | ξ 1. From Lemma 2.2 one obtains |k | ≤ C η 1 . Hence, we have only a ?nite number of grading subspace, thus, at least some of them must be in?nite dimensional. ? From now on we identify any element ξ of La,K (g) with the corresponding mapping ξ 1 from R to g omitting the tilde. Similarly, we identify each element of Di? K (S ) with the ? again omitting the tilde. An element X of DerK C ∞ (S 1 ) is identi?ed corresponding mapping f with the vector ?eld on R, which we denote again by X . One has X = v d/ds, where the function v satis?es the relation v (σ + 2π/K ) = v (σ ). Proposition 4.7 Let the twisted loop Lie algebra La,K (g) be endowed with an integrable Zgradation with ?nite dimensional grading subspaces, and Q be the corresponding grading operator, which has the form described in Proposition 4.6. For any diffeomorphism f ∈ Di? K S 1 one has 1 Af QA? f ξ = ?if? X (ξ ) + i[η, ξ ], where Af is the automorphism of La,K (g) induced by f . Here the diffeomorphism f can be chosen so that f? X = κ d/ds for some nonzero real constant κ. Proof. The ?rst statement of the proposition follows from the well known equality f? X (?) = f ?1? X (f ? ?) valid for any ? ∈ C ∞ (S 1 ). Writing the vector ?eld X as v d/ds, in accordance with Proposition 4.5 we conclude that the function v has no zeros. Thus we can consider a diffeomorphism f of Di? K (S 1 ) with

σ

f (σ ) = κ

0

dσ ′ . v (σ ′ )

Here the constant κ is ?xed by f (2π/K ) = 2π/K . It is easy to verify that f? X = κ d/ds. Thus, the second statement of the proposition is true. 18

Without any loss of generality one can assume that the constant κ of the above proposition is positive. Indeed, if it is not the case one can do so performing the mapping ξ (σ ) → ξ (?σ ) which maps La,K (g) isomorphically onto La?1 ,K (g). Let G be a simply connected Lie group whose Lie algebra coincides with g. Denote the automorphism of G corresponding to the automorphism a of g by the same letter a. Since G is a complex simple Lie group, we will consider it as a linear group. The following proposition is evident. Proposition 4.8 Let the twisted loop Lie algebra La,K (g) be endowed with an integrable Zgradation, and Q be the corresponding grading operator, which has the form described in Proposition 4.6. Let γ be a smooth mapping from R to G satisfying the relation γ (σ + 2π/K ) = a(g γ (σ )) for some g ∈ G. Consider a linear mapping Aγ acting on any element ξ ∈ La,K (g) as Aγ ξ = γ ξγ ?1. The mapping Aγ is an isomorphism from La,K (g) to the Lie algebra of smooth mappings ξ from R to g satisfying the equality ξ (σ + 2π/K ) = a(g ξ (σ )g ?1). This isomorphism conjugates the Z-gradation of La,K (g) and the Z-gradation generated by the 1 grading operator Aγ QA? γ which acts as

1 ?1 Aγ QA? + X (γ )γ ?1 , ξ ]. γ ξ = ?i X (ξ ) + i[γ η γ

Now we are able to prove our main theorem. Theorem 4.1 An integrable Z-gradation of a twisted loop Lie algbera La,K (g) with ?nite dimensional grading subspaces is conjugated by an isomorphism to a Z-gradation of an appropriate twisted loop Lie algebra La′ ,K ′ (g) generated by grading operator Q′ ξ = ?idξ/ds. Here the automorphisms a and a′ differ by an inner automorphism of g. Proof. In accordance with Proposition 4.6 the grading operator of an integrable Z-gradation of La,K (g) with ?nite dimensional grading subspaces is speci?ed by the choice of a vector ?eld X ∈ DerK C ∞ (S 1 ) and by an element η ∈ La,K (g). Having in mind Proposition 4.7 and the discussion given just below it, we assume without loss of generality that X = κ d/ds for some positive real constant κ. Let a mapping γ : R → G be a solution of the equation κγ ?1 dγ/ds = ?η. It is well known that this equation always has solutions, all its solutions are smooth, and if γ and γ ′ are two solutions then γ ′ = gγ 19

for some g ∈ G. Using the equality η (σ + 2π/K ) = a(η (σ )), one concludes that, if γ is a solution, then the mapping γ ′ de?ned by the equality γ ′ (σ ) = a?1 (γ (σ + 2π/K )) is also a solution. Hence, for some g ∈ G one has γ (σ + 2π/K ) = a(gγ (σ )). The mapping Aγ , described in Proposition 4.8, accompanied by the transformation σ → σ/K maps La,K (g) isomorphically onto the Fr? echet Lie algebra G formed by smooth mappings ξ from R to g satisfying the condition ξ (σ + 2π ) = a′ (ξ (σ )), where a′ = a ? Ad(g ). Denote the grading operator generating the corresponding conjugated Z-gradation again by Q. In accordance with Proposition 4.8 the operator Q acts on an element ξ as Qξ = ?iK ′ dξ/ds, where K ′ = κK . Suppose that for some integer k the grading subspace Gk is nontrivial and ξ ∈ Gk is not equal to zero, then ξ = exp(i ks/K ′ ) ξ (0) with ξ (0) = 0. Since ξ is an element of G, one should have a′ (ξ (0)) = exp(2π i k/K ′ ) ξ (0). For any integer l the mapping ξ ′ de?ned by ξ ′ = exp(i ls) ξ is a nonzero element of G. The action of the grading operator Q on ξ ′ gives (K ′ l + k )ξ ′ . The number K ′ l + k should be an integer. Since l is an arbitrary integer, it is possible only if K ′ is an integer. Actually, due to the remark given after the proof of Proposition 4.7, one can assume without any loss of generality that it is a positive integer. For any integer k denote by [k ]K ′ the element of the ring ZK ′ corresponding to k . Let x be an arbitrary element of g and ξ be an element of G such that ξ (0) = x. Expanding ξ over the grading subspaces, ξ= ξk ,

k ∈Z

one obtains x=

m∈ZK ′

xm ,

where xm =

k ∈Z [ k ] K ′ =m

ξk (0).

20

Here for any m ∈ ZK ′ we have a′ (xm ) = exp(2π i k/K ′ ) xm , where k is an arbitrary integer such that [k ]K ′ = m. Hence, the automorphism a′ is semisimple ′ and a′K = idg. The change σ → K ′ σ induces an isomorphism from G to La′ ,K ′ (g) which conjugates the Z-gradation of G under consideration with the Z-gradation of La′ ,K ′ (g) generated by grading operator Q′ = ?id/ds. That was to be proved. It follows from the above theorem that to classify all Z-gradations of the twisted loop Lie algebra La,K (g) it suf?ces to classify the automorphisms of g of ?nite order. The solution of the latter problem can be found, for example in [11, 5], or in [8]. Note here that classi?cation of the automorphisms of g of ?nite order is equivalent to classi?cation of ZK -gradations of g. Let us have two Z-gradations of La,K (g) which are conjugated to standard Z-gradations of Lie algebras La′ ,K ′ (g) and La′′ ,K ′′ (g). It is clear that the initial Z-gradations are congugated by an isomorphism of La,K (g) if and only if K ′ = K ′′ and the automorphisms a′ and a′′ are conjugated. Acknowledgments. Kh.S.N. is grateful to the Max-Planck-Institut f¨ ur Gravitationsphysik – Albert-Einstein-Institut in Potsdam for hospitality and friendly atmosphere. His work was supported by the Alexander von Humboldt-Stiftung, under a follow-up fellowship program. The work of A.V.R. was supported in part by the Russian Foundation for Basic Research (Grant No. 04–01–00352).

A Diffeomorphism groups

Let M and N be two ?nite dimensional manifolds, and M be compact. The space C ∞ (M, N ) of all smooth mappings from M to N can be supplied with the structure of a smooth manifold modelled on Fr? echet spaces (see, for example, [12, 13, 14]). Let K , M , N be three ?nite dimensional manifolds, and let M be compact. Consider a smooth mapping ? from K to C ∞ (M, N ). This mapping induces a mapping Φ from K × M to N de?ned by the equality Φ(p, q ) = (?(p))(q ). One can prove that the mapping Φ is smooth. Conversely, if one has a smooth mapping from K × M to N , reversing the above equality one can de?ne a mapping from K to C ∞ (M, N ), and this mapping is also smooth. Thus, we have the following canonical identi?cation C ∞ (K, C ∞ (M, N )) = C ∞ (K × M, N ). This fact is called the exponential law or the Cartesian clousedness (see, for example, [13, 14]). Let M be a compact ?nite dimensional manifold. The group Di?(M ) of smooth diffeomorphisms of M is an open submanifold of the manifold C ∞ (M, M ). Here Di?(M ) is a Lie group. The Lie algebra of Di?(M ) is the vector space Der C ∞ (M ) of all smooth vector ?elds on M equipped with the negative of the usual Lie bracket (see, for example, [15, 13, 14]). Let λ : R → Di?(M ) be a smooth curve through the point idM . For each p ∈ M the curve λ induces a curve τ ∈ R → (λ(τ ))(p) in M through p. The tangent vector to this curve at the point p is an element of Tp (M ). In this way we obtain a vector ?eld on M which is the tangent vector to the curve λ at the point idM . 21

Let now λ : R → Di?(M ) be a one-parameter subgroup of Di?(M ). This means that λ is a smooth curve in Di?(M ) which satis?es the equality λ(0) = idM , and the relation λ(τ1 ) ? λ(τ2 ) = λ(τ1 + τ2 ). The mapping λ is an element of C ∞ (R, C ∞ (M, M )). Denote the corresponding element of C ∞ (R × M, M ) by Λ. The mapping Λ satis?es the equality Λ(0, p) = p and the relation Λ(τ1 , Λ(τ2 , p)) = Λ(τ1 + τ2 , p). Hence, the mapping Λ is a ?ow on M . Here the tangent vector to the curve λ at idM is the vector ?eld generating the ?ow Λ. Since M is a compact manifold, then for each vector ?eld X there is a ?ow ΛX generated by X . This ?ow induces the one-parameter subgroup λX of Di?(M ). It is clear that λX (τ ) = exp(τ X ). Therefore, in such a way we realize the exponential mapping for Di?(M ).

B Distributions on S 1 and generalisations

A continuous linear functional on the Fr? echet space C ∞ (S 1 ) = C ∞ (S 1 , C) is said to be a distribution on S 1 . For a general presenation of the theory of distributions we refer to the book by Rudin [10]. The support of a function ? ∈ C ∞ (S 1 ) is de?ned as the closure of the set where ? does not vanish and denoted as supp ?. We say that a distribution T vanishes on an open set U if T (?) = 0 whenever supp ? ? U . Then the support of T is de?ned as the complement of the union of all open sets where T vanishes. It is clear that the support of a distribution on S 1 is a closed set. If the support of a distribution T coincides with a one-point set {p}, then

n

T (? ) =

m=0

cm ?(m) (p).

for some nonnegative integer n and constants cm . Let now T be a continuous linear mapping from L(g) to g. Given a basis (ei ) of g, denote by (?i) the dual basis of g? . For any element x of g one has x=

i

ei ?i (x).

Using this equality, one can write T (ξ ) =

i

ei ?i (T (ξ )). 22

Representing a general element ξ of L(g) as ?i (T (ξ )) =

j

j

ej ξ j , one obtains ?i (T (ej ξ j )).

Introduce a matrix of distributions (T i j ) on S 1 de?ned by the relation T i j (?) = ?i (T (ej ?)). Here ? is a smooth function on S 1 . Now one can write T (ξ ) =

i,j

ei T i j (ξ j ).

Thus, the matrix (T i j ) completely determines the mapping T . The support supp ξ of the element ξ of L(g) is de?ned as the closure of the set where ξ does not take zero value. Representing ξ as i ei ξ i one concludes that supp ξ = i supp ξ i. We say that a continuous smooth mapping T from L(g) to g vanishes on an open set U if T (ξ ) = 0 whenever supp ξ ? U . The support of T is de?ned as the complement of the union of all open sets where T vanishes. It is clear that supp T = i,j supp T i j , where (T i j ) is the matrix of distributions on S 1 which determines the mapping T for given dual bases (ei ) and (?i ) of g and g? respectively. If the support of T is a one-point set {p}, then one can easily demonstrate that

n

T (ξ ) =

m=0

cm (ξ (m) (p)).

for some nonnegative integer n and endomorphisms cm of g. Consider now continuous linear mappings from La (g) to g for the case when g is a semisimple Lie algebra and a is an automorphism of g satisfying the relation aK = idg for some positive integer K . In this case La (g) can be considered as a subalgebra of L(g) formed by the elements ξ satisfying the condition ξ (εK p) = a(ξ (p)). We denote this subalgebra as La,K (g). De?ne a linear operator A in L(g) acting on an element ξ in accordance with the relation

1 Aξ (p) = a(ξ (ε? K p)).

An element ξ ∈ L(g) belongs to La,K (g) if Aξ = ξ . For an arbitrary element ξ ∈ L(g) the element ξ de?ned as ξ= 1 K

K ?1

A?m ξ

m=0

belongs to La,K (g), and one can extend a continuous linear mapping T from La,K (g) to g to a continuous linear mapping T from L(g) to g assuming that T (ξ ) = T (ξ ). One can easily show that T ? A = T, 23

and that supp T = supp T. It is clear that if the support of an element ξ ∈ La,K (g) contains a point p ∈ S 1 , then it contains also the point εK p. Therefore, the support of an element of La,K (g) is the union of sets of the form K ?1 {p} = {p, εK p, . . . , εK p} , p ∈ S 1. The same is true for the support of an arbitrary continuous linear mapping from La,K (g) to g. Let T be a continuous linear mapping from La,K (g) to g whose support is {p}. The corresponding mapping T has the same support and is invariant with respect to the action of the automorphism A. Using these facts one can obtain that T (ξ ) = 1 K

K ?1 n

cm (a?l (ξ (m) (εl K p)))

l=0 m=0

for some nonnegative integer n and endomorphisms cm of g. Restricting the mapping T again to La,K (g) one has

n

T (ξ ) =

m=0

cm (ξ (m) (p)),

for any ξ ∈ La,K (g).

C Convergence and series in Fr? echet spaces

A set D is said to be directed if it is supplied with a binary relation properties: (a) for any element α ∈ D one has α (b) if α β and β γ , then α γ; α and α; satisfying the following

(c) for any two elements α, β ∈ D , there exists an element γ ∈ D such that γ γ β.

The relation is called a direction in D . Below we use the notation D for a general directed set. Given a countable set S , we denote by D (S ) the set of all ?nite subspaces of S , considered as a directed set, where α β if and only if α ? β . A mapping from a directed set D to a topological space X is called a net in X . A net (xα )α∈D in a topological space X is said to converge to an element x ∈ X , or has limit x, if for any neighbourhood U of x there is an element A ∈ D such that xα ∈ U for all α A. Here one also says that the net {xα }α∈D is convergent. If D = N and is the ordinary order relation , nets are sequences with the usual de?nition of convergence. Let X and Y be topological spaces, and f be a mapping from X to Y . The mapping f is continuous if and only if for any net (xα )α∈D which converges to x ∈ X the net (f (xα ))α∈D converges to f (x) ∈ Y . Let X be a topological vector space, I be some countable set, and (xi )i∈I be a collection of elements of X indexed by I . The symbol i∈I xi is called a series in X . Consider a net (sα )α∈D(I ) , where sα = xi .

i∈α

24

If the net (sα ) converges to an element s ∈ X we say that the series ditionally to s and write s= xi .

i∈I

i∈I

xi converges uncon-

Here the element s is called the sum of the series i∈I xi . The next proposition is a direct generalisation of the corresponding proposition for series in normed spaces (see, for example, [16]). Proposition C.1 Let X be a Fr? echet space whose topology is induced by a countable collection of seminorms ( · m ), and i∈I xi be a series in X . If for each m the series i∈I xi m converges unconditionally, then the series i∈I xi also converges unconditionally. Let X be a topological vector space whose topology is induced by a countable family of seminorms ( · )m . If a series i∈I xi in X converges unconditionally and for each m the series i∈I xi m also converges unconditionally, one says that the series i∈I xi converges absolutely. The above proposition says that for complete X unconditional convergence of the series i∈I xi m leads to unconditional convergence of the series i∈I xi . For a series whose terms are positive real numbers, unconditional convergence is equivalent to absolute convergence. Therefore, in this case it is customary to say simply about convergence. As for the case of a general series, one sees that absolute convergence, by de?nition, implies unconditional convergence, but in accordance with the Dvoretzky–Rogers theorem [17], for in?nite dimensional topological vector spaces there are series which converge unconditionally, but do not converge absolutely. The following proposition can be proved along the lines of the proof of the corresponding proposition for series in normed spaces (see, for example, [16]). Proposition C.2 Let a series i∈I xi in a Fr? echet space converge absolutely. Assume that the set I is represented as the union of a countable number of nonempty nonintersecting sets Ij , j ∈ J . For any j ∈ J the series i∈Ij xi converges absolutely and the series j ∈J yj , where yj =

i∈Ij

xi ,

converges absolutely. Moreover, one has xi =

i∈I j ∈J

? ?

i∈Ij

xi ? .

?

References

[1] M. A. Semenov–Tian–Shansky, Integrable systems and factorization problems, In: Factorization and Integrable Systems, eds. I. Gohberg, N. Manojlovic and A. Ferreira dos Santos, Birkh¨ auser, Boston, 2003, p. 155–218. [2] A. N. Leznov and M. V. Saveliev, Group-theoretical Methods for Integration of Nonlinear Dynamical Systems, Birkh¨ auser, Basel, 1992. [3] A. V. Razumov and M. V. Saveliev, Lie Algebras, Geometry, and Toda-type Systems, Cambridge University Press, Cambridge, 1997. 25

[4] A. V. Razumov and M. V. Saveliev, Multi-dimensional Toda-type systems, Theor. Math. Phys. 112 (1997) 999–1022 [arXiv:hep-th/9609031]. [5] V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras, III. Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, vol. 41, Springer, Berlin, 1994. [6] A. V. Razumov, M. V. Saveliev and A. B. Zuevsky, Non-abelian Toda equations associated with classical Lie groups, In: Symmetries and Integrable Systems, ed. A. N. Sissakian, JINR, Dubna, 1999, p. 190–203 [arXiv:math-ph/9909008]. [7] Kh. S. Nirov and A. V. Razumov, On classi?cation of non-abelian Toda systems, In: Geometrical an Topological Ideas in Modern Physics, ed. V. A. Petrov, Protvino, 2002, p. 213-221 [arXiv:nlin.SI/0305023]. [8] V. G. Kac, In?nite dimensional Lie algebras, Cambridge University Press, Cambridge, 1994. [9] A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford, 1986. [10] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. [11] A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer, Berlin, 1990. [12] R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Am. Math. Soc. 7 (1982) 65–222. [13] A. Kriegl and P. Michor, Aspects of the theory of in?nite dimensional manifolds, Diff. Geom. Appl. 1 (1991) 159–176. [14] A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997. [15] J. Milnor, Remarks on in?nite-dimensional Lie groups, In: Relativity, Groups and Topology II, eds. B. S. DeWitt and R. Stora, North-Holland, Amsterdam, 1984, p. 1007–1057. [16] J. Dieudonn? e, Foundations of Modern Analysis, Academic Press, New York, 1960. [17] A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed spaces, Proc. Nat. Acad. Sci. USA 36 (1950) 192–197.

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