aa r X i v : . [ m a t h . R A ] M a r LECTURES ON EXTENDED AFFINE LIE ALGEBRAS
ERHARD NEHER
Abstract.
We give an introduction to the structure theory of extended affineLie algebras, which provide a common framework for finite-dimensional semisim-ple, affine and toroidal Lie algebras. The notes are based on a lecture seriesgiven during the Fields Institute summer school at the University of Ottawain June 2009.
Contents
Introduction 21. Affine Lie algebras and some generalizations 41.1. Realization (construction of affine Kac-Moody Lie algebras) 41.2. Multiloop and toroidal Lie algebras 61.3. Appendix on central extensions of Lie algebras 102. Extended affine Lie algebras: Definition and first examples 142.1. Definition of an extended affine Lie algebra 152.2. Some elementary properties of extended affine Lie algebras 172.3. Extended affine Lie algebras of nullity 0 192.4. Affine Kac-Moody Lie algebras again 192.5. Higher nullity examples 223. The structure of the roots of an EALA 233.1. Affine reflection systems: Definition 233.2. Examples of affine reflection systems 253.3. The Structure Theorem of affine reflection systems 313.4. Extended affine root systems 364. The core and centreless core of an EALA 384.1. Lie tori: Definition 384.2. Some basic properties of Lie tori 404.3. The core of an EALA 434.4. Lie tori of type A l , l ≥ Date : October 24, 2018.1991
Mathematics Subject Classification.
Primary 17B.This work is partially supported by the Natural Sciences and Engineering Research Council(NSERC) of Canada through the author’s Discovery Grant.
Introduction
Extended affine Lie algebras form a category of Lie algebras containing finite-dimensional semisimple, affine, toroidal and some other interesting classes of Liealgebras.Like finite-dimensional simple Lie algebras, extended affine Lie algebras are de-fined by a set of axioms prescribing their internal structure, rather than a potentiallyelusive presentation. The structure of extended affine Lie algebras is now well un-derstood, and is quite similar to the construction of affine Lie algebras: They areobtained from a generalized loop algebra, a so-called invariant Lie torus, by takinga central extension and adding some derivations:central extension of L (another Lie torus) addderivations / / /o/o/o/o/o/o/o/o/o/o/o extended affine Lie algebrainvariant Lie torus L O O O(cid:15)O(cid:15)O(cid:15)
Invariant Lie tori have been classified. Although there are some rather sophisticatedexamples, many of them have a concrete matrix realization or can be described interms of familiar objects like finite-dimensional simple Lie algebras and Laurentpolynomial rings. This makes extended affine Lie algebras easily accessible. Sincethey are an emerging new area, there are many open questions, opportunities forresearch and applications, for example in physics. A short history of extended affineLie algebras is given in section 2.1, in particular it describes the role physicists haveplayed.The goal of these notes is to provide a survey of the structure theory of extendedaffine Lie algebras, accessible to graduate students. The emphasis is on examples,and not on an exposition containing all proofs. Such an exposition will appearelsewhere. Thus, while we have endeavored to present a complete picture of thetheory by giving precise definitions and theorems, most of the proofs have been leftout. But references to proofs are provided, as far as possible.
Outline.
Section 1 reviews the construction of affine Kac-Moody algebras anddiscusses some natural generalizations, like toroidal algebras. It also contains anexposition of central extensions of Lie algebras, which are crucial for the theory.The following section 2 starts with the definition of an extended affine Lie algebraand then presents some easily proven properties. We also give examples of extendedaffine Lie algebras: finite-dimensional split simple, affine Kac-Moody and untwistedmulti-loop algebras. Part of the axioms for an extended affine Lie algebra is theexistence of a root space decomposition. Section 3 describes the structure of theroots occurring in an extended affine Lie algebra, naturally called extended affineroot systems. They turn out to be special types of so-called affine reflection systems.In section 4 we reverse the picture above: We start with an extended affine Liealgebra and, using the structure of affine reflection systems, we associate to it agraded ideal, the so-called core, and its central quotient, the centreless core. Bothare Lie tori. This section also presents properties of Lie tori and examples. Finally,in section 5 we survey the general construction of extended affine Lie algebras, assummarized in the picture above.
ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 3
Prerequisites.
We assume that the reader is familiar with the basic structuretheory of complex finite-dimensional semisimple Lie algebras, as for example de-veloped in [Hu]. Some familiarity with affine Kac-Moody algebras, e.g. chapters7 and 8 of [Kac], is helpful but not essential, since section 1.1 will give a shortreview of the necessary background. Similarly, knowing split simple Lie algebraswill facilitate reading the notes, but is not required. A short summary of the factsused here is presented in section 2.3.
Notation and setting.
With some rare exceptions (in 1.1, 5.1 and 5.2), all vectorspaces and algebras are defined over a field F of characteristic 0. We will notassume that F is algebraically closed , since this is not needed and would not doproper justice to the theory to be explained here. Thus, F could be, but need notbe the field C of complex numbers or the field R of real numbers or the field ofrational numbers Q or ... Unless specified otherwise, linear maps will always be F -linear. All unadorned tensor products will be over F .The symbol g will always denote a split simple finite-dimensional Lie algebra.We let Z ( L ) = { z ∈ L : [ z, L ] = 0 } denote the centre of a Lie algebra L . We willsay that L is centreless if Z ( L ) = 0. If K is a subspace of a Lie algebra E , the centralizer of K in E is C E ( K ) = { c ∈ E : [ c, K ] = 0 } .With the exception of some remarks, all algebras will be associative or Lie al-gebras. For an F -algebra A we denote by Der F ( L ) the Lie algebra of all deriva-tions of L (recall that an F -linear map d : L → L is a derivation if d ([ l , l ]) =[ d ( l ) , l ] + [ l , d ( l )] holds for all l , l ∈ L ).The algebras considered here will often be graded by some abelian group, usuallydenoted Λ and always written additively. A Λ -grading of a vector space V by theabelian group Λ is a decomposition V = L λ ∈ Λ V λ into subspaces V λ . Suppose V is such a Λ-graded vector space. Then the Λ -support of V is defined as supp Λ V = { λ ∈ Λ : V λ = 0 } . A graded subspace of V is a subspace U of V satisfying U = L λ ( U ∩ V λ ). We will say that V has finite bounded dimension if there existsa constant M such that for all λ ∈ Λ we have dim V λ ≤ M . Note that this is astronger condition than requiring that V has finite homogeneous dimension , whichby definition just means that every V λ , λ ∈ Λ, is finite-dimensional.Given two Λ-graded vector spaces V = L λ V λ and W = L λ W λ , we say an F -linear map f : V → W has degree λ if f ( V µ ) ⊂ W λ + µ holds for all µ ∈ Λ. Wedenote by Hom F ( V, W ) λ the linear maps of degree λ and putgrHom F ( V, W ) = L λ ∈ Λ Hom F ( V, W ) λ and grEnd F ( V ) = grHom F ( V, V ) . We note that grEnd F ( V ) is a Λ-graded associative algebra with respect to compo-sition of maps. We give F the trivial grading F = F and define the graded dualspace of V as V gr ∗ = grHom F ( V, F ) = L λ ∈ Λ ( V gr ∗ ) λ . Observe that ( V gr ∗ ) λ consists of those linear forms ϕ : V → F which satisfy ϕ ( V µ ) = 0 whenever λ + µ = 0 and can therefore be identified with the usualdual space ( V − λ ) ∗ .Given a symmetric bilinear form on a vector space V , an endomorphism d of V is called skew-symmetric if ( d ( v ) | v ) = 0 for all v ∈ V . Since we assume that ourbase field has characteristic 0, this is equivalent to the condition ( d ( v ) | v ) + ( v | d ( v )) = 0 for all v , v ∈ V . A bilinear form is nondegenerate if ( v | u ) = 0 for all u ∈ V implies v = 0. ERHARD NEHER If A is an algebra, a Λ -grading of the algebra A is a Λ-grading of the underlyingvector space A , say A = L λ ∈ Λ A λ , for which in addition A λ A µ ⊂ A λ + µ holds for all λ, µ ∈ Λ. Since we will often deal with algebras with two gradings, it is convenientto use superscripts and subscripts to distinguish them.These notes grew out of my notes for a lecture series during the Fields Insti-tute summer school on Geometric Representation Theory and Extended Affine LieAlgebras, held at the University of Ottawa in June 2009. I would like to thankall the participants of the summer school for their interest and questions. I alsothank Bruce Allison and Juana S´anchez Ortega for their careful reading of an earlierversion of these notes.1.
Affine Lie algebras and some generalizations
We will always assume that F is a field of characteristic 0. Occasionally we willneed some roots of unity in F , so certainly an algebraically closed field like C willdo.We denote by g a split simple finite-dimensional Lie algebra over F . For example,if F is algebraically closed then this just means that g is a simple and finite-dimensional. Their structure theory is explained in most standard textbooks, forexample in [Hu]. For more general fields, an example of a split simple g is the Liealgebra sl n ( F ) of n × n -matrices over F which have trace 0. These types of Liealgebra are investigated in [Bou3, Ch. VII], [D, Ch. 1] or [J, Ch. IV].1.1. Realization (construction of affine Kac-Moody Lie algebras).
Let ζ ∈ F be a primitive m th root of 1. In other words, the multiplicative subgroup of F generated by ζ is isomorphic to Z /m Z . For example, in F = C we can take ζ = exp(2 πi/m ).Let σ be an automorphism of g of finite order m ∈ N . Thus, the subgroup h σ i ofthe automorphism group of g is isomorphic to Z /m Z . For example, if g = sl n ( F )an example of such an automorphism is σ ( x ) = axa − , where a is an n × n -matrixof order m , and an example of such a matrix is a = ζE n where E n is the n × n identity matrix.Observe that σ is diagonalizable. Indeed, its minimal polynomial divides thepolynomial t m = 1 and therefore has no multiple roots in F . For a general field F this would of course only say that σ is a semisimple endomorphism. But since aswe assumed that F contains all roots of unity which we need, σ is diagonalizableover F . To describe its eigenspaces we need some notation. In anticipation of thelater developments we put Λ = Z and ¯Λ = Z /m Z , and denote the canonical map Λ → ¯Λ by λ ¯ λ . That σ is diagonalizable, means(1.1) g = L ¯ λ ∈ ¯Λ g ¯ λ for g ¯ λ = { x ∈ g : σ ( x ) = ζ λ x } Of course, some of the g ¯ λ could be zero. The eigenspaces of σ are precisely thenon-zero among the subspaces g ¯ λ . It is also appropriate to note that g ¯ λ is well-defined: if ¯ λ = ¯ µ then ζ λ = ζ µ . Finally we point out that the decomposition (1.1)is a ¯Λ-grading, which means that it satisfies(1.2) [ g ¯ λ , g ¯ µ ] ⊂ g ¯ λ +¯ µ for all ¯ λ, ¯ µ ∈ ¯Λ. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 5
Let F [ t ± ] be the ring of Laurent polynomials. This is a unital associativecommutative F -algebra with F -basis { t λ : λ ∈ Z } and multiplication rule t λ t µ = t λ + µ .The loop algebra associated to the data ( g , σ ) is the Lie algebra(1.3) L = L ( g , σ ) = L λ ∈ Λ g ¯ λ ⊗ F t λ with product(1.4) [ u ¯ λ ⊗ t λ , v ¯ µ ⊗ t µ ] = [ u ¯ λ , v ¯ µ ] ⊗ t λ + µ . We will sometimes use more precise terminology: If σ = Id, i.e., m = 1, we will call L ( g , Id) = g ⊗ F [ t ± ] the untwisted loop algebra , and we will call L ( g , σ ) a twistedloop algebra if it is clear that σ = Id and we want to emphasize this.We point out that we consider L ( g , σ ) as a Lie algebra over F . It is thereforeinfinite-dimensional. It is also important to note that L is a Λ-graded algebra,whose homogenous spaces are L λ = g ¯ λ ⊗ F t λ for λ ∈ Λ. For the reader with somebackground in algebraic geometry, a more geometric definition of L ( g , σ ) is thefollowing: It is (isomorphic to) the Lie algebra of equivariant maps F × → g , where σ acts on F × by σ ( x ) = ζx .)Let κ be the Killing form of g , i.e., κ ( u, v ) = tr(ad u ◦ ad v ), and define(1.5) ψ : L × L →
F, ψ ( u ⊗ t λ , v ⊗ t µ ) = λ δ λ, − µ κ ( u, v )where δ λ, − µ is the Kronecker delta: It has the value 1 if λ = − µ and is zerootherwise. Exercise 1.1.
Check that the map ψ of (1.5) is a 2 -cocycle of L , i.e. an F -bilinearmap satisfying(1.6) ψ ( l, l ) = 0 = ψ ([ l , l ] , l ) + ψ ([ l , l ] , l ) + ψ ([ l , l ] , l )for l, l i ∈ L .A consequence of Exercise 1.1 is that we can enlarge our Lie algebra L by ad-joining a 1-dimensional space, denoted F c here:(1.7) ˜ L = ˜ L ( g , σ ) = L ( g , σ ) ⊕ F c is a Lie algebra over F with respect to the product[ l ⊕ s c, l ⊕ s c ] ˜ L = [ l , l ] L ⊕ ψ ( l , l ) c for l i ∈ L and s i ∈ F . We have added subscripts on the products to emphasizewhere the product is calculated, in ˜ L or in L . It is obvious from the productformula, that it is important to know in which Lie algebra the product is beingcalculated. But in the future we will leave out the subscripts, if it is clear in whichalgebra the product is calculated.The equations (1.6) are exactly what is needed to make ˜ L a Lie algebra. Themap u : ˜ L → L , u ( l ⊕ sc ) = l is a surjective Lie algebra homomorphism with kernel Ker( u ) = F c = Z ( ˜ L ), thecentre of ˜ L . In other words, u is a central extension (see 1.3 for a short reviewof central extensions). In fact, u is the “biggest” central extension, the so-called universal central extension , see [G] and [Wi] for a proof. ERHARD NEHER
The Lie algebra ˜ L has a canonical derivation d , the so-called degree derivation (1.8) d (cid:0) ( u ⊗ t λ ) ⊕ sc (cid:1) = λu ⊗ t λ , ( λ ∈ Z , u ∈ g ¯ λ , s ∈ F ) . Hence we can form the semidirect product ˆ L = L ( g , σ )ˆ = ˜ L ⋊ F d with product[˜ l ⊕ s d, ˜ l ⊕ s d ] ˆ L = [˜ l , ˜ l ] ˜ L + s d (˜ l ) − s d (˜ l )for ˜ l i ∈ ˜ L and s i ∈ F . In untangled form,(1.9) ˆ L = (cid:0) L λ ∈ Z ( g ¯ λ ⊗ F t λ ) (cid:1) ⊕ F c ⊕ F d is the Lie algebra with product(1.10) [ u ¯ λ ⊗ t λ ⊕ s c ⊕ s ′ d, v ¯ µ ⊗ t µ ⊕ s c ⊕ s ′ d ]= (cid:0) [ u ¯ λ , v ¯ µ ] ⊗ t λ + µ + µs ′ v ¯ µ ⊗ t µ − λs ′ u ¯ λ ⊗ t λ (cid:1) ⊕ λ δ λ, − µ κ ( u ¯ λ , v ¯ µ ) c. Exercise 1.2.
Show [ ˆ L , ˆ L ] = ˜ L and Z ( ˜ L ) = F c = Z ( ˆ L ).The importance of the Lie algebras L ( g , σ )ˆ stems from the following. Theorem 1.3. ( Realization Theorem [Kac, Th. 7.4, Th. 8.3, Th. 8.5])
Suppose F is algebraically closed. (a) The Lie algebra L ( g , σ )ˆ is an affine Kac-Moody Lie algebra, and every affineKac-Moody Lie algebra is isomorphic (as F -algebra) to some L ( g , σ )ˆ . (b) L ( g , σ )ˆ ∼ = L ( g , σ ′ )ˆ where σ ′ is a diagram automorphism with respect tosome Cartan subalgebra of g . We note that diagram automorphisms have order 1 , g of type D .1.2. Multiloop and toroidal Lie algebras.
We will discuss some (straightfor-ward) generalizations of L = L ( g , σ ), the central extension ˜ L and the big Lie algebraˆ L . The first idea is to replace the Laurent polynomial ring F [ t ± ] by a ring withsimilar properties. Instead of one variable we will use the Laurent polynomial ring F [ t ± , . . . , t ± n ] in n variables. This ring has indeed very similar properties to thering F [ t ± ]. We put Λ = Z n and define t λ = t λ · · · t λ n n for λ = ( λ , . . . , λ n ) ∈ ΛThen { t λ : λ ∈ Λ } is an F -basis of F [ t ± , . . . , t ± n ] and the multiplication rule in F [ t ± , . . . , t ± n ] is t λ t µ = t λ + µ , which is the “same” as in the 1-variable case. Also, F [ t ± , . . . , t ± n ] is still a unital commutative associative F -algebra. We can thereforedefine the untwisted multiloop algebra , the “several variable” generalization of theuntwisted loop algebra of 1.1 as(1.11) L ( g ) = g ⊗ F [ t ± , . . . , t ± n ] , which becomes a Lie algebra with respect to the product[ u ⊗ t λ , v ⊗ t µ ] = [ u, v ] ⊗ t λ + µ for u, v ∈ g and λ, µ ∈ Z n . We will meet this Lie algebra again in Example 4.31.To continue the analogy we let σ = ( σ , . . . , σ n ) be a family of n commutingfinite order automorphisms of g , say σ i has order m i ∈ N + . Let ζ i ∈ F be a ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 7 primitive m i -th root of 1 (recall that we assumed that F has an ample supply ofthem). We put ¯Λ = ( Z /m Z ) ⊕ · · · ⊕ ( Z /m n Z )and let λ ¯ λ be the obvious map. The automorphisms σ i are simultaneouslydiagonalizable:(1.12) g = L ¯ λ ∈ ¯Λ g ¯ λ , g ¯ λ = { u ∈ g : σ i ( u ) = ζ λ i i u, ≤ i ≤ n } . As in the one-variable case, the decomposition (1.12) is a ¯Λ-grading: [ g ¯ λ , g ¯ µ ] ⊂ g ¯ λ +¯ µ for ¯ λ, ¯ µ ∈ ¯Λ. It follows from this that(1.13) L ( g , σ ) = L λ ∈ Λ g ¯ λ ⊗ F t λ is a subalgebra of g ⊗ F [ t ± , . . . , t ± n ], called the multiloop algebra associated to g and σ . If all σ i = Id g we will (of course) call it an untwisted multiloop algebra . Multiloopalgebras are investigated in the papers [ABFP1], [ABFP2], [ABP1], [ABP2] and[ABP3].Following our procedure in section 1.1 we should now make a central extensionto get a bigger Lie algebra ˜ L and then add some derivations:(1.14) ˜ L = L ⊕ C (cid:31) (cid:127) addderivations / / centralextension (cid:15) (cid:15) ˆ L = ˜ L ⋊ D L = L ( g , σ ) To define the Lie algebra product on ˜ L we would use a 2-cocycle ψ : L × L → C where C is some vector space and then put(1.15) [ l ⊗ c , l ⊗ c ] ˜ L = [ l , l ] L ⊕ ψ ( l , l )for l i ∈ L and c i ∈ C . The Lie algebra ˆ L should be a semidirect product with D acting on ˜ L by derivations.But here is where the problems start, or things become interesting depending onone’s taste. In the one-variable case the 2-cocycle ψ of (1.5) was the only possiblechoice up to scalars, i.e., the universal central extension ˜ L of L had a 1-dimensionalcentre C = F c . This is no longer true in the case of several variables. It is not sosurprising that there exists a 2-cocycle with values in F n : We can simply use thesame formula as in (1.5). Exercise 1.4.
Let L = L ( g , σ ) be a multiloop algebra and embed Λ ⊂ F n canon-ically. Then ψ : L × L → F n , given by(1.16) ψ ( u ⊗ t λ , v ⊗ t µ ) = δ λ + µ, κ ( u, v ) λ , is a 2-cocycle of L .However, this is still not the “biggest” possible. Rather, the centre of the univer-sal central extension is infinite-dimensional and the so-called universal -cocycle ,i.e., the 2-cocycle used in (1.15) to describe the universal central extension ˆ L of L ,is described in the following result. ERHARD NEHER
Theorem 1.5 ([Ne6]) . Let L = L ( g , σ ) be a multiloop algebra. We embed Λ ⊂ F n canonically, put Γ = m Z ⊕ · · · ⊕ m n Z and let C = L γ ∈ Γ C γ where C γ = F n /F γ .Then the universal -cocycle is ψ u : L × L → C for which the γ -component of ψ u is (1.17) ψ u ( u ⊗ t λ , y ⊗ t µ ) γ = κ ( u, v ) δ λ + µ, − γ ¯ λ ∈ C γ . Observe that (1.16) is just the 0-component of (1.17). The theorem is well-knownin the untwisted case (all σ i = Id g , so Γ = Λ), in which it can be deduced fromthe description of the universal central extension of the Lie algebra g ⊗ A where A is any unital commutative associative F -algebra, see [Kas] and [MRK]. (In thesereferences the centre C of the universal central extension is described as Ω A /dA where Ω A is the module of K¨ahler differentials, which is also the same as the firstcyclic homology group HC ( A ).)In the untwisted case, the universal central extension ˆ L was termed the n -toroidalLie algebra based on g . The reader should however be warned that this terminologyis not standard. It is sometimes used for the Lie algebra ˜ L with the 2-cocycle ofexercise 1.4, and sometimes also for the Lie algebras of the form ˆ L = L ⊕ C ⊕ D for an appropriate subalgebra D of derivations, e.g. in [DFP].Thus, there are many possibilities for C in the diagram (1.14), and it is notclear which one is the best possible choice. (In fact, we will later allow any centralextension).Assuming that we have settled for some C , which D should we take? For sim-plicity we will discuss this only in the untwisted case. If n = 1 we added the degreederivation d described in (1.8). This is far from being an arbitrary derivation. Thefull derivation algebra of the Lie algebra g ⊗ A for A is described in [BM, Th. 1]:Der F ( g ⊗ A ) = (cid:0) Der F ( g ) ⊗ A (cid:1) ⊕ (cid:0) F Id ⊗ Der F ( A ) (cid:1) = IDer( g ⊗ A ) ⊕ F Id ⊗ Der F ( A ) . (1.18)where Der F ( g ) ⊗ A and F ⊗ Der F ( A ) = F Id ⊗ Der F ( A ) act on g ⊗ A in the obviousway.Since g ⊗ A is perfect, up to a canonical isomorphism, this is then also thederivation algebra of the universal central extension of g ⊗ A (see for example [BM,Th. 2.2]). From Der F [ t ± ] = F [ t ± ] d we see that we added a rather special derivation, one which can be used to definethe Λ-grading of L (see also Ex. 1.7).We can do something similar in multi-variable case. Define the i -th degree deriva-tion ∂ i of L ( g ) ⊕ C by(1.19) ∂ i ( u ⊗ t λ ⊕ c ) = λ i u ⊗ t λ for λ = ( λ , . . . , λ n ) ∈ Λ = Z n and put D = span F { ∂ i : 1 ≤ i ≤ n } , the space of degree derivations . Possible (interesting) choices for D are:(1) D = D ,(2) F [ t ± , . . . , t ± n ] D (in physics parlance: “all vector fields”), and(3) L λ ∈ Λ F t λ { P ni =1 s i ∂ i : P i s i = 0 } (the “divergence 0 vector fields”). ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 9
It will turn out that for the Lie algebras which we are going to study in the nextchapters, the choices (1) and (3) are the correct ones. In addition, there will be asurprise: semidirect products in (1.14) will not be enough!
Exercise 1.6.
Recall that a bilinear form ( ·|· ) on a Lie algebra L is called invariant if ([ l , l ] | l ) = ( l | [ l , l ]) holds for all l i ∈ L . Show:(a) The set IF( L ) of invariant bilinear forms on L is a vector space with respectto the obvious scalar multiplication and addition defined by ( β + β )( l , l ) = β ( l , l ) + β ( l , l ) for β i ∈ IF( L ).(b) If L is perfect, any invariant bilinear form is symmetric.(c) Let S be a unital associative F -algebra. A bilinear form b on S is called invariant if b ( s s , s ) = b ( s , s s ) = b ( s , s s ) for s i ∈ S .(i) The set IF( S ) of invariant bilinear forms on S is a vector space with respectto the obvious operations.(ii) Any linear form λ ∈ S ∗ with λ ([ S, S ]) = 0 gives rise to an invariant bilinearform b λ on S , defined by b λ ( s , s ) = λ ( s s ).(iii) The map ( S/ [ S, S ]) ∗ → IF( S ), given by λ b λ , is a vector space isomor-phism.(d) Let L be a perfect Lie algebra with a 1-dimensional space IF( L ), say IF( L ) = F κ . Also, let S be a unital associative commutative F -algebra. We consider L ⊗ S as Lie algebra with respect to the product [ l ⊗ s , l ⊗ s ] = [ l , l ] ⊗ s s , cf. (1.5).For λ ∈ IF( S ) define a bilinear form κ ⊗ λ on L ⊗ S by( κ ⊗ λ ) ( l ⊗ s , l ⊗ s ) = κ ( l , l ) λ ( s , s ) . Then κ ⊗ λ ∈ IF( L ⊗ S ) and the map IF( S ) → IF( L ⊗ S ), given by λ κ ⊗ λ , isan isomorphism of vector spaces. Exercise 1.7.
Define the i th degree derivation ∂ i of the Laurent polynomial ring S = F [ t ± , . . . , t ± n ] by ∂ i ( t λ ) = λ i t λ , so that the ∂ i of (1.19) becomes ∂ i ( u ⊗ t λ ) = u ⊗ ∂ i ( t λ ) = (Id ⊗ ∂ i )( u ⊗ t λ ) (this double meaning of ∂ i should not create anyconfusion). Show:(a) The derivation algebra Der F ( S ) of S is given byDer F ( S ) = S D = L λ ∈ Z n F t λ D where, as above, D = span F { ∂ i : 1 ≤ i ≤ n } . The derivation algebra is a Z n -gradedLie algebra with Lie algebra product determined by[ t λ ∂ i , t µ ∂ j ] = t λ + µ ( µ i ∂ j − λ j ∂ i ) . Thus, for n = 1 we obtain the usual Witt algebra , see for example [MP, 1.4].(b) ( t λ | t µ ) = δ λ + µ, defines a nondegenerate symmetric bilinear form ( ·|· ) on S which is invariant in the sense that ( ab | c ) = ( a | bc ) for all a, b, c ∈ S .(c) Let SDer F ( S ) be the subalgebra of derivations of S , which are skew-sym-metric with respect to the form ( ·|· ) of (b). ThenSDer F ( S ) = L λ ∈ Z n F t λ (cid:8) P ni =1 s i ∂ i : P i s i λ i = 0 (cid:9) . In particular, for n = 1 we get SDer F ( F [ t ± ]) = F d for d = ∂ . Appendix on central extensions of Lie algebras.
Central extensions willturn out to be an important tool in the construction of extended affine Lie algebras.Although this provides one with a bigger and hence potentially more complicatedLie algebra, central extensions turn up naturally in the general theory and thebiggest of them (the universal central extension) is in fact quite “nice”. For example,universal central extensions often have a simpler presentation and a much richerrepresentation theory than the original Lie algebra. In this appendix we review thenecessary background.
Definition 1.8 (Extensions) . An extension of a Lie algebra L is a surjective ho-momorphism f : K → L of Lie algebras. A homomorphism from an extension f : K → L to another extension f ′ : K ′ → L is a Lie algebra homomorphism g : K → K ′ satisfying f = f ′ ◦ g . In other words, the diagram below is commuta-tive.(1.20) K g / / f (cid:31) (cid:31) @@@@@@@ K ′ f ′ ~ ~ }}}}}}}} L We will use abelian extensions , i.e., extensions f : K → L with Ker f an abelianideal in the construction of an extended affine Lie algebra in section 5.4. Definition 1.9 (central extensions) . A central extension of L is an extension f : K → L whose kernel Ker f is contained in the centre Z ( K ) of K . A centralextension f : K → L is called a covering if K is perfect, i.e., K = [ K, K ]. It istraditional (but not always advisable) to not specify the morphism f and simplysay that K is a central extension of L or a covering .A central extension u : L → L is called a universal central extension if there existsa unique homomorphism from u : L → L to any other central extension f : K → L of L . It is obvious from the universal property that two universal central extensionsof L are isomorphic as central extensions and hence in particular their underlyingLie algebras are isomorphic. We denote the universal central extension of L by u : uce ( L ) → L or simply uce ( L ). Theorem 1.10 ([vdK, Prop. 1.3], [G, § . A Lie algebra L has a universal centralextension if and only if L is perfect. In this case, the universal central extension u : uce ( L ) → L is perfect too, i.e., u is a covering. The process of taking universal central extensions stops at uce ( L ), due to thefollowing equivalent conditions for a Lie algebra L :(i) Id : L → L is a universal central extension, i.e., uce ( L ) = L ,(ii) every central extension f : K → L is direct product K = ˜ L × Ker f suchthat f | ˜ L is an isomorphism between ˜ L and L .If (i) and (ii) hold, one calls L centrally closed . Examples 1.11. (a) It is an immediate corollary of the Levi-Malcev Theorem thatevery finite-dimensional semisimple Lie algebra is centrally closed ([Bou3, VII, § F ( F [ t ± ]), see for example [MP, ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 11
I.9, Prop. 4]. Hence the Virasoro algebra is centrally closed, while Der F ( F [ t ± ])is not. On the other hand, the higher rank Witt algebra Der F ( F [ t ± , . . . , t ± n ]), n >
1, is centrally closed ([RSS, V, Th. 5.1]).
Definition 1.12 (Central extensions via 2-cocycles.) . We have already seen in § L by using 2 -cocycles ,which, we recall, are bilinear maps ψ : L × L → C into a vector space C satisfyingfor all l, l , l , l ∈ L (1.21) ψ ( l, l ) = 0 and ψ ([ l , l ] , l ) + ψ ([ l , l ] , l ) + ψ ([ l , l ] , l ) = 0 . The first equation is of course equivalent to ψ ( l , l ) = − ψ ( l , l ). Given a 2-cocycle ψ : L × L → C , the algebra(1.22) K = L ⊕ C by [ l ⊕ c , l ⊕ c ] K = [ l , l ] L ⊕ ψ ( l , l )( l i ∈ L , c i ∈ C ) is a Lie algebra and pr L : K → L , pr L ( l ⊕ c ) = l , is a centralextension of L , which we will denote by E( L, C, ψ ) or E(
L, ψ ) for short.Conversely, given a central extension f : K → L , let s : L → K be a sectionof f in the category of vector spaces, i.e. a linear map s : L → K such that f ◦ s = Id L . Such a section always exists: We can choose a subspace L ′ of K ,which is complementary to C = Ker f , and take s = ( f | L ′ ) − which makes sensesince ( f | L ′ ) : L ′ → L is an invertible linear map (but in general not a Lie algebrahomomorphism since L ′ need not be a subalgebra). Given a section s , the map(1.23) ψ s : L × L → C, ψ s ( l , l ) = [ s ( l ) , s ( l )] K − s ([ l , l ] L )turns out to be a 2-cocycle. Moreover, the map K → L ⊕ C, x f ( x ) ⊕ (cid:0) x − ( s ◦ f )( x ) (cid:1) = f ( x ) ⊕ x C , where x C is the C -component of x ∈ K , is an isomorphism from the central exten-sion f : K → L to the central extension E( L, Ker f, ψ s ). To summarize, modulosome verifications left as an exercise, we have proven the following well-known re-sult. Proposition 1.13.
For any -cocycle ψ the construction (1.22) is a central ex-tension E( L, ψ ) of L and, conversely, every central extension L is isomorphic ascentral extension to some E( L, ψ ) . Exercise 1.14.
Let ψ : L × L → C be a 2-cocycle and let C ′ be a subspace of C satisfying ψ ( L, L ) := span F { ψ ( l , l ) : l i ∈ L } ⊂ C ′ . Then E( L, C ′ , ψ ) is also acentral extension, and if E( L, C, ψ ) is a covering then C = ψ ( L, L ). Examples 1.15. (a) Any Lie algebra L has many uninteresting central extensions.One can simply take the direct product of L with an abelian Lie algebra, i.e., L × C with product [( l , c ) , ( l , c )] = ([ l , l ] ,
0) for l i ∈ L , c i ∈ C , and consider thecanonical projection pr L : L × L → L , which is a central extension (but not acovering, unless L is perfect and C = { } ). Observe that the canonical inclusioninc : L → L × C is a section of pr L , not only in the category of vector spaces, buteven in the category of Lie algebras. Its associated 2-cocycle ψ inc = 0.(b) Let h : L → C be a linear map into some vector space C . Then β h : L × L → C , β h ( l , l ) = h ([ l , l ]) is a 2-cocycle, a so-called 2 -coboundary .The two examples are related in the following exercise. Exercise 1.16.
For a central extension f : K → L of L with C = Ker f thefollowing are equivalent:(i) The extension f : K → L is split in the category of Lie algebras, i.e, thereexists a section L → K of f , which is a Lie algebra homomorphism.(ii) For any section s of f the associated 2-cocycle ψ s is a 2-coboundary.(iii) There exists a section s of f , for which the associated 2-cocycle ψ s is a2-coboundary.(iv) As central extension, f is isomorphic to the central extension pr L : L ⊕ C → L .If these conditions are fulfilled, one calls f a split extension . Exercise 1.17.
Let ψ : L × L → C be a 2-cocycle and let π : C → C ′ be a linearmap. Show:(a) ψ ′ = π ◦ ψ is a 2-cocycle of L and the mapE( π ) : E( L, C, ψ ) → E( L, C ′ , ψ ′ ) , l ⊕ c l ⊕ π ( c )is a homomorphism of central extensions of L :E( L, C, ψ ) E( π ) / / pr L $ $ IIIIIIIIII E( L, C ′ , ψ ′ ) pr L y y tttttttttt L (b) If π is surjective, the map E( π ) is a central extension of L ′ = E( L, C ′ , π ◦ ψ ),which as central extension of L ′ has the form E( L ′ , C ′′ , ψ ′′ ) for ψ ′′ ( l ⊕ c ′ , l ⊕ c ′ ) = (cid:0) (Id − γ ◦ π ) ◦ ψ (cid:1) ( l , l ) , where γ : C ′ → C is a section of π with γ ( C ′ ) = C ′′ .(c) Conversely, suppose f ′ : L ′ → L is a central extension and f : E( L, C, ψ ) ։ L ′ is a surjective homomorphism of central extensions. Then π = f | C maps C onto C ′ = Ker f ′ and there exists a unique isomorphism of extensions Φ : L ′ → E( L, C ′ , ψ ′ ), ψ ′ = π ◦ ψ such that all triangles in the diagram below commute:E( L, C, ψ ) f / / E( π ) ' ' OOOOOOOOOOO pr L (cid:31) (cid:31) ??????????????????? L ′ Φ z z tttttttttt f ′ (cid:3) (cid:3) (cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6) E( L, C ′ , ψ ′ ) pr L (cid:15) (cid:15) L Exercise 1.18.
Let C = C ⊕ C be a vector space direct sum and denote by π i : C → C i the canonical projections. Let ψ : L × L → C be a 2-cocycle with theproperty that π ◦ ψ is a 2-coboundary. Then for ψ = π ◦ ψ ,E( L, C, ψ ) ∼ = E( L, C , ψ ) × C as central extensions of L (even as central extensions of the Lie algebra E( L, C , ψ )). ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 13
Example 1.19.
Let ( ·|· ) : L × L → F be a symmetric bilinear form, which isinvariant, see Ex. 1.6. We denote by SDer F ( L ) the subalgebra of Der F ( L ) whichconsists of all skew-symmetric derivations, where a derivation d ∈ Der F ( L ) is called skew-symmetric if ( d ( l ) | l ) + ( l | d ( l )) = 0 for all l , l ∈ L . Observe thatIDer( L ) = { ad l : l ∈ L } ⊳ SDer F ( L ) . Let D be a subspace of SDer F ( L ) and let D ∗ be its dual space. Then the map ψ D : L × L → D ∗ , given by(1.24) ψ D ( l , l )( d ) = (cid:0) d ( l ) | l )for l i ∈ L and d ∈ D , is a 2-cocycle of L . Exercise 1.20.
Show: (a) (1.24) defines indeed a 2-cocycle.(b) ψ D for D ⊂ IDer( L ) is a 2-coboundary.(c) If ˜ D is a subspace of D , then the central extension E( L, D ∗ , ψ D ) of L factorsthrough the central extension E( L, ˜ D ∗ , ψ ˜ D ) of L ,E( L, D ∗ , ψ D ) ։ E( L, ˜ D ∗ , ψ ˜ D ) ։ L. Definition 1.21 (Graded central extensions) . Let Λ be an abelian group, and let L = L λ ∈ Λ L λ be a Λ-graded Lie algebra. We say that f : K → L is a Λ -gradedcentral extension of L if K is a Λ-graded Lie algebra and f is a central extensionwhich is at the same time a homomorphism of Λ-graded algebras: f ( K λ ) ⊂ L λ forall λ ∈ Λ. A Λ-graded central extension f : K → L is called a Λ -covering , if f is acovering, i.e., K is perfect. We note that an arbitrary central extension of a gradedLie algebra need not be a graded central extension.A homomorphism of a Λ-graded central extension f : K → L to another Λ-graded central extension f ′ : K ′ → L is a homomorphism g : K → K ′ of Λ-gradedLie algebras satisfying f = f ′ ◦ g , cf. 1.20.To define graded central extensions of a Λ-graded Lie algebra L via a 2-cocycle,we need (obviously) a Λ -graded -cocycle , i.e., a 2-cocycle ψ : L × L → C into aΛ-graded vector space C = L λ ∈ Λ C λ which is graded of degree 0, ψ ( L λ , L µ ) ⊂ C λ + µ for all λ, µ .For a graded 2-cocycle ψ the Lie algebra K = L ⊕ C of (1.22) is naturally Λ-gradedby K λ = L λ ⊕ C λ and the central extension pr L : K → L is a Λ-graded central extension. Conversely,if f : K → L is a Λ-graded central extension, we can choose a section s : L → K of the underlying vector spaces of degree 0, meaning s ( L λ ) ⊂ K λ . The 2-cocycleassociated to s in (1.23) is then a graded 2-cocycle. Thus, Prop. 1.13 holds in ananalogous way for graded central extensions.The following proposition also shows that one does not have to introduce a newobject of a “graded universal central extension”. Proposition 1.22 ([Ne2, 1.16]) . Let L = L λ ∈ Λ L λ be a Λ -graded perfect Liealgebra. Then its universal central extension u : uce ( L ) → L is Λ -graded, hence a Λ -covering. Moreover, Ker u is a graded subspace of uce ( L ) . Example 1.23.
We also have the graded versions of the Example 1.15 (details leftto the reader) and the Example 1.19, whose details follow.
Let L = L λ ∈ Λ L λ be a Λ-graded Lie algebra and let ( ·|· ) be an invariant bilinearform on L , which is Λ -graded in the following sense:( L λ | L µ ) = 0 if λ + µ = 0 . We define the Λ-graded subalgebra of grEnd F ( L )(1.25) grSDer F ( L ) = grEnd F ( L ) ∩ SDer F ( L ) = L λ ∈ Λ (cid:0) SDer F ( L ) (cid:1) λ where (SDer F ( L )) λ consists of all skew-symmetric derivations of degree λ . If D ⊂ grSDer F ( L ) is a graded subspace of grSDer F ( L ), the 2-cocycle ψ D of (1.24) is Λ-graded and maps L × L into D gr ∗ , thus giving rise to a graded central extensionE( L, D gr ∗ , ψ D ) of L . Exercise 1.24.
Show that the 2-cocycles ψ of (1.5), (1.16) and (1.17) can beobtained in the form (1.24), i.e., find an invariant bilinear form on L = L ( g , σ )resp. L = L ( g , σ ) and a subspace D ⊂ SDer F ( L ) such that ψ and ψ D yieldisomorphic central extensions of L .It is not so surprising that the 2-cocycles we used in sections 1.1 and 1.2 canall be obtained in the form ψ D for D ⊂ grSDer F ( L ). This is a special case of thefollowing general result. Theorem 1.25 ([Ne6]) . Let L = L λ ∈ Λ L λ be a Λ -graded Lie algebra, which (i) is perfect and finitely generated as Lie algebra, (ii) has finite homogeneous dimension: dim L λ < ∞ for all λ ∈ Λ , and (iii) has an invariant nondegenerate Λ -graded symmetric bilinear form. (a) Then
Der F ( L ) = grDer F ( L ) is Λ -graded and has finite homogeneous dimen-sion, whence the same is true for SDer F ( L ) . (b) The universal central extension uce ( L ) has finite homogenous dimension withrespect to the Λ -grading of . Moreover, uce ( L ) ∼ = E( L, D gr ∗ , ψ D ) as central extensions of L , where D is any graded subspace of SDer F ( L ) whichcomplements IDer( L ) in SDer F ( L ) , and ψ D is the -cocycle of (1.24) . Remarks 1.26. (a) Th. 1.5 is an application of Th. 1.25, as is Th. 4.13(c).(b) The Exercise 1.18 gives some indication why it is sufficient to take a subspaceof SDer F ( L ) complementing IDer( L ) and not an arbitrary subspace of SDer F ( L ). Exercise 1.27.
In the setting of Th. 1.25, every Λ-graded central covering of L is isomorphic as central extension to a central extension E( L, B gr ∗ , ψ B ) for somegraded subspace B of D .2. Extended affine Lie algebras: Definition and first examples
Rather than constructing Lie algebras in a concrete way as we have done inLecture 1, in this chapter we will define extended affine Lie algebras by a set ofaxioms and give examples. We will see that these examples encompass all theexamples of Lecture 1 (with the exception of the choice 2. for D in 1.2).As before we will consider Lie algebras over an arbitrary field F of characteristic0, but we will no longer assume that F has enough roots of unity (multiloop algebraswill not be play a role here), except in § F = C ). ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 15
Definition of an extended affine Lie algebra. An extended affine Liealgebra , or EALA for short, is a pair ( E, H ) consisting of a Lie algebra E over F and subalgebra H satisfying the following axioms (EA1) – (EA6). (EA1): E has an invariant nondegenerate symmetric bilinear form ( ·|· ) . (EA2): H is nontrivial finite-dimensional toral and self-centralizing subalge-bra of E . Before we can state the other four axioms, we need to draw some consequences ofthe axioms (EA1) and (EA2). But first we give explanations of some of the notionsused. The term invariant (= associative ) means that ( ·|· ) satisfies ([ e , e ] | e ) =( e | [ e , e ]) for all e i ∈ E , and ( ·|· ) is nondegenerate if ( e | E ) = 0 = ⇒ e = 0. Inthe context of above, a toral subalgebra , sometimes also called an ad -diagonalizablesubalgebra is a subalgebra H which induces a decomposition of E via the adjointrepresentation of H : E = L α ∈ H ∗ E α ,E α = { e ∈ E : [ h, e ] = α ( h ) e for all h ∈ H } . (2.1)Such a subalgebra is necessarily abelian, whence H ⊂ E = { e ∈ E : [ h, e ] =0 for all h ∈ H } . That H is also required to be self-centralizing means H = E . Now to the consequences of (EA1) and (EA2). Because of invariance of the bilinearform ( ·|· ), we have(2.2) ( E α | E β ) = 0 if α + β = 0 , in particular the restriction of the bilinear form to E = H is nondegenerate. Be-cause of this and finite-dimensionality of H , every linear form α ∈ H ∗ is representedby a unique t α ∈ H , defined by the condition that ( t α | h ) = α ( h ) holds for all h ∈ H . This allows us to transport the restricted form ( ·|· ) | H × H to a symmetricbilinear form on H ∗ , also denoted ( ·|· ) and defined by(2.3) ( α | β ) = ( t α | t β ) , α, β ∈ H ∗ . This transport of bilinear forms is a standard procedure in the theory of semisimpleLie algebras, see for example [Hu, § R = { α ∈ H ∗ : E α = 0 } ( set of roots of ( E, H )) ,R = { α ∈ R : ( α | α ) = 0 } ( null roots ) ,R an = { α ∈ R : ( α | α ) = 0 } ( anisotropic roots ) . (2.4)We prefer to call R the set of roots of ( E, H ) and not the “root system” since wewant to restrict the latter term for root systems in the usual sense, see 3.3. Wepoint out that by definition 0 is a root,0 ∈ R ⊂ R. This is the customary convention for EALAs and has some notational advantages.We define the core of ( E, H ) as the subalgebra E c of E generated by all anisotropicroot spaces: E c = h S α ∈ R an E α i subalg We can now state the remaining four axioms. (EA3):
For every α ∈ R an and x α ∈ E α , the operator ad x α is locally nilpo-tent on E . (EA4): R an is connected in the sense that for any decomposition R an = R ∪ R with ( R | R ) = 0 we have R = ∅ or R = ∅ . (EA5): The centralizer of the core E c of E is contained in E c : { e ∈ E :[ e, E c ] = 0 } ⊂ E c . (EA6): The subgroup
Λ = span Z ( R ) ⊂ H ∗ generated by R in ( H ∗ , +) is afree abelian group of finite rank. In other words, Λ ∼ = Z n for some n ∈ N (including n = 0!).The term locally nilpotent means that for every e ∈ E there exists an n ∈ N ,possibly depending on e , such that (ad x α ) n ( e ) = 0. The property (EA5) is called tameness . The condition [ e, E c ] = 0 is of course equivalent to [ e, E α ] = 0 for all α ∈ R an . The rationale for this axiom is the following. The subalgebra E c is infact an ideal of E (Th. 4.14). Hence we have a representation ρ of E on E c , givenby ρ ( e )( x c ) = [ e, x c ] for e ∈ E and x c ∈ E c . The kernel of the representation ρ isthe centralizer of E c in E . Hence tameness means that Ker ρ ⊂ E c . The idea hereis that the core E c should control E . We will make this more precise in section5.4. The rank of the free abelian group Λ in axiom (EA6) is called the nullity of( E, H ). It is invariant under isomorphisms. We will describe EALAs of nullity 0and 1 below.Although the structure of an EALA requires the existence of an invariant non-degenerate symmetric bilinear form ( ·|· ) in the axiom (EA1), which is then usedto define the anisotropic roots, it turns out that this bilinear form is really not soimportant. Because of this, we have defined an EALA as a pair (
E, H ) and not as atriple (
E, H, ( ·|· )) as it is for example done in [AF]. Consequently, an isomorphism from an EALA ( E, H ) to another EALA ( E ′ , H ′ ) is a Lie algebra isomorphism f : E → E ′ such that f ( H ) = H ′ . It is immediate that any isomorphism induces abijection f ′ between the set of roots R and R ′ of ( E, H ) and ( E ′ , H ′ ) respectively.It then follows that f ′ maps R an onto R ′ an , whence also R onto R ′ . One canthen show that f ′ preserves the forms on X = span F ( R ) and X ′ = span F ( R ′ ) upto scalars.For F = C one can define a special class of EALAs. We call a pair ( E, H ) a discrete EALA if it satisfies the axioms (EA1) – (EA5) and in addition (DE): R is a discrete subset of H ∗ with respect to the natural topology ofthe finite-dimensional complex vector space H ∗ .It is justified to call a discrete EALA an EALA, since one can show that a discreteEALA also satisfies (EA6). Indeed, this follows from Prop. 3.21 and Th. 3.22.However, not every EALA over C is a discrete EALA (see [Ne5, 6.17]). Some historical comments.
Although there were some precursors (papersby Saito and Slodowy for nullity 2), it was in the paper [HT] by the physicistsHøegh-Krohn and Torr´esani that the class of discrete extended affine Lie algebraswas introduced, however not under this name. Rather, they were called “irreduciblequasi-simple Lie algebras” and later ([BGK, BGKN]) “elliptic quasi-simple Lie al-gebras”. The stated goal of the paper [HT] was applications in quantum gaugetheory. The theory developed there did however not stand up to the scrutiny ofmathematicians. The errors of [HT] were corrected in the AMS memoir [AABGP]
ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 17 by Allison, Azam, Berman, Gao and Pianzola. There also the name “extendedaffine Lie algebras” appears for the first time. But not in the sense as definedabove. Rather, the authors develop the basic theory of what here are called dis-crete EALAs. Nevertheless, [AABGP] has become the standard reference even forthe more general extended affine Lie algebras, since many of the results presentedthere for discrete extended affine Lie algebras easily extend to the more generalsetting. The definition of an extended affine Lie algebra given above is due to theauthor ([Ne4]) and was motivated by the fact that all the examples presented in[AABGP] did make sense over an arbitrary base field F and not just over C only.Before [Ne4] the tameness axiom (EA5) was not part of the definition of an EALA.However, as examples show ([BGK, §
3] or [Ne5, 6.10]), it seems impossible to clas-sify EALAs without (EA5). After [Ne4], several generalizations of EALAs havebeen proposed. They are surveyed in [Ne5].2.2.
Some elementary properties of extended affine Lie algebras.
The fol-lowing chapters will (hopefully) show that extended affine Lie algebras share manyproperties with familiar Lie algebras, like finite-dimensional split simple Lie alge-bras or affine Kac-Moody Lie algebras. Some of these properties are immediateconsequences of the axioms. The following (strongly recommended!) exercise givesan incomplete list of such properties.
Exercise 2.1.
Let (
E, H ) be an EALA. We use the notation of above. Show:(a) For α, β ∈ R we have(2.5) [ E α , E β ] ⊂ E α + β . Thus the root space decomposition (2.1) is a grading by the abelian groupspan Z ( R ).(b) H is a Cartan subalgebra , defined as a nilpotent subalgebra which is self-normalizing: H = { e ∈ E : [ e, H ] ⊂ H } .(c) For α, β ∈ R we have ( E α | E β ) = 0 unless α + β = 0. The restriction of thebilinear form ( ·|· ) to E α × E − α is nondegenerate, i.e., if x α ∈ E α satisfies( x α | E − α ) = 0 then x α = 0. In particular, R = − R .(d) For α ∈ R and x α ∈ E α and y − α ∈ E − α ,(2.6) [ x α , y − α ] = ( x α | y − α ) t α . In particular, [ E α , E − α ] = F t α , and if α ∈ R an then(2.7) [[ E α , E − α ] , E α ] = E α . (e) The core E c satisfies(2.8) E c = (cid:16) ⊕ α ∈ R an E α (cid:17) ⊕ (cid:16) L α ∈ R ( E c ∩ E α ) (cid:17) . We will now show that EALAs are built out of “little” sl ’s and Heisenberg’s(albeit in a complicated way). Proposition 2.2.
Let ( E, H ) be an extended affine Lie algebra, with anisotropicroot R an and null roots R . (a) Let α ∈ R an . Then dim E α = 1 , and for any e α ∈ E α there exists f α ∈ E − α such that ( e α , h α = [ e α , f α ] , f α ) ∈ E α × H × E − α is an sl -triple: E α ⊕ [ E α , E − α ] ⊕ E − α = F e α ⊕ F h α ⊕ F f α ∼ = sl ( F ) . (b) Let α ∈ R . Then for any = x α ∈ E α there exists y α ∈ E − α such that [ x α , y α ] = t α and F x α ⊕ F t α ⊕ F y α ∼ = h , the -dimensional Heisenberg algebra. It is not true that dim E α = 1 if α ∈ R (this is already not true in the examplesof sections 2.3 and 2.4). But we will show in Th. 4.18 that all root spaces E α arefinite-dimensional in a rather strong way.The following exercise shows that one can “extend” the 3-dimensional Heisenbergsubalgebras in (b) above. Exercise 2.3.
In the setting and notation of Prop. 2.2(b) show that there exists d α ∈ H such that [ d α , x α ] = x α and [ d α , y α ] = − y α . Hence
F x α ⊕ F t α ⊕ F d α ⊕ F y α is a 4-dimensional subalgebra. It is 2-step solvable, not nilpotent and isomorphicto the subalgebra n (cid:16) a b c d (cid:17) : a, b, c, d ∈ F o of gl ( F ).And now an exercise which implies that in an EALA one can produce manyso-called elementary automorphisms. Exercise 2.4.
Let M be an F -vector space. Once calls an endomorphism f ∈ End F ( M ) locally nilpotent if for every m ∈ M there exists n ∈ N , possibly depend-ing on m , such that f n ( m ) = 0.(a) Show that the following conditions are equivalent for f ∈ End F ( M ):(i) f is locally nilpotent,(ii) for every finitely spanned subspace N of M there exists a finite-dimensional subspace P of M such that N ⊂ P and f ( P ) ⊂ P ,(iii) f is nilpotent on every finite-dimensional and f -invariant subspace of M .(b) Let f ∈ End F ( M ) be locally nilpotent and define the exponential exp f of f by (exp f )( m ) = P n ∈ N n ! f n ( m ) , for m ∈ M (note that the sum on the right is always finite). Show that exp f is aninvertible endomorphism of M with inverse given by (exp f ) − = exp( − f ).(c) Let L be a Lie algebra and let d be a locally nilpotent derivation of L . Showthat then exp d is an automorphism of L .We will next present some examples of EALAs. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 19
Extended affine Lie algebras of nullity . Let g be a finite-dimensionalsplit simple Lie algebra with splitting Cartan subalgebra h , for example sl l ( F ) ora finite-dimensional simple Lie algebra over an algebraically closed field. We willshow that then ( g , h ) is an EALA of nullity 0. The facts needed to prove this canbe found in [Bou3, VIII, §
2] or in [Hu, §
8] for F algebraically closed.(EA1) Up to a scalar, there exists only one invariant nondegenerate symmetricbilinear form on g , the Killing form κ . Hence we can (and will) take ( ·|· ) = κ .(EA2) By definition of a splitting Cartan subalgebra, the Lie algebra g has aroot space decomposition g = g ⊕ (cid:0) L α ∈ Φ g α (cid:1) , g = h , where Φ is the root system of ( g , h ) (which is a reduced root system in the usualsense, see 3.3) and where the root spaces g α are defined as in 2.1. Hence the set ofroots R of ( g , h ) is(2.9) R = { } ∪ Φ . It is a basic fact that κ ( t α , t α ) = 0 for t α ∈ h representing α ∈ Φ via κ ( t α , h ) = α ( h )for all h ∈ h . Hence, the anisotropic and null roots are R an = Φ and R = { } . (EA3) is now obvious: From [ g α , g β ] ⊂ g α + β for α ∈ Φ and β ∈ R and finite-dimensionality of g , it is clear that ad x α for x α ∈ g α is not only locally nilpotentbut even (globally) nilpotent.(EA4) is another way of saying that Φ is an irreducible root system. This isindeed the case and follows from simplicity of g .(EA5) We first need to determine the core g c of g . By definition, g c is thesubalgebra of g generated by L α ∈ Φ g α . Since h = P α ∈ Φ [ g α , g − α ] we have g c = g . It is now a tautology that (EA5) holds, i.e., that the centralizer of the core g c iscontained in g c = g . Of course, we know even more: The centralizer of the coreequals the centre of g , and is therefore { } .(EA6) We have Λ = h R i = h{ }i = { } .We have now shown:(2.10) A finite-dimensional split simple Lie algebra is an EALA of nullity . We will see in Prop. 3.24 that the converse of (2.10) is true too. We thus knowall the nullity 0 examples of EALAs, and can therefore focus on the higher nullityexamples. We will answer the case of nullity 1 in the next section.2.4.
Affine Kac-Moody Lie algebras again.
To justify the name extended affine Lie algebra , we will now show that any affine Kac-Moody Lie algebra isan extended affine Lie algebra. To do so, we will need some basic facts about affineKac-Moody Lie algebras. All of them can be found in Kac’s book [Kac]. Since thisreference uses C as base field, we will do the same in this section. But everythingwe say here holds true for arbitrary algebraically closed fields of characteristic 0.Thus we let L = L n ∈ Z g ¯ n ⊗ C t n ˆ L = ˆ L ( g , σ ) = L ⊕ C c ⊕ C d be the complex Lie algebra described in (1.9) and (1.10). Recall that g is a finite-dimensional simple Lie algebra over C and σ is a diagram automorphism of g . Welet m ∈ { , , } be the order of σ , and denote the canonical map Z → Z /m Z by n ¯ n . Recall from (1.1) and (1.2) that σ induces a Z /m Z -grading of g , namely g = g ¯0 ⊕ · · · ⊕ g m − where g ¯ n = { x ∈ g : σ ( x ) = ζ n x } for a primitive m th root of unity ζ . For example,for m = 2 we get a Z / Z -grading g = g ¯0 ⊕ g ¯1 with g ¯0 = { x ∈ g : σ ( x ) = x } and g ¯1 = { x ∈ g : σ ( x ) = − x } . We identify g ¯0 ≡ g ¯0 ⊗ C t .We now verify the axioms (EA1) – (EA5) and (DE) which, we recall, implies(EA6).(EA1) We let κ be the Killing form of g and define a bilinear form ( ·|· ) on ˆ L ,using the notation of (1.10), (cid:0) u ¯ λ ⊗ t λ ⊕ s c ⊕ s ′ d | v ¯ µ ⊗ t µ ⊕ s c ⊕ s ′ d (cid:1) = κ ( u ¯ λ , v ¯ µ ) δ λ, − µ + s s ′ + s s ′ . (2.11)The form is visibly symmetric. The reader is invited in Exercise 2.6 to show thatit is in fact an invariant nondegenerate symmetric bilinear form on ˆ L , as requiredin (EA1). In anticipation of the later developments, we point out that ( ·|· ) has thefollowing features: • ˆ L is an orthogonal sum of L and C c ⊕ C d : ˆ L = L ⊥ ( C c ⊕ C d ), • C c ⊕ C d is a hyperbolic plane, i.e., ( c | c ) = 0 = ( d | d ) while ( c | d ) = 1. • The Laurent polynomial ring C [ t ± ] has a nondegenerate symmetric bilinearform ǫ given by ǫ ( t λ , t µ ) = δ λ, − µ . It is invariant in the sense that ǫ ( pq, r ) = ǫ ( p, qr ) for p, q, r ∈ C [ t ± ], and is graded in the sense that ǫ ( t λ , t µ ) = 0unless λ + µ = 0. For σ = Id g , the bilinear form on the loop algebra g ⊗ C [ t ± ] is simply the tensor product form κ ⊗ ǫ , and for a general σ theform is obtained by restriction.(EA2) To construct a subalgebra H as required in axiom (EA2) we start with aCartan subalgebra h of g . Since σ is a diagram automorphism, it leaves h invariant.We let h ¯0 = h ∩ g ¯0 = { h ∈ h : σ ( h ) = h } and put H = h ¯0 ⊕ C c ⊕ C d. One knows that g ¯0 is a simple Lie algebra with Cartan subalgebra h ¯0 ([Kac,Prop. 7.9]). The grading property implies that [ g ¯0 , g ¯ n ] ⊂ g ¯ n for n ∈ Z . Hence g ¯0 acts on g ¯ n by the adjoint action. Let ∆ ¯ n be the set of weights of the g ¯0 -module g ¯ n with respect to h ¯0 : g ¯ n = L γ ∈ ∆ ¯ n g ¯ n,γ g ¯ n,γ = { x ∈ g ¯ n : [ h ¯0 , x ] = γ ( h ¯0 ) x for all h ¯0 ∈ h ¯0 } . In particular, ∆ ¯0 \ { } is the root system of g ¯0 with respect to h ¯0 and h ¯0 = g ¯0 , .We extend ∆ ¯ n ⊂ h ∗ ¯0 to a linear form on H by zero, i.e., for γ ∈ ∆ ¯ n we put γ ( h ¯0 ⊕ sc ⊕ s ′ d ) = γ ( h ¯0 )and define a linear form δ on H by δ ( h ¯0 ⊕ sc ⊕ s ′ d ) = s ′ . ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 21
Then for γ ∈ ∆ ¯ n , n ∈ Z , we haveˆ L γ ⊕ nδ = { u ∈ ˆ L : [ h, u ] = ( γ ⊕ nδ )( h ) u for all h ∈ H } = ( g ¯ n,γ ⊗ t n , γ ⊕ nδ = 0 ,H, γ ⊕ nδ = 0 , (2.12)whence ˆ L = L α ∈ R ˆ L α has a root space decomposition with respect to H with setof roots(2.13) R = { γ ⊕ nδ : γ ∈ ∆ ¯ s , ¯ n = ¯ s, ≤ s < m } . This establishes (EA2).To check the other axioms we first need to determine which of the roots in R arethe null respectively anisotropic roots. Following the procedure in § ·|· ) to H . With obvious notation this is (cid:0) h ¯0 ⊕ s c ⊕ s ′ d | h ′ ¯0 ⊕ s c ⊕ s ′ d ) = κ ( h ¯0 , h ′ ¯0 ) + s s ′ + s s ′ . Since κ | h ¯0 × h ¯0 is nondegenerate, this is indeed a nondegenerate symmetric bilinearform on H , as it should be. Let t γ ∈ h ¯0 be the element representing γ ∈ h ∗ ¯0 : κ ( t γ , h ¯0 ) = γ ( h ¯0 ) for all h ¯0 ∈ h ¯0 . For the canonical extension of γ to a linear formof H , also denoted by γ , we then get ( t γ | h ) = γ ( h ) for all h ∈ H . Moreover( c | h ¯0 ⊕ sc ⊕ s ′ d ) = s ′ = δ ( h ¯0 ⊕ sc ⊕ s ′ d ) shows that δ is represented by t δ = c ∈ H .Therefore α = γ ⊕ nδ ∈ R is represented by t γ ⊕ nδ = t γ ⊕ nc. Now observe ( t γ ⊕ nδ | t γ ⊕ nδ ) = ( t γ ⊕ nc | t γ ⊕ nc ) = κ ( t γ , t γ ). It is of course well-known that κ ( t γ , t γ ) = 0 for 0 = γ ∈ ∆ ¯0 . But one can (easily) show that this alsoholds for any 0 = γ ∈ ∆ ¯ n . We therefore get(2.14) R an = { γ ⊕ nδ ∈ R : γ = 0 } and R = Z δ, which in the theory of affine Kac-Moody algebras are usually called real and imag-inary roots . We are now set for the verification of the remaining axioms.(EA3) holds in the stronger form: ad ˆ L λ , α ∈ R an , is nilpotent. (We have alreadyseen the same phenomenon in the Example 2.3 of a finite-dimensional split simpleLie algebra. Perhaps the reader wonders if this is true in general. The answer isyes.)(EA4) The verification of (EA4) is left to the reader.(EA5) The core of ˆ L is ˆ L c = (cid:0) L n ∈ Z g ¯ n ⊗ C t n (cid:1) ⊕ C c , and therefore equals thederived algebra [ ˆ L , ˆ L ] of ˆ L . The centralizer of ˆ L c in ˆ L , in fact the centre of ˆ L is C c ⊂ ˆ L c , see Exercise 1.2.(DE) In this example the subgroup Λ = h R i equals R = Z d and is a discretesubset of H ∗ .We have now shown one implication of the following result. Theorem 2.5 ([ABGP]) . A complex Lie algebra E is a discrete EALA of nullity if and only if E is an affine Kac-Moody Lie algebra. Exercise 2.6.
Check the following details of the construction above.(a) (2.11) defines an invariant symmetric bilinear form on ˆ L .(b) ˆ L has a root space decomposition whose root spaces are given by (2.12) andwhose set of roots is (2.13). (c) κ ( t γ , t γ ) = 0 for any 0 = γ ∈ ∆ ¯ n .(d) (EA4) holds for ( ˆ L , H ).2.5. Higher nullity examples.
We have seen all examples of EALAs of nullity0 and 1. In this section we will construct examples of higher nullity. To simplifythings we consider untwisted algebras (no non-trivial finite order automorphism areinvolved). We can therefore go back to our standard setting: g is a split simple Liealgebra over a field F of characteristic 0.As in § F [ t ± , . . . , t ± nn ] be the Laurent polynomial ring in n variables andlet L = L ( g ) = g ⊗ F [ t ± , . . . , t ± nn ]be the associated untwisted multiloop algebra. We have seen in Exercise 1.4 that L has a 2-cocycle ψ : L × L → F n =: C , given by (1.16): ψ ( u ⊗ t λ , v ⊗ t µ ) = δ λ + µ, κ ( u, v ) λ . We can therefore define the central extension K = L ⊕ C with product (1.22). In (1.19) we have defined degree derivations ∂ i , i = 1 , . . . , n ,of K . Let(2.15) D = span F { ∂ , . . . , ∂ n } and define the Lie algebra E as the semidirect product, E = (cid:0) L ( g ) ⊕ C (cid:1) ⋊ D . Let h be a Cartan subalgebra of g and put H = h ⊕ C ⊕ D . We claim that (
E, H ) is an EALA of nullity n . (EA1) We will mimic the construction of an invariant nondegenerate symmetricbilinear form in § • ( L ( g ) | C ⊕ D ) = 0. • C ⊕ D is a hyperbolic space with ( C | C ) = 0 = ( D | D ) and( P i s i c i | P i s ′ i ∂ i ) = P i s i s ′ i , where c , . . . , c n is the canonical basis of F n . Thus C ⊕ D is the orthogonalsum of the n hyperbolic planes F c i ⊕ F ∂ i . • On L ( g ) the form is the tensor product form of the Killing form κ of g andthe natural invariant bilinear form on F [ t ± , . . . , t ± n ].Putting all these requirements together, we arrive at the global formula which iscompletely analogous to (2.11): (cid:0) u ⊗ t λ ⊕ P i s i c i ⊕ P j s ′ j ∂ j | v ⊗ t µ ⊕ P i t i c i ⊕ P j t ′ j ∂ j (cid:1) = κ ( u, v ) δ λ, − µ + P i ( s i t ′ i + t i s ′ i ) . (2.16)(EA2) Let h be a splitting Cartan subalgebra and let Φ be the usual root systemof ( g , h ), thus 0 Φ. We put ∆ = { } ∪ Φ and then have the root space decompo-sition g = L γ ∈ ∆ g γ with g = h . We embed ∆ ֒ → H ∗ by requiring γ | C ⊕ D = 0for γ ∈ ∆. Also we embed Λ = Z n ֒ → H ∗ by λ ( h ⊕ C ) = 0 and λ ( ∂ i ) = λ i for ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 23 λ = ( λ , . . . , λ n ) ∈ Λ. Then E has the root space decomposition E = L α ∈ R E α with root spaces E γ ⊕ λ = g γ ⊗ t λ ( γ ⊕ λ = 0) , E = H, (2.17) R an = Φ × Λ , R = Λ . (2.18)It is now not difficult to verify (EA3) – (EA5) and (DE). Thus: Lemma 2.7.
The pair ( E, H ) constructed above is a discrete EALA of nullity n . There is however no analogue of Prop. 2.10 and Th. 2.5: There are many moreEALAs of nullity n ≥
2. We have just seen the “tip of the iceberg”! Otherexamples can be found in Ch.III of [AABGP], some of them involving heavy-dutynonassociative algebras, like octonion algebras and Jordan algebras over Laurentpolynomial rings!
Exercise 2.8.
Supply the missing details of the proof that (
E, H ) above is adiscrete EALA of nullity n . In particular, prove:(a) (2.16) defines an invariant nondegenerate symmetric bilinear form on E .(b) The root spaces of ( E, H ) and the anisotropic and null roots are as statedin (2.17) and (2.18).3.
The structure of the roots of an EALA
In this chapter we will describe the structure of the set of roots R of an EALA( E, H ), defined in (2.4). We have already seen some examples: R can be a finiteirreducible reduced root system, (2.9), R can be an affine root system (2.13), i.e.,the set of roots of an affine Kac-Moody Lie algebra, or R can be of the form R = S × Z n where S \ { } is a finite irreducible reduced root system (2.18). Thusany description of the general case has to encompass all these different examples.It turns out that the roots of an EALA form an extended affine root system andthat the latter is naturally described as a special case of affine reflection systems.We therefore first introduce the latter, describe their structure and then specializelater to extended affine root systems. Affine reflection systems are themselvesspecial cases of reflection systems, whose theory is developed in [LN].3.1. Affine reflection systems: Definition.
Throughout this section we workwith a triple (
R, X, ( ·|· )) where • X is a finite-dimensional vector space over a field F of characteristic 0, • ( ·|· ) is a symmetric bilinear form on X and • R ⊂ X .For any such triple ( R, X, ( ·|· )) we define X = { x ∈ X : ( x | X ) = 0 } , the radical of ( ·|· ) ,R = { α ∈ R : ( α | α ) = 0 } , ( null roots ) R an = { α ∈ R : ( α | α ) = 0 } , ( anisotropic roots ) h x, α ∨ i = 2 ( x | α )( α | α ) , ( x ∈ X and α ∈ R an ) s α ( x ) = x − h x, α ∨ i α. (3.1) By definition we therefore have R = R ∪ R an . The map s α : X → X is a reflectionin α , i.e., s α = Id X and { x ∈ X : s α ( x ) = − x } = F α . It is also orthogonal withrespect to ( ·|· ): ( s α ( x ) | s α ( y )) = ( x | y ) for all x, y ∈ X .We call ( R, X, ( ·|· )), or just R for short, an affine reflection system if (AR1): ∈ R and R spans X , (AR2): s α ( R ) = R for all α ∈ R an , (AR3): for every α ∈ R an the set h R, α ∨ i is finite and contained in Z , and (AR4): R = R ∩ X . An affine reflection system is said to be • reduced if for every α ∈ R an and c ∈ F : cα ∈ R an ⇐⇒ c = ± • connected if for any decomposition R an = R ∪ R with ( R | R ) = 0 wehave R = ∅ or R = ∅ .The nullity of ( R, X ) is the rank of the torsion-free abelian group Z [ R ] = span Z ( R )generated by R in ( X, +). Thus, by definition,nullity of ( R, X ) = dim Q ( Z [ R ] ⊗ Z Q ) = dim F ( Z [ R ] ⊗ Z F ) . Since the vector space Z [ R ] ⊗ Z F maps onto span F ( R ), the nullity of ( R, Z )is bounded below by dim F span F ( R ). It is in general not equal to it. But thisis of course so for nullity 0: ( R, X ) has nullity 0 if and only if R = { } ⇐⇒ dim F span F ( R ) = 0. Remarks 3.1. - For a large part of the theory it is not necessary that X befinite-dimensional, see [LN]. But assuming this right from the start, simplifies thepresentation.- We need the bilinear form ( ·|· ) to define R and the reflections. But although wewill sometimes write ( R, X, ( ·|· )), we will not consider ( ·|· ) as part of the structureof an affine reflection system. For example, in the definition of an isomorphismbelow we will not require that the bilinear forms are preserved. See [LN], wherethis point of view is emphasized.- The requirement 0 ∈ R is in line with the previous chapter, in which 0 wasconsidered a root of an EALA. This conflicts with the traditional approach to rootsystems in which 0 is not a root, see for example [Bou2], [Hu] or [Kac]. The questionwhether 0 is a root or is not a root, has lead to heated debates. In the author’sopinion, there are some advantages of considering 0 as a root, which however canonly be fully seen when one develops the theory for affine reflection systems. Butperhaps the reader can be convinced by the natural) example ( R, X ) = ( { } , { } )of an affine reflection system.- The condition h β, α ∨ i ∈ Z in axiom (AR3) makes sense since every field ofcharacteristic 0 contains (an isomorphic copy of) the field of rational numbers,which allows us to identify Z ≡ Z F .- By definition h X , α ∨ i = 0 for all α ∈ R an . Hence s α ( x ) = x for x ∈ X .Also, the inclusion R ∩ X ⊂ R in (AR4) is always true. Therefore the axioms(AR2)–(AR4) can be replaced by the following conditions(AR2) ′ s α ( R an ) = R an for all α ∈ R an , (AR3) ′ for every α ∈ R an the set h R an , α ∨ i ⊂ Z is finite,(AR4) ′ R ⊂ X . This new set of axioms makes it (even more) clear that the conditions on R arerather weak: We (may) need R to span X from (AR1), we need R ⊂ X for ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 25 (AR4) ′ and we need 0 ∈ R , which is no condition since one can always add 0 to R .We will see this phenomena re-appearing in the examples, e.g., in Example 3.2, andin the definition of an extension datum in 3.9.- The definition of a connected affine reflection system is the same as the axiom(EA4) in the definition of an EALA.- The definition of an affine reflection system given in [LN] is not the same asthe one given here. The equivalence of two definitions follows from [LN, Prop. 5.4].An isomorphism from an affine reflection system ( R, X, ( ·|· )) to another affinereflection system ( R ′ , X ′ ( ·|· ) ′ ) is a vector space isomorphism f : X → X ′ satisfying f ( R an ) = R ′ an and f ( R ) = R ′ If such a map exists, (
R, X, ( ·|· )) and ( R ′ , X, ( ·|· ) ′ ) are called isomorphic . One canshow, as a corollary of the Structure Theorem 3.10, that an isomorphism f alsosatisfies f ◦ s α = s f ( α ) ◦ f for all α ∈ R an , equivalently, h x, α ∨ i = h f ( x ) , f ( α ) ∨ i for all x ∈ X and α ∈ R an . This is always fulfilled if f is an isometry for ( ·|· ) and( ·|· ) ′ respectively. But in general an isomorphism is not necessarily an isometry.For example, one can always multiply the bilinear form ( ·|· ) by a non-zero scalarwithout changing h x, α ∨ i .Since a reflection s α is an isometry, it follows from (AR2) and (AR4) that s α leaves R an and R invariant and is thus an automorphism of ( R, X ). The subgroup W ( R ) of the automorphism group of ( R, X ) generated by all reflections s α , α ∈ R an ,is (obviously) called the Weyl group of (
R, X ). (It will not play a big role in thischapter.)3.2.
Examples of affine reflection systems.
We will now give some immediateexamples of affine reflection systems.
Example 3.2 ( The real part of an affine reflections system ) . Let (
R, X, ( ·|· )) bean affine reflection system. ThenRe( R ) = { } ∪ R an , Re( X ) = span F ( R an ) , ( ·|· ) Re = ( ·|· ) Re( X ) × Re( X ) defines an affine reflection system, called the real part of ( R, X ), withRe( R ) an = R an , Re( R ) = { } , in particular Re( R ) has nullity 0.Observe that ( ·|· ) Re need not be nondegenerate, see Example 3.4 for an example.The fact that one can “throw away” the non-zero null roots and still have anaffine reflection system indicates that one has little control over the null roots in ageneral affine reflection system. This will be made even more evident in the conceptof an extension datum 3.9, used in the general Structure Theorem 3.10 for affinereflection systems. It is therefore natural to define subclasses of affine reflectionsystem by imposing conditions on the null roots. For example, we will do so whenwe define extended affine root systems in 3.4.In [Ne5, 3.6] the author claimed that an affine reflection system of nullity 0 is afinite root system. The example above show that this is far from being true. Butwhat remains true is the converse, also claimed in [Ne5, 3.6]: A finite root systemis an affine reflection system of nullity 0, as we will show now. Example 3.3 ( Finite root systems ) . Let Φ be a (finite) root system `a la Bourbaki[Bou2, VI, § F -vector space Y satisfying the axioms (RS1)–(RS3) below. (RS1): Φ is finite, 0 Φ and Φ spans Y . (RS2): For every α ∈ Φ there exists a linear form α ∨ ∈ Y ∗ such that α ∨ ( α ) =2 and s α (Φ) = Φ, where s α is the reflection of Y defined by s α ( y ) = y − α ∨ ( y ) α , (RS3): for every α ∈ Φ the set α ∨ (Φ) is contained in Z .Observe that the reflection s α defined in (RS2) satisfies s α ( α ) = − α and s α ( y ) = y for α ∨ ( y ) = 0. It therefore seems to depend on α and the linear form α ∨ . However,since Φ is finite, there exists at most one reflection s with s (Φ) = Φ and s ( α ) = − α ([Bou2, VI, § α ∨ in thenotation of s α .Note that we do not assume that Φ is reduced. This more general concept of aroot system is necessary for the Structure Theorem of affine root systems (3.10).The reader who is only familiar with the theory of reduced finite root systems,as for example developed in [Hu, Ch. III], can perhaps be comforted by the factthat the difference is not very big. Indeed, every finite root system is a directsum of connected (= irreducible) root systems and there is only one irreduciblenon-reduced root system of rank l , namelyBC l = B l ∪ C l = {± ε i : 1 ≤ i ≤ l } ∪ {± ε i ± ε j : 1 ≤ i, j ≤ l } where here and in the following ε , . . . , ε l is the standard basis of F l . (Note 0 ∈ BC l in anticipation of the convention introduced below.)In the context of finite-dimensional Lie algebras, non-reduced root systems arisenaturally as the roots of a finite-dimensional semisimple Lie algebra L with respectto a maximal ad-diagonalizable subalgebra H ⊂ L which is not self-centralizing,hence not a Cartan algebra. In particular, non-reduced root systems do not occurover an algebraically closed field. However, they do occur in the context of infinite-dimensional Lie algebras, even over algebraically closed fields, see Ex. 3.6.Given a finite root system (Φ , Y ), define(3.2) S = { } ∪ Φ and ( x | y ) = P α ∈ Φ α ∨ ( x ) α ∨ ( y )for x, y ∈ Y . Then ( ·|· ) is a nondegenerate symmetric bilinear form on Y with re-spect to which all reflections s α are isometric ([Bou2, VI, § α | α ) is a positive integer for every α ∈ Φ (viewing Q ⊂ F canonically) and h y, α ∨ i = α ∨ ( y ) = 2 ( y | α )( α | α ) for all y ∈ Y . Hence s α as defined in (RS2) is also given by the formula (3.1). Wehave S = { } = X = X ∩ S . Since h Φ , α ∨ i ⊂ Z we have shown that( S, Y, ( ·|· )) as defined in (3.2) is a finite affine reflection system of nullity . We will characterize finite root systems within the category of affine reflectionsystems in Cor. 3.11.In the following we will always assume that a finite root system contains 0. Wewill usually use the symbol S for a finite root system, and put S × = S \ { } = Φ . ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 27
We will also need the following subsets of roots of a finite root system S : S div is the set of divisible roots, where α ∈ S is called divisible if α/ ∈ S . Inparticular 0 ∈ S div . We put S × div = S div ∩ S × = S div \ { } . S ind = S \ S × div , the subsystem of indivisible roots .We also need the fact that there exists a unique symmetric bilinear form ( ·|· ) u on Y which is invariant under the Weyl group W ( S ) and which satisfies 2 ∈ { ( α | α ) u :0 = α ∈ C } ⊂ { , , , } for every connected component C of S . This follows easilyfrom [Bou2, Prop. 7]. Observe that S × div = { α ∈ S : ( α | α ) u = 8 } . We use ( ·|· ) u to define short and long roots: S sh = { α ∈ S : ( α | α ) u = 2 } is the set of short roots . S lg = { α ∈ S : ( α | α ) u ∈ { , }} is the set of long roots in S .Thus S lg = S \ ( S sh ∪ S div ). For example, for S = BC l we haveBC l, sh = {± ε i : 1 ≤ i ≤ l } , BC × l, div = {± ε i : 1 ≤ i ≤ l } , BC l, lg = {± ε i ± ε j : 1 ≤ i = j ≤ l } , in particular BC , lg = ∅ , and if S is simply laced , i.e., S × = S sh , then S div = { } and S lg = ∅ . Example 3.4 ( Untwisted affine reflection systems ) . Let (
S, Y, ( ·|· ) Y ) be a finiteroot system. Hence 0 ∈ S and Φ = S \ { } , as stipulated in Example 3.3. Also, let Z be an n -dimensional F -vector space, say with a basis ε , . . . , ε n . We define X = Y ⊕ Z, Λ = Z ε ⊕ · · · ⊕ Z ε n ⊂ Z,R = S ξ ∈ S { ξ ⊕ λ : λ ∈ Λ } ⊂ Y ⊕ Z, ( x | x ) X = ( y | y ) Y for x i = y i ⊕ z i with y i ∈ Y and z i ∈ Z .By construction we then have X = Z, R = Λ , R an = S ξ ∈ Φ ξ ⊕ Λwhere of course ξ ⊕ Λ = { ξ ⊕ λ : λ ∈ Λ } . For α = ξ ⊕ λ ∈ R an with ξ ∈ S and λ ∈ Λ the reflection s α satisfies(3.3) s α ( y ⊕ z ) = s ξ ( y ) ⊕ ( z − h y, ξ ∨ i λ ) . We will leave it to the reader to verify that(3.4) (
R, X ) is an affine reflection system of nullity n . Observe that (
R, X ) is the set of roots of the EALA constructed in 2.5, see inparticular (2.18).Observe that span F ( R an ) = X = Re( X ) in case S = { } . This shows that theform ( ·|· ) Re of the real part ℜ ( R ) of R need not be nondegenerate. Exercise 3.5.
Show the claim in (3.4), and also that (
R, X ) is reduced resp.connected if and only if (
S, Y ) is so.
Example 3.6 ( Affine root systems ) . By definition, an affine root system is the setof roots of an affine Kac-Moody Lie algebra, which we studied in § § an affine root system is an affine reflection system of nullity . Let us first collect the data necessary to prove this. We use the notation establishedin 2.4. Thus, ˆ L = ˆ L ( g , σ ) is an affine Kac-Moody Lie algebra over C , σ is a diagramautomorphism of the simple finite-dimensional Lie algebra g of order m ∈ { , , } ,and ∆ ¯ s denotes the set of weights of the ( g ¯0 , h ¯0 )-module g ¯ s ⊂ g , s = 0 , . . . , m − ¯0 is a reduced irreducible root system in h ∗ ¯0 =: Y . The roots ofˆ L with respect to H = h ¯0 ⊕ C c ⊕ C d are R = { γ ⊕ nδ : γ ∈ ∆ ¯ s , ¯ n = ¯ s, ≤ s < m } , see (2.13), hence X = span C ( R ) = Y ⊕ C δ. The bilinear form ( ·|· ) X used to determine the (an)isotropic roots in R has the form( x | x ) X = ( y | y ) Y where x i = y i ⊕ a i δ with y i ∈ Y and a i ∈ C , and where ( ·|· ) Y is the nondegeneratesymmetric bilinear form on Y , obtained by transporting the Killing form κ | h ¯0 × h ¯0 from h ¯0 to Y . It follows that X = C δ and R an = { γ ⊕ nδ ∈ R : γ = 0 } . We can now verify the axioms (AR1)–(AR4).(AR1) holds by definition. (AR2) is a consequence of [Kac, Prop. 3.7(b)]. Con-cerning (AR3), it follows from the structure of ( ·|· ) X that(3.6) h x, α ∨ i = h y, γ ∨ i for x = y ⊕ aδ ∈ X and α = γ ⊕ nδ ∈ R an .This implies that h R, α ∨ i is a finite set since S = ∆ ¯0 ∪ · · · ∪ ∆ m − is a finite set ( S is actually a finite root system; for m > h R, α ∨ i ⊂ Z because ˆ L is an integrable ˆ L -module ([Kac, Lemma 3.5]).Thus (AR3) holds, and (AR4) follows from (3.6) and ( γ | γ ) = 0 ⇔ γ = 0 for γ ∈ S .This proves (3.5).To motivate the definition of extension data in Def. 3.9 and the Structure The-orem 3.10 for affine reflection systems, we will now look at R and S more closely.In the untwisted case, i.e., m = 1, we have of course∆ ¯0 = S, R = S × Z δ ( m = 1) . Thus R is an untwisted affine root system of nullity 1, a special case of the Exam-ple 3.4. For m = 2 , ¯ s and S is summarized in the table below. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 29
Proofs can be extracted from [Kac, 7.9, 7.8, 8.3].(3.7) ( g , m ) ∆ ¯0 ∆ ¯1 S (A l , , l ≥ or B l ( l ≥
2) ∆ ¯0 ∪ {± ε i : 1 ≤ i ≤ l } BC l ( A l − , , l ≥ l { } ∪ C l, sh C l (D l +1 , , l ≥ l { } ∪ B l, sh B l (E ,
2) F { } ∪ F , sh F (D ,
3) G { } ∪ G , sh G We can now rewrite R . For subsets T ⊂ S and Ξ ⊂ Z δ we put T ⊕ Ξ = { τ ⊕ nδ : τ ∈ T, nδ ∈ Z δ } and abbreviate B = A = { , ± α } . For m = 2 we get R = (∆ ¯0 ⊕ Z δ ) ∪ (cid:0) ∆ ¯1 ⊕ (1 + 2 Z ) δ (cid:1) = ( ( { } ⊕ Z δ ) ∪ (cid:0) (B l \{ } ⊕ Z δ (cid:1) ∪ (cid:0) BC × div ⊕ (1 + 2 Z ) δ (cid:1) , g = A l ( { } ⊕ Z δ ) ∪ ( S sh ⊕ Z δ ) ∪ ( S lg ⊕ Z δ ) , g = A l . For ( g , m ) = (D ,
3) one knows ∆ ¯1 = ∆ ¯2 = { } ∪ G , sh , whence R = (∆ ¯0 ⊕ Z δ ) ∪ (cid:0) ∆ ¯1 ⊕ (1 + 3 Z ) δ (cid:1) ∪ (cid:0) ∆ ¯2 ⊕ (2 + 3 Z ) δ (cid:1) = ( { } ⊕ Z δ ) ∪ ( S sh ⊕ Z δ ) ∪ ( S lg ⊕ Z δ ) . In all three cases R has a simultaneous description in terms of the root system S and subsets Λ sh , Λ lg , Λ div ⊂ Z δ as(3.8) R = R ∪ ( S sh ⊕ Λ sh ) ∪ ( S lg ⊕ Λ lg ) ∪ ( S × div ⊕ Λ div )where(3.9) Λ sh = Z δ = R , Λ div = (1 + 2 Z ) δ, Λ lg = Z δ, g = A l , m = 2 , Z δ, g = A l , m = 2 , Z δ, m = 3 . If we define Λ ξ for ξ ∈ S by Λ ξ ∈ { Λ = R , Λ sh , Λ lg , Λ div } according to ξ belongto the corresponding subset of S , then (3.8) becomes(3.10) R = S ξ ∈ S ξ ⊕ Λ ξ . Note that we also recover [Kac, Th. 5.6(b)]: R ∩ X = R ∩ Z δ. Example 3.7 ( Type A generalized ) . We consider a final example of an affinereflection system to motivate the definition of an extension datum in 3.9 below.Let Z be a finite-dimensional F -vector space and define the vector space X anda symmetric bilinear form on X by X = F α ⊕ Z, ( a α ⊕ z | a α ⊕ z ) = a a , where 0 = α and a i ∈ F . We define R ⊂ X in terms of three non-empty subsetsΛ , Λ α , Λ − α ⊂ Z as follows:(3.11) R = Λ ∪ ( α ⊕ Λ α ) ∪ ( − α ⊕ Λ − α ) . It is then immediate that X = Z, R = Λ , R an = ( α ⊕ Λ α ) ∪ ( − α ⊕ Λ − α ) . We will now discuss under which conditions (
R, X, ( ·|· )) is an affine reflection sys-tem. Let us start with (AR2). For s i ∈ {± } , µ ∈ Λ s α and λ ∈ Λ s α we have h s α ⊕ µ, ( s α ⊕ λ ) ∨ i = 2 s s ,s s α ⊕ λ ( s α ⊕ µ ) = − s α ⊕ ( µ − s s λ )Hence, all reflections s s α ⊕ λ leave R invariant if and only if µ − s s λ ∈ Λ − s α for µ, λ as above, i.e., in obvious short form Λ s α − s s Λ s α ⊂ Λ − s α . In particular,Λ α − α ⊂ Λ − α , Λ − α − − α ⊂ Λ α for s = s ,Λ − α + 2Λ α ⊂ Λ α for s = − − s . For λ ∈ Λ α we therefore get λ − λ = − λ ∈ Λ − α , whence − Λ α ⊂ Λ − α and,analogously, Λ − α ⊂ − Λ α . We therefore obtain(3.12) Λ − α = − Λ α , or with the notation of above Λ s α = s Λ α . It is now easy to see that (AR2) isequivalent to the two conditions (3.12) and(3.13) 2Λ α − Λ α ⊂ Λ α . It then follows that R is an affine reflection system if and only if(i) (3.12) and (3.13) hold,(ii) 0 ∈ Λ , and(iii) Z = span F (Λ ∪ Λ α ∪ Λ − α ).Observe the similarity with the previous examples: R has the form R = S ξ ∈ S ξ ⊕ Λ ξ where S = { , ± α } is a finite root system and (Λ ξ : ξ ∈ S ) is a family of subsets in X . However, in the previous examples the Λ ξ were subgroups of ( Z, +) while herewe only have the condition (3.13). Does this imply that Λ α is a subgroup? Theanswer is no! For example, in Z = F the subset Λ α = 1 + 2 Z ⊂ F satisfies (3.13).A subset A of an abelian group ( Z, +) is called a reflection subspace if 2 a − a ∈ A for all a i ∈ A (see [L] or [Ne5, 3.3] for a justification for this terminology). Hence,(3.13) just says that Λ α is a reflection subspace. The structure of two special typesof reflection subspaces is described in Exercise 3.8 below.While in general Λ α is far from being a subgroup, one can always “re-coordina-tize” R to at least get 0 ∈ Λ α . Namely, for a fixed λ ∈ Λ α we have α + Λ α =( α + λ ) + (Λ α − λ ). Hence, with ˜ α = α + λ and Λ ˜ α = Λ α − λ , we obtain(3.14) R = Λ ∪ (˜ α + Λ ˜ α ) ∪ (cid:0) − (˜ α + Λ ˜ α ) (cid:1) , where now Λ ˜ α not only satisfies (3.13) but also 0 ∈ Λ ˜ α . In other words, Λ ˜ α isa pointed reflection subspace as defined in Lemma 3.8 and therefore also satisfiesΛ ˜ α = − Λ ˜ α .The process of re-coordinatization works well in this example. The reason is thatthe finite root system S in (3.14) is reduced. Re-coordinatization will not work if S is not reduced, as for example in the case ( g , m ) = (A l ,
2) of Example 3.6. This“explains” why in the property (ED2) of an extension datum in 3.9 we require0 ∈ Λ ξ only for an indivisible root ξ ∈ S . ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 31
Exercise 3.8.
Let A be a subset of an abelian group ( Z, +). As above we put2 A − A = { a − a : a i ∈ A } . We denote by Λ = span Z ( A ) the Z -span of A in Z .A subset A ⊂ Z is called symmetric if A = − A .(a) The following equivalent conditions characterize symmetric reflection sub-spaces A ⊂ Z :(i) 2 A − A ⊂ A and A = − A ,(ii) 2 λ + a ∈ A for every λ ∈ Λ and a ∈ A ,(iii) A is a union of cosets modulo 2Λ,(iv) a − a ∈ A for all a i ∈ A .(b) The following are equivalent for A ⊂ Z :(i) 0 ∈ A and A − A ⊂ A ,(ii) 0 ∈ A and 2 A − A ⊂ A ,(iii) 2 Z [ A ] ⊂ A and 2 Z [ A ] − A ⊂ A ,(iv) A is a union of cosets modulo 2 Z [ A ], including the trivial coset 2 Z [ A ].In this case A is called a pointed reflection subspace.(c) Every pointed reflection subspace is symmetric.(d) If A is a symmetric reflection subspace then A + A is a pointed reflectionsubspace.3.3. The Structure Theorem of affine reflection systems.
After the manyexamples in 3.2, the following definition should not be too surprising.
Definition 3.9.
Let S be a finite root system as defined in 3.3. Recall S × = S \{ } and S ind = { } ∪ { α ∈ S : α/ S } = S \ S × div . Also, let Z be a finite-dimensional F -vector space. An extension datum of type ( S, Z ), sometimes simply called an extension datum , is a family (Λ ξ : ξ ∈ S ) of subset Λ ξ ⊂ Z satisfying the axioms(ED1)–(ED3) below. (ED1): For η, ξ ∈ S × , µ ∈ Λ η and λ ∈ Λ ξ we have µ − h η, ξ ∨ i ξ ∈ Λ s ξ ( η ) , inobvious short form Λ η − h η, ξ ∨ i Λ ξ ⊂ Λ s ξ ( η ) . (ED2): ∈ Λ ξ for ξ ∈ S ind , and Λ ξ = ∅ for ξ ∈ S \ S ind = S × div . (ED3): Z = span F (cid:0) S ξ ∈ S Λ ξ (cid:1) .The axiom (ED1) is trivially true for η = 0 since h η, ξ ∨ i = 0 and s ξ (0) = 0. Also,if S × div = ∅ , then there is no Λ ξ for ξ ∈ S × div and so the second condition in (ED2)is trivially fulfilled. (ED3) simply serves to determine Z . If it does not hold, onecan simply replace Z by span Z ( S ξ ∈ S Λ ξ ).The definition of an extension datum above is a special case of the notion of anextension datum for a pre-reflection system, introduced in [LN, 4.2]. (The readerwill note that the axiom (ED1) in [LN] simplifies since in our setting the subset S re of [LN] is S re = S an = S \ { } .) The Structure Theorem 3.10 below is proven in[LN, Th. 4.6] for extensions of pre-reflection systems. Affine reflection systems arespecial types of such extensions, namely finite-dimensional extensions of finite rootsystems.The rationale for the concept of an extension datum is the following StructureTheorem for affine reflection systems. Theorem 3.10 ( Structure Theorem for affine reflection systems ) . (a) Let ( S, Y, ( ·|· ) Y ) be a finite root system and let L = (Λ ξ : ξ ∈ S ) be an extensiondatum of type ( S, Z ) . Define ( R, X, ( ·|· ) X ) by X = Y ⊕ ZR = S ξ ∈ S ξ ⊕ Λ ξ ⊂ Y ⊕ Z = X, ( y ⊕ z | y ⊕ z ) X = ( y | y ) Y for y i ∈ Y and z i ∈ Z . Then ( R, X, ( ·|· ) X ) is an affine reflection system, denoted A ( S, L ) , with R = Λ , X = Z and R an = S = ξ ∈ S ξ ⊕ Λ ξ . For α = ξ ⊕ λ ∈ R an and x = y ⊕ z ∈ X the reflection s α is given by s α ( x ) = s ξ ( y ) ⊕ ( z − h y, ξ ∨ i λ )(b) Conversely, let ( R, X, ( ·|· ) X ) be an affine reflection system. (i) Let f : X → X/X =: Y be the canonical map, put S = f ( R ) and let ( ·|· ) Y be the induced bilinear form on Y , that is ( f ( x ) | f ( x )) Y = ( x | x ) X .Then ( S, Y, ( ·|· ) Y ) is a finite root system, the so-called quotient root system of ( R, X ) . (ii) There exists a linear map g : Y → X satisfying f ◦ g = Id Y and g ( S ind ) ⊂ R . (iii) For g as in (ii) and ξ ∈ S define Λ ξ ⊂ Ker( f ) =: Z by (3.15) R ∩ f − ( ξ ) = g ( ξ ) ⊕ Λ ξ . Then L = (Λ ξ : ξ ∈ S ) is an extension datum of type ( S, Z ) . (iv) ( R, X ) is isomorphic to the affine reflection system A ( S, L ) constructed in (a) . Let us note that it is not reasonable to expect g ( S ) ⊂ R in (b.ii) above, since R may be reduced while S is not, see for example the case ( g , A l ) in 3.6. Thequotient root system S is uniquely determined, but not so the extension datum,see [LN, Th. 4.6(c)]. Corollary 3.11.
An affine reflection system ( R, X, ( ·|· )) is nondegenerate in thesense that ( ·|· ) is nondegenerate if and only if R is a finite root system.Proof. If (
R, X, ( ·|· )) is an affine reflection system with a nondegenerate form ( ·|· ),then { } = X = Ker f , so f is the identity. We have seen the other direction inExample 3.3. (cid:3) Corollary 3.12 ([LN, Cor. 5.5]) . Let ( R, X, ( ·|· )) be an affine reflection system over F = R . Then there exists a positive semidefinite symmetric bilinear form ( ·|· ) ≥ on X such that ( R, X, ( ·|· ) ≥ ) is an affine reflection system with the same (an)isotropicroots and reflections. The morale of the Structure Theorem isaffine reflection system = finite root system + extension datumThus properties of an affine reflection system can be described in terms of propertiesof its quotient root system and the associated extension datum. Some examples ofthis philosophy are given in the Proposition 3.13 and the Exercise 3.14 below.
ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 33
Proposition 3.13 ([LN, Cor. 5.2]) . Let R be an affine reflection system, let S beits quotient root system and let (Λ ξ : ξ ∈ S ) be the associated extension datum. Wedefine Λ diff = S = ξ ∈ S Λ ξ − Λ ξ . Then Z Λ diff = Λ diff . Moreover: (a) R is tame in the sense that R ⊂ R an − R an if and only if R ⊂ Λ diff . (b) All root strings S ( β, α ) = R ∩ ( β + Z α ) , ( β ∈ R, α ∈ R an ) are unbroken , i.e., Z ( β, α ) = { n ∈ Z : β + nα ∈ R } is either a finite interval in Z or equals Z , if and only if Λ diff ⊂ R . (c) A tame affine reflection system with unbroken root strings is symmetric.
Exercise 3.14.
Let R be an affine reflection system, let S be its quotient rootsystem and let (Λ ξ : ξ ∈ S ) be the associated extension datum. We use thenotation of Prop. 3.13. Prove:(a) R is reduced if and only if for all 0 = ξ ∈ S with 2 ξ ∈ S we haveΛ ξ ∩ ξ = ∅ . In particular, S need not be reduced for R to be reduced!(b) R is connected iff S is connected (= irreducible).(c) R is symmetric, i.e., R = − R , iff Λ is symmetric.(d) For all α ∈ R an and β ∈ R the α -string through β , i.e., S ( β, α ) has length | S ( β, α ) | ≤ α, β ) ∈ R an × R and define d, u ∈ N by put − d = min Z ( β, α ) and u = max Z ( β, α ). Then u − d = h β, α ∨ i .We will now describe how the examples of affine reflection systems of section 3.2fit into the general scheme of the Structure Theorem above. Examples 3.15. (a) Let L = (Λ ξ : ξ ∈ S ) be an extension datum of type ( S, Z ).Observe that the only conditions on Λ are 0 ∈ Λ from (ED2) and that Λ togetherwith the other Λ ξ ’s spans Z from (ED3). This is in line with our earlier observationthat one has little control over the null roots R of an affine reflection system.Following the Example 3.2 we define a new extension datum Re( L ) = (Re(Λ ξ ) : ξ ∈ S ) of type ( S, Re( Z )) byRe( Z ) = span F (cid:0) S = ξ ∈ S Λ ξ (cid:1) , Re(Λ ξ ) = ( { } for ξ = 0,Λ ξ for ξ = 0.If L is the extension datum associated to the affine reflection system ( R, X ), thenRe( L ) is the extension datum associated to the affine reflection system Re( R ).(b) All Λ ξ = { } , whence Z = { } , defines a trivial extension datum for anyroot system S . It is “used” when we view S as an affine reflection system, as donein Example 3.3.(c) Let Λ be a subgroup of a finite-dimensional vector space Z such that span F (Λ)= Z . Then for any finite root system S the family (Λ ξ ≡ Λ : ξ ∈ S ) is an extensiondatum of type ( S, Z ). It is used to construct the untwisted affine reflection systemsof Example 3.4. (d) Let R be an affine root system. We have seen that R is an affine reflectionsystem. Its quotient root system S and associated extension datum (Λ ξ : ξ ∈ S )are described in Example 3.6 using the very same symbols, see the formulas (3.10)and (3.15).(e) The family ˜ L = (Λ , Λ ˜ α , Λ − ˜ α ) in Example 3.7 is an extension datum, butnot necessarily L = (Λ , Λ α , Λ − α ) since 0 need not lie in Λ ± α . In fact, replacing L by ˜ L was the rationale for the re-coordinatization in 3.7.To describe the classification of affine reflection systems we need some moreproperties of the subsets Λ ξ of an extension datum. They are given in the followingexercise (just do it!). Recall from Exercise 3.8 that a reflection subspace A is calledsymmetric if A = − A and is called pointed if 0 ∈ A . Exercise 3.16.
Let (Λ η : η ∈ S ) be an extension datum of type ( S, Z ). Show:(a) Every Λ ξ for 0 = ξ ∈ S is a symmetric reflection subspace and is even apointed reflection subspace if ξ ∈ S × ind .(b) For w ∈ W ( S ), the Weyl group of S , we have(3.16) Λ ξ = Λ w ( ξ ) . In particular, Λ ξ = Λ − ξ = − Λ ξ .(c) Whenever 0 = ξ ∈ S and 2 ξ ∈ S , thenΛ ξ ⊂ Λ ξ . (d) Z Λ ξ ⊂ Λ ξ for ξ ∈ S × ind .Let (Λ ξ : ξ ∈ S ) be an extension datum where S is irreducible. Then W ( S )acts transitively on the roots of the same length ([Bou2, VI, § { } , S sh , S lg and S div (some of these sets might be empty). Because of (3.16),there are therefore at most four different subsets Λ , Λ sh , Λ lg and Λ div among theΛ ξ , defined by(3.17) Λ ξ = Λ , ξ = 0;Λ sh , ξ ∈ S sh ;Λ lg , ξ ∈ S lg ;Λ div , ξ ∈ S × div . Of course, Λ lg or Λ div only exists if the corresponding subset of roots exits. Theassertions below referring to Λ lg or Λ div should be interpreted correspondingly.We have seen in Exercise 3.16 (did you do it?), that the subsets Λ sh and Λ lg are pointed reflection subspaces and that Λ div is a symmetric reflection subspace.Assuming only these properties, does however not give an extension datum, sinceonly parts of the axiom (ED1) are fulfilled, namely those with η = ± ξ . We alsoneed to evaluate what happens for η = ± ξ with h η, ξ ∨ i 6 = 0. We will do this in thefollowing examples. Examples 3.17.
Let S be an irreducible root system. We suppose that we aregiven a pointed reflection subspace Λ sh of a finite-dimensional vector space, and if S lg = ∅ or S × div = ∅ then also a pointed reflection subspace Λ lg and a symmetricreflection subspace Λ div . We define Λ ξ , ξ ∈ S , by (3.17) and ask, when is the family L min defined in this way an extension datum in Z = span (cid:0) S ξ ∈ S Λ ξ (cid:1) ? Note thatwe only have to check (ED1). We will consider some examples of S . ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 35 (a) S = A : In this case there are no further conditions, so L min describes allpossible extension data for A with Λ = { } .(b) S = A : In this case there exists roots η, ξ with h η, ξ ∨ i = 1, namely thosefor which ∠ ( η, ξ ) = π . Evaluating (ED1) for those gives Λ sh − Λ sh ⊂ Λ sh , forcingΛ sh to be a subgroup of Z . Thus, L min is an extension datum for S = A iff Λ sh isa subgroup.(c) S simply laced, rank S ≥
2: The argument in (b) works whenever S sh containsroots η, ξ with h η, ξ ∨ i = 1. Since this is the case here, we get that L min is anextension datum iff Λ sh is a subgroup of Z .(d) S = B = {± ε , ± ε , ± ε ± ε } : Here we have pointed reflection subspacesΛ sh and Λ lg . Since non-zero roots of the same length are either proportional ororthogonal, (ED1) is fulfilled for them. Because (ED1) is invariant under signchanges, we are left to evaluate the case of two roots η, ξ of different lengths formingan obtuse angle of π , for example η = ε , ξ = ε − ǫ .If h η, ξ ∨ i = −
2, then η is long, ξ is short and so (ED1) becomes Λ lg + 2Λ sh ⊂ Λ lg .If η is short, ξ is long, we have h η, ξ ∨ i = − sh + Λ lg ⊂ Λ sh . To summarize: L min is an extension datum for S = B iff Λ sh and Λ lg arepointed reflection subspaces satisfying (3.18) Λ lg + 2Λ sh ⊂ Λ lg and Λ sh + Λ lg ⊂ Λ sh . Note that (3.18) implies 2Λ sh ⊂ Λ lg ⊂ Λ sh .(e) S = B l , l ≥
3. Recall S = {± ǫ i : 1 ≤ i ≤ l } ∪ {± ε i ± ε j : 1 ≤ i = j ≤ l } .Since the short roots in S are either proportional or orthogonal, (ED1) is fulfilledfor all short roots η, ξ . But there exist long roots η, ξ ∈ S with ∠ ( η, ξ ) = π , whence h η, ξ ∨ i = 1 and so (ED1) reads Λ lg − Λ lg ⊂ Λ lg . This forces Λ lg to be a subgroup.As for S = B , (ED1) for roots of different lengths leads to the condition (3.18). Itis then easy to check that L min is an extension datum for S = B l , l ≥ , iff Λ sh isa pointed reflection subspace, Λ lg is a subgroup and (3.18) holds. Continuing in this way, one arrives at the following.
Theorem 3.18 ( Structure of extension data).
Let S be an irreducible finiteroot system and define a family L min as in (3.17) with Λ = { } . Then L min is anextension datum if and only if Λ sh and Λ lg are pointed reflection subspaces, Λ div isa symmetric reflection subspace and the following conditions, depending on S , hold. (i) S is simply laced, rank S ≥ No further condition for S = A , but Λ sh isa subgroup if rank S ≥ . (ii) S = B l ( l ≥ , C l ( l ≥ , F : Λ sh and Λ lg satisfy Λ lg + 2Λ sh ⊂ Λ lg and Λ sh + Λ lg ⊂ Λ sh . Moreover, • Λ lg is a subgroup if S = B l , l ≥ or S = F , and • Λ sh is a subgroup if S = C l or S = F . (iii) S = G : Λ sh and Λ lg are subgroups satisfying Λ lg + 3Λ sh ⊂ Λ lg and Λ sh + Λ lg ⊂ Λ sh . (iv) S = BC : Λ sh and Λ div satisfy Λ div + 4Λ sh ⊂ Λ div and Λ sh + Λ div ⊂ Λ sh . (v) S = BC l ( l ≥
2) : Λ sh , Λ lg and Λ div satisfy Λ lg + 2Λ sh ⊂ Λ lg , Λ sh + Λ lg ⊂ Λ sh , Λ div + 2Λ lg ⊂ Λ div , Λ lg + Λ div ⊂ Λ lg , Λ div + 4Λ sh ⊂ Λ div , Λ sh + Λ div ⊂ Λ sh . In addition, if l ≥ then Λ lg is a subgroup. The inclusions Λ div + 4Λ sh ⊂ Λ div and Λ sh + Λ div ⊂ Λ sh in case (v) above areconsequences of the other inclusions. Since 0 lies in Λ sh and also in Λ lg if it exists,the displayed inclusions in the Structure Theorem above imply(3.19) Λ div ⊂ Λ lg ⊂ Λ sh . The details of this theorem are given in [AABGP, II, §
2] for the special caseof extended affine root systems and then in [Y3] in general (it follows from theStructure Theorem 3.10 that an affine reflection system is the same as a “rootsystem extended by a torsion-free abelian group of finite rank” in the sense of[Y3]). The reference [AABGP] also contains a classification of discrete extensiondata for extended affine root systems of low nullity.
Exercise 3.19.
Without looking at [AABGP] or [Y3], work out some of the casesabove.3.4.
Extended affine root systems.
Let us come back to the beginning of thischapter. Our goal was to describe the structure of the set of roots occurring in anextended affine Lie algebra. After all the preparations in 3.1–3.3, this is now easy.We start with the same setting as in 3.1, i.e., X is a finite-dimensional vectorspace over a field F of characteristic 0, R is a subset of X and ( ·|· ) is a symmetricbilinear form on X . As in 3.1 we define R = { α ∈ R : ( α | α ) = 0 } , R an = { α ∈ R :( α | α ) = 0 } and h x, α ∨ i = 2 ( x | α )( α | α ) , ( x ∈ X and α ∈ R an ) . Definition 3.20.
A triple (
R, X, ( ·|· )) as above is called an extended affine rootsystem or EARS for short, if the following seven axioms (EARS1)–(EARS7) arefulfilled. (EARS1): ∈ R and R spans X , (EARS2): R has unbroken finite root strings , i.e., for every α ∈ R an and β ∈ R there exist d, u ∈ N = { , , , . . . } such that { β + nα : n ∈ Z } ∩ R = { β − dα, . . . , β + uα } and d − u = h β, α ∨ i . ( d stands for “down” and u for “up”.) (EARS3): R = R ∩ X . (EARS4): R is reduced as defined in 3.1: for every α ∈ R an we have F α ∩ R an = {± α } . (EARS5): R is connected in the sense of 3.1: whenever R an = R ∪ R with( R | R ) = 0, then R = ∅ or R = ∅ . (EARS6): R is tame, i.e., R ⊂ R an + R an . (EARS7): The abelian group span Z ( R ) is free of finite rank.In analogy with the concept of discrete EALAs we call ( R, X, ( ·|· )) for F = C or F = R a discrete extended affine root system if (EARS1)–(EARS6) hold and inaddition (DE): R is a discrete subset of X , equipped with the natural topology. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 37
As for EALAs, a discrete extended affine root system necessarily satisfies (EARS7),see Proposition 3.21(c) below, so that it is justified to call it an EARS.We will immediately connect EARS to affine reflection systems:
Proposition 3.21. (a)
A pair ( R, X ) satisfying (EARS1)–(EARS3) is an affinereflection system. In particular: (i) An extended affine root system is an affine reflection system which is re-duced, connected, symmetric, tame and which has unbroken root strings. (ii) If F = R we can assume that ( ·|· ) is positive semidefinite. (b) Let ( R, X ) be an affine reflection system with quotient root system S andextension datum L . Then ( R, X ) is an extended affine root system if and only if (i) S is irreducible, hence L = (Λ , Λ sh , Λ lg , Λ div ) , (ii) Λ = Λ sh + Λ sh , Λ div ∩ sh = ∅ , and (iii) (EARS7) holds. (c) For an extended affine root system ( R, X ) over F = R or F = C the followingare equivalent: (i) R is discrete; (ii) R is a discrete subset of X ; (iii) span Z ( R ) is a discrete subgroup of X .In this case, all reflection subspaces Λ ξ of (b) are discrete too, and span Z ( R ) is afree abelian group of finite rank.Proof. (a) Obviously (AR1) = (EARS1) and (AR4) = (EARS3). The axiom (AR2),i.e., s α ( R ) = R , follows from (EARS2): s α ( β ) = β + ( u − d ) α and − d ≤ u − d ≤ u .Also (AR3) is immediate from (EARS2).In any affine reflection system the root strings R ∩ ( β + Z α ) for ( β, α ) ∈ R × R an are finite. This is Exercise 3.14(d) and an immediate consequence of the StructureTheorem 3.10. Also, a tame affine reflection system with unbroken roots stringsis necessarily symmetric by Prop. 3.13. The characterization of an EARS in (i) isnow clear. (ii) follows from Cor. 3.12.(b) follows from Prop. 3.13 and Exercise 3.14, since for a connected R = irre-ducible S the formula (3.19) implies Λ diff = Λ sh + Λ sh .(c) (i) ⇒ (ii) is obvious. Suppose (ii) holds. We know from (b) that R = Λ =Λ sh + Λ sh . Since Λ sh is a pointed reflection subspace, so is Λ (Exercise 3.8). Hence2 span Z (Λ ) ⊂ Λ is discrete. But then so is span Z (Λ ). Thus (ii) ⇒ (iii).By (3.19) and (b.ii), Λ div ⊂ Λ lg ⊂ Λ sh ⊂ Λ . Hence, if (iii) holds, then all Λ ξ are discrete subsets of X . But then so is is R , as a finite union of discrete subsets.This shows (iii) → (i).It is well-known fact that every discrete subgroup of a finite-dimensional realvector space is free of finite rank. (cid:3) A quick comparison of [AABGP, Definition 2.1] and our Definition 3.20 togetherwith Prop. 3.21(a) will convince the reader that an extended affine root system inthe sense of [AABGP] is the same as a discrete extended affine root system over R in our sense. The reason for the generalization and the change of name is the sameas the one justifying our more general notion of extended affine Lie algebras: Weare considering EALAs over arbitrary fields of characteristic 0 and the set of rootsof an EALA will not be an extended affine root system in the sense of [AABGP].Finally, here is the result which brings us back to EALAs. Theorem 3.22.
Let ( E, H ) be an EALA over F and let R ⊂ H ∗ be its set of roots.Put X = span F ( R ) and let ( ·|· ) X be the restriction of the bilinear form (2.3) to X .Then ( R, X, ( ·|· ) X ) is an extended affine root system. If F = C , then ( E, H ) is adiscrete EALA if and only if ( R, X, ( ·|· ) X ) is a discrete extended affine root system. Remarks 3.23. (a) For (
E, H ) a discrete EALA over F = C , the theorem is provenin [AABGP, I, Th. 2.16], using discreteness. The generalization to arbitrary EALAsis due to the author, see [Ne4, Prop. 3]. It has been further generalized to otherclasses of Lie algebras, the so-called invariant affine reflection algebras, see [Ne5,Th. 6.6 and Th. 6.8]. Special cases have also been proven in [Az1] and [MY]. That R is symmetric, is an easy exercise, namely Exercise 2.1(c).(b) In view of the theorem above, one can ask if every extended affine root systemis the set of roots of some extended affine Lie algebra. This is however not the case,see [AG, Th. 6.2] for a detailed discussion of this question.As a first application of this theorem, we can now completely characterize EALAsof nullity 0. Proposition 3.24.
The following are equivalent: (i) (
E, H ) is an EALA of nullity , (ii) ( E, H ) is an EALA with a finite-dimensional E , (iii) E is a finite-dimensional split simple Lie algebra with splitting Cartan sub-algebra H .In this case, E equals its core and the set of roots R coincides with the quotientroot system S of R and is an irreducible reduced finite root system. The core and centreless core of an EALA
In the previous chapter we have studied affine reflection systems per se. Therationale for doing so became clear only in the end, when we saw in Th. 3.22 thatthe set of roots R of an EALA ( E, H ) is an extended affine root system, a specialtype of an affine reflection system.In this chapter we start by drawing consequences of the Structure Theorem 3.10of affine reflection systems and the description of extended affine root systems inProp. 3.21. The examples in § § E c and centrelesscore E cc = E c /Z ( E c ) of an extended affine Lie algebra ( E, H ) really are the “core”of the matter. We will show in Th. 4.14 and in Cor. 4.16 that both are so-calledLie tori, a new class of Lie algebras which we will introduce in 4.1. We will presentsome basic properties of Lie tori in 4.2 and describe some examples in 4.4 and 4.5.With some justification, this chapter could therefore also be entitled “On Lietori”. But the reader can be re-assured that we are not getting side-tracked toomuch: In the next chapter we will see that Lie tori are precisely what is needed toconstruct EALAs.4.1.
Lie tori: Definition.
Lie tori are special objects in the following category ofgraded Lie algebras.
Definition 4.1.
Let (
S, Y ) be a finite irreducible, but not necessarily reduced rootsystem, as defined in Example 3.3. We denote by Q ( S ) = span Z ( S ) ⊂ Y the rootlattice of S . To avoid some degeneracies we will always assume that S = { } . LetΛ be an abelian group. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 39
A ( Q ( S ) , Λ) -graded Lie algebra is a Lie algebra L with compatible Q ( S )- andΛ-gradings. It is convenient (and helpful) to use subscripts for the Q ( S )-gradingand superscripts for the Λ-grading. Thus, L = L q ∈ Q ( S ) L q = L λ ∈ Λ L λ are Q ( S )- and Λ-gradings of L , and compatibility means L = L q ∈ Q ( S ) , λ ∈ Λ L λq for L λq = L q ∩ L λ . Hence for λ, µ ∈ Λ and p, q ∈ Q ( S ) L λ = L q ∈ Q ( S ) L λq , L q = L λ ∈ Λ L λq and [ L λq , L µp ] ⊂ L λ + µq + p . Thus, L has three gradings, by Q ( S ), Λ and Q ( S ) ⊕ Λ whose interplay will be crucialin the following. Corresponding to these three different gradings are three supportsets : supp Q ( S ) L = { q ∈ Q ( S ) : L q = 0 } , supp Λ L = { λ ∈ L : L λ = 0 } , andsupp Q ( S ) ⊕ Λ L = { ( q, λ ) ∈ ( Q ( S ) , Λ) : L λq = 0 } . Definition 4.2.
We keep the notation of the Def. 4.1. A
Lie torus of type ( S, Λ)is a ( Q ( S ) , Λ)-graded Lie algebra L over F , a field of characteristic 0, satisfying theaxioms (LT1)–(LT3) below. (LT1): supp Q ( S ) L ⊂ S , hence L = L ξ ∈ S L ξ . (LT2): If L λξ = 0 and ξ = 0, then there exist e λξ ∈ L λξ and f λξ ∈ L − λ − ξ suchthat(4.1) L λξ = F e λξ , L − λ − ξ = F f λξ , and for x τ ∈ L τ we have(4.2) [[ e λξ , f λξ ] , x τ ] = h τ, ξ ∨ i x τ . (LT3): (a) L ξ = 0 if ξ ∈ S × ind , i.e., 0 = ξ ∈ S and ξ/ S .(b) As a Lie algebra, L is generated by S = ξ ∈ S L ξ .(c) Λ = span Z (supp Λ L ).We will say that L is a Lie torus if L is a Lie torus for some pair ( S, Λ).A Lie torus is called invariant , if L has an invariant nondegenerate symmetricbilinear form ( ·|· ) which is graded in the sense that(4.3) ( L λξ | L µτ ) = 0 if λ + µ = 0 or ξ + τ = 0.Two Lie tori L and ˜ L , both of type ( S, Λ), are called graded-isomorphic if thereexists a Lie algebra isomorphism f : L → ˜ L such that f ( L λξ ) = ˜ L λξ for all ( ξ, λ ) ∈ S × Λ. Thus, a graded-isomorphism is an isomorphism in the category of gradedLie algebras. But we will use the term “graded-isomorphism” to emphasize thatLie tori are graded algebras.
Remarks 4.3. (a) Let L be a Lie torus. Hence, by (LT1), L = L ξ ∈ S L ξ = L ξ ∈ S, λ ∈ Λ L λξ . We will determine supp Q ( S ) L in Cor. 4.6 below. The axiom (LT2)implies that(4.4) dim L λξ = 1 if 0 = ξ and L λξ = 0and that(4.5) ( e λξ , h λξ , f λξ ) with h λξ = [ e λξ , f λξ ] ∈ L
000 ERHARD NEHER is an sl -triple. The condition (LT3.a) together with (LT2) ensures that a Lie torushas enough sl -triples.The other two conditions in (LT3) are not really serious; they just serve to nor-malize things: If (LT3.c) does not hold, one can simply replace Λ by span Z (supp Λ L ).Also,(4.6) (LT3.b) ⇐⇒ L λ = P = ξ ∈ S P µ ∈ Λ [ L µξ , L λ − µ − ξ ]for all λ ∈ Λ. If one has a Lie algebra, for which all axioms except (4.6) hold, onecan replace the subspaces L λ by the right hand side of (4.6) and then gets a Lietorus. Observe that (4.6) for λ = 0 together with (LT2) yields(4.7) L = P F h λξ where the sum in (4.7) is taken over all pairs ( ξ, λ ) for which h λξ exists, i.e., thosewith L λξ = 0 and ξ = 0.(b) A Lie torus is a special type of a so-called division-( S, Λ)-graded Lie alge-bras, or more generally of a root-graded Lie algebra. This and also the differentapproaches to root-graded Lie algebras are discussed in [Ne5, § e and f of the sl -triple (4.5) as invertible elements of L , then a Lie torusis a graded Lie algebra in which most of the non-zero homogenous elements areinvertible. It is certainly unusual to speak of “invertible elements” in a Lie algebra.But the examples below will provide some justification to that: We will see that theinvertible elements of L are given by invertible elements of its coordinate algebra.Besides the analogy with the already existing concepts of “tori” in categories of(non)associative algebras, the fact that a toroidal Lie algebra is a Lie torus , see4.31, reinforces the choice of the name Lie torus.4.2.
Some basic properties of Lie tori.
Throughout this section L is a Lie torusof type ( S, Λ). We use the notation of Def. 4.2. We describe some basic propertiesof Lie tori and prove some of them, in particular those for which there does not yetexist a published proof.We first show that the homogeneous subspaces of the Q ( S )-grading of L areweight spaces for the ad-diagonalizable subalgebra h = span F { h ξ : ξ ∈ S × ind } . Lemma 4.4.
The subspaces L τ , τ ∈ S , are given by (4.8) L τ = { l ∈ L : [ h ξ , l ] = h τ, ξ ∨ i l for all ξ ∈ S × ind } . Proof.
The inclusion from left to right holds by (4.2). For the proof of the otherinclusion we write l ∈ L as l = P α l α with l α ∈ L α . Then l satisfies [ h ξ , l ] = h τ, ξ ∨ i l for all ξ ∈ S × ind if and only if for every α ∈ S we have h α − τ, ξ ∨ i l α = 0 forall ξ ∈ S × ind . Since span F ( S ind ) = Y and the bilinear form on Y associated with the ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 41 root system S is nondegenerate, for every pair ( α, τ ) ∈ S with α = τ there exists ξ ∈ S × ind with h α − τ, ξ ∨ i 6 = 0. Hence, any l belonging to the set on the right handside of (4.8) has l α = 0 for α = τ , proving l ∈ L τ . (cid:3) Proposition 4.5.
For every ( ξ, λ ) ∈ supp Q ( S ) ⊕ Λ L with ξ = 0 the map ϕ λξ = exp (cid:0) ad( e λξ ) (cid:1) exp (cid:0) ad( − f λξ ) (cid:1) exp (cid:0) ad( e λξ ) (cid:1) is a well-defined automorphism of the Lie algebra L with the property (4.9) ϕ λξ ( L µτ ) = L µ −h µ,ξ ∨ i λs ξ ( τ ) . Moreover, for every w ∈ W ( S ) , the Weyl group of S , there exists an automorphism ϕ w of the Lie algebra L such that ϕ w ( L µτ ) = L µw ( τ ) for all τ ∈ S and µ ∈ L . This proposition can be proven in the same way as [AABGP, Prop. 1.27].
Corollary 4.6 ([ABFP2, Lemma 1.10]) . The Q ( S ) -support of L satisfies supp Q ( S ) L = ( S if S is reduced ,S or S ind if S is non-reduced. As a consequence of this corollary, supp Q ( S ) L is always a finite irreducible rootsystem. It is immediate that L is also a Lie torus of type (supp Q ( S ) , Λ). Withoutloss of generality we can therefore assume that S = supp Q ( S ) L if this is convenient. Proposition 4.7.
For ξ ∈ S define Λ ξ = { λ ∈ Λ : L λξ = 0 } , so that supp Λ L = S ξ ∈ S Λ ξ . Then the family (Λ ξ : ξ ∈ S ) satisfies the axioms (ED1) and (ED2) of Def. 3.9 , Λ η − h η, ξ ∨ i Λ ξ ⊂ Λ s ξ ( η ) (ED1) 0 ∈ Λ ξ for ξ ∈ S ind and Λ ξ = ∅ for ξ ∈ S × div (ED2) Hence Λ ξ is a pointed reflection subspace for ξ ∈ S × ind , a symmetric reflectionsubspace for ξ ∈ S × div and (4.10) Λ ξ = Λ w ( ξ ) for all w ∈ W ( S ) . Defining Λ sh , Λ lg and Λ div as in , we have Λ sh ⊃ Λ lg ⊃ Λ div , (4.11) supp Λ L = Λ = Λ sh + Λ sh , (4.12) ∅ = 2Λ sh ∩ Λ div , (4.13) Λ = span Z (Λ sh ) . (4.14) The support families (Λ ξ : ξ ∈ S ) are the same for L and L/Z ( L ) .Proof. (ED1) is a consequence of Prop. 4.5 and (ED2) of (LT3.a). It then followsas in section 3 that Λ ξ , ξ ∈ S × , are pointed respectively symmetric reflectionsubspaces such that (4.10) and (4.11) hold. (4.12) and (4.13) are proven in [Y3,Th. 5.1] and [ABFP2, Lemma 1.1.12]. (4.14) follows from (4.11) and (4.12). Thelast claim is also proven in [Y3, Th. 5.1]. (cid:3) Proposition 4.8.
Define g = subalgebra generated by { L ξ : ξ ∈ S × ind } , h = span F { h ξ : ξ ∈ S × ind } . (a) Then g is a finite-dimensional split simple Lie algebra with splitting Cartansubalgebra h . (b) The root system S ind and the root system of ( g , h ) are canonically isomorphic.Namely, for every ξ ∈ S × ind there exists a unique ˜ ξ ∈ h ∗ , defined by ˜ ξ ( h η ) = h ξ, η ∨ i for η ∈ S × ind , such that the map ξ ˜ ξ extends to an isomorphism between the rootsystem S ind and the root system of ( g , h ) .Proof. This is a special case of a result for arbitrary root-graded Lie algebras, see[Ne1, Remark 2 of § § (cid:3) The following Ex. 4.10 lists some more basic properties of Lie tori. You will needEx. 4.9(a) in part (d) of 4.10.
Exercise 4.9. (a) Let K be a perfect Lie algebra. Then K/Z ( K ) is a perfect andcentreless Lie algebra.(b) Let E be a Lie algebra with an invariant nondegenerate symmetric bilinearform ( ·|· ), and let K be an ideal of E with K = [ E, K ]. Then { z ∈ K : ( z | K ) =0 } = Z ( K ). Exercise 4.10.
Let L be a Lie torus of type ( S, Λ). Show:(a) L λ is given by the formula (4.6).(b) L is perfect.(c) The centre satisfies Z ( L ) = L λ ∈ Λ Z ( L ) λ for Z ( L ) λ = Z ( L ) ∩ L λ .(d) Let Y = L λ ∈ Λ Y λ , Y λ = Y ∩ L λ , be a graded subspace of Z ( L ). Then L/Y is a Lie torus with respect to the subspaces (
L/Y ) λξ = L λξ /Y λξ , where for ξ = 0 weput Y λξ = { } and thus have ( L/Y ) λξ ∼ = L λξ as vector spaces. In particular, L/Z ( L )is a centreless Lie torus.(e) For λ, µ ∈ Λ ξ , ξ ∈ S × , we have h λξ ≡ h µξ mod Z ( L ).(f) L = g ⊕ Z ( L ) and L = h ⊕ Z ( L ) .(g) Let I be a Λ-graded ideal of L , whence I = L λ ∈ Λ I λ for I λ = I ∩ L λ . Theneither I = L or I ⊂ Z ( L ). In particular, a centreless Lie torus is graded-simplewith respect to the Λ-grading of L .Since a Lie torus is perfect by part (b) of the exercise above, it has a universalcentral extension (Th. 1.10). Theorem 4.11 ([Ne3, § . Let u : uce ( L ) → L be a universal centralextension of a Lie torus L = L ξ,λ L λξ of type ( S, Λ) . Then uce ( L ) is also a Lietorus of type ( S, Λ) , say uce ( L ) = L ξ,λ uce ( L ) λξ , and u maps uce ( L ) λξ onto L λξ . Remark 4.12.
It follows from this theorem and the exercise above that in orderto describe Lie tori up to graded isomorphism, one can proceed in two steps:(A) Classify centreless Lie tori, up to graded isomorphism. We will discuss someexamples in 4.4 and 4.5.
ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 43 (B) Describe the universal central extension of the centreless Lie tori from (A).They are unique up to isomorphism. We will not say anything about thishere. The reader can find some results in [BGK, BGKN, Ne4, Ne6] forLie tori arising from EALAs and in [ABG1, ABG2, BS, Ne6] for generalroot-graded Lie algebras.Once (A) and (B) completed, an arbitrary Lie torus of type ( S, Λ) is then obtainedas uce ( L ) /C where L is taken from the list in (A) and where C is a graded subspaceof the centre of uce ( L ).The following results only holds for special types of Lie tori. Theorem 4.13 ([Ne3, Th. 5], proven in [Ne6]) . Let L = L ξ ∈ S, λ ∈ Λ L λξ be a Lietorus of type ( S, Λ) where Λ is a finitely generated abelian group. (a) Then L is finitely generated as Lie algebra and has bounded homogeneousdimension with respect to the Q ( S ) ⊕ Λ -grading of L . (b) Moreover, the Lie algebra
Der F ( L ) = grDer F ( L ) where grDer F ( L ) is natu-rally Q ( S ) ⊕ Λ -graded and has bounded homogeneous dimension with respect to thisgrading. (c) If L is invariant, its universal central extension is isomorphic to the cen-tral extension E( L, D gr ∗ , ψ D ) where D is any graded complement of IDer( L ) in SDer F ( L ) . Part (c) of this theorem is an immediate corollary of parts (a) and (b) and ofTh. 1.25.4.3.
The core of an EALA.
We will now connect extended affine Lie algebrasand Lie tori, and first introduce some notation. Let (
E, H ) be an EALA with set ofroots R . We have seen in Th. 3.22 that R is an extended affine root system, hencean affine reflection system. We can therefore apply the Structure Theorem 3.10.Recall the following data describing the structure of R : • X = span F ( R ) ⊂ H ∗ , X = { x ∈ X : ( x | X ) = 0 } = { x ∈ X : ( x | R ) =0 } , f : X → X/X = Y the canonical projection, • S = f ( R ) the quotient root system, a finite irreducible but possibly non-reduced root system, and S ind = { α ∈ S : α/ S } ∪ { } , • g : Y → X a linear map satisfying f ◦ g = Id Y and g ( S ind ) ⊂ R , • (Λ ξ : ξ ∈ S ) the associated extension datum, defined by R ∩ f − ( ξ ) = g ( ξ ) ⊕ Λ ξ and Λ ξ ⊂ X , • Λ = span Z (cid:0) S ξ ∈ S Λ ξ (cid:1) , a free abelian group of finite rank (this is axiom(EARS7)).Hence R = S ξ ∈ S (cid:0) g ( ξ ) ⊕ Λ ξ (cid:1) ⊂ g ( Y ) ⊕ X ,R an = S ξ ∈ S × (cid:0) g ( ξ ) ⊕ Λ ξ (cid:1) ,R = 0 ⊕ Λ = R ∩ X . Theorem 4.14 ([AG] for F = C ) . Let K = E c be the core of an EALA ( E, H ) .We use the notation of above and define subspaces (4.15) K λξ = K ∩ E g ( ξ ) ⊕ λ = ( E g ( ξ ) ⊕ λ ξ = 0 ,K ∩ E ⊕ λ , ξ = 0 . (a) Then K = L ξ, λ K λξ is a Lie torus of type ( S, Λ) , where Λ is free abelian offinite rank. (b) K is a perfect ideal of E . (c) Let ( ·|· ) be a nondegenerate invariant bilinear form on E , whose existenceis guaranteed by the axiom (EA1) . Then the radical of the restricted bilinear form ( ·|· ) | K × K equals the centre Z ( K ) , that is (4.16) { z ∈ K : ( z | K ) = 0 } = Z ( K ) = L λ ∈ Λ Z ( K ) ∩ K λ . Remark 4.15.
The subspaces K λξ in (4.15) and hence the Lie torus structure of K depend on the section g . A different choice of g leads to a so-called isotope of K , see [AF] and [Ne5, Prop. 6.4]. Corollary 4.16.
We use the notation of
Th. 4.14 , and put E cc = K/Z ( K ) = L ,the centreless core of ( E, H ) . Then L is an invariant centreless Lie torus of type ( S, Λ) with respect to the homogeneous subspaces L λξ = K λξ (cid:14)(cid:0) Z ( K ) ∩ K λξ (cid:1) and the bilinear form ( ·|· ) L defined by (¯ x | ¯ y ) L = ( x | y ) where x, y ∈ K , ¯ x and ¯ y are the canonical images in L and ( ·|· ) is the bilinear formof . Remark 4.17.
Yoshii ([Y4]) has shown that any Lie torus of type ( S, Λ) with Λa torsion-free abelian group admits a non-zero graded invariant symmetric bilin-ear form. This implies that the existence of a nondegenerate such form on E cc .However, Yoshii’s proof uses the existence of invariant nondegenerate symmetricbilinear forms on Jordan tori ([NY]) and hence relies on the classification of Jordantori.We can now show that all root spaces of an EALA are finite-dimensional in astrong form. Proposition 4.18 ([Ne4, Prop. 3]) . An EALA has finite bounded dimension. Thesame is true for its core and centreless core.Proof.
Let K = E c be the core of the EALA ( E, H ). By Th. 4.14, K is a Lie torusof type ( S, Λ), where Λ is a free abelian group of finite rank. Hence, by Th. 4.13,one knows that K has finite bounded dimension with respect to its double grading,say dim K λξ ≤ M for all pairs ( ξ, λ ). By the same reference, one also knows thatthe Lie algebra Der F ( K ) of all F -linear derivations of K has a double grading by Q ( S ) and Λ, Der F ( K ) = L ξ ∈ S, λ ∈ Λ (Der F K ) λξ , where (Der F K ) λξ is the subspace of those derivations mapping K µτ to K λ + µξ + τ , andthat Der F ( K ) has finite bounded dimension with respect to this grading, saydim F (Der F K ) λξ ≤ M for all pairs ( ξ, λ ).Since K is an ideal, we have a Lie algebra homomorphism ρ : E → Der F ( K ),given by ρ ( e ) = ad e | K . It is homogenous of degree 0, i.e., ρ ( E α ) ⊂ (Der F K ) λξ for α = g ( ξ ) ⊕ λ as in (4.15). Moreover, by the tameness axiom (EA5) for an EALAwe know that Ker ρ ⊂ K (whence Ker ρ = Z ( K ), but we won’t need this). It now ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 45 follows that dim E α = dim Ker( ρ | E α ) + dim ρ ( E α ) ≤ dim K λξ + dim(Der F K ) λξ ≤ M + M . (cid:3) Lie tori of type A l , l ≥ . As explained in Rem. 4.12, in classifying Lie torione can restrict one’s attention to the case of centreless Lie tori, at least modulothe solution of problem (B) in 4.12. In this section we will describe centreless Lietori of type A.The reader will expect that this will have something to do with trace-0-matrices.This turns out to be correct, but only with the proper interpretation of “trace-0”.It will not be sufficient to consider trace-0-matrices over F . We will see in 4.28 thatthey will only lead to nullity 0-examples. Rather, one must allow matrices withentries from a possibly non-commutative algebra.To avoid some degeneracies, in this section we let N be a natural number with N ≥
3. We start with an arbitrary associative unital F -algebra A . In particular, A need not be commutative, and hence[ A, A ] = span F { a a − a a : a , a ∈ A } is in general non-zero. As usual, gl N ( A ) is the Lie algebra of all N × N matriceswith entries in A and Lie algebra product [ x, y ] = xy − yx , the usual commutatorof the matrices x and y . We define the special linear Lie algebra sl N ( A ) as thederived algebra sl N ( A ) = [ gl N ( A ) , gl N ( A )] . of gl N ( A ). In particular, sl N ( A ) is an ideal of gl N ( A ). To analyze the structure of sl N ( A ) we use the matrix units E ij , i.e., the N × N matrices with 1 at the position( ij ) and 0 at all other positions. They satisfy the basic multiplication rule(4.17) [ aE ij , bE mn ] = δ jm ab E in − δ ni ba E mj where δ ∗ is the usual Kronecker delta. We put E N = P Ni =1 E ii . Some properties ofthe Lie algebra sl N ( A ) are listed in the following (very worthwhile) exercise. Exercise 4.19. (a) sl N ( A ) = { x ∈ gl N ( A ) : tr( x ) ∈ [ A, A ] } .(b) As a vector space, sl N ( A ) decomposes as sl N ( A ) = sl N ( A ) ⊕ (cid:0) L i = j AE ij (cid:1) , where(4.18) sl N ( A ) = sl N ( A ) ∩ (cid:0) L Ni =1 AE ii (cid:1) = P i = j [ AE ij , AE ji ] = P i = j span F { abE ii − baE jj : a, b ∈ A } = { cE N : c ∈ [ A, A ] } ⊕ (cid:0) L N − i =1 { a ( E ii − E i +1 ,i +1 ) : a ∈ A } (cid:1) (c) For a commutative A : sl N ( A ) = { x ∈ gl N ( A ) : tr( x ) = 0 } = sl N ( F ) ⊗ F A. (d) The centre of sl N ( A ) is Z ( sl N ( A )) = { zE N : z ∈ Z ( A ) ∩ [ A, A ] } where Z ( A ) = { z ∈ A : za = az for all a ∈ A } is the centre of A .(e) For a, b ∈ A and i = j , ( aE ij , E ii − E jj , bE ji )is an sl -triple if and only if a is invertible and b = a − . The exercise shows that the structure of a general sl N ( A ) is quite similar to thatof sl N ( F ). In particular, the decomposition (4.18) is a Q (A l )-grading whereA l = { ε i − ε j : 1 ≤ i, j ≤ N } , l = N − . is the root system of type A l and sl N ( A ) ε i − ε j = AE ij for i = j .In fact, sl N ( A ) is the prototype of an A l -graded Lie algebra (see [BerM]). At thislevel of generality we are far from the structure of a Lie torus. Most importantly, weare missing a compatible Λ-grading of sl N ( A ). We will use gradings of A , definedas follows. Definition 4.20.
Let A = L λ ∈ Λ A λ be a unital associative Λ-graded F -algebra.Then A is called an associative torus of type Λ if it satisfies (AT1)–(AT3) below. (AT1): if every non-zero A λ contains an invertible element, (AT2): dim A λ ≤ λ ∈ Λ, and (AT3): span Z (supp Λ A ) = Λ.One calls A simply an associative torus if A is an associative torus of type Λ forsome abelian group Λ. See 4.23 for a short discussion of associative tori.These definitions are justified by the following exercise, describing when sl N ( A )is a Lie torus of type (A l , Λ).
Exercise 4.21. (a) The Lie algebra sl N ( A ) has a Λ-grading compatible with the Q (A l )-grading (4.18) if and only if A is Λ-graded. In this case, the compatibleΛ-grading of sl N ( A ) is given by sl N ( A ) = L λ ∈ Λ sl N ( A ) λ where sl N ( A ) λ consistsof matrices in sl N ( A ), which have all their entries in A λ .(b) With respect to the compatible gradings of (a), the Lie algebra sl N ( A ) is aLie torus of type (A N − , Λ) if and only if A is an associative torus of type Λ.(c) The Lie torus sl N ( A ) is invariant with respect to the bilinear form ( ·|· ) sl given by ( P i,j x ij E ij | P p,q y pq E pq ) sl = P i,j ( x ij y ji ) where a for a ∈ A denotesthe A -component of a .But we not only have an example of a Lie torus of type (A l , Λ), we actually haveall centreless examples.
Theorem 4.22.
Let l ≥ . A Lie algebra L is a centreless Lie torus of type (A l , Λ) if and only if L is graded-isomorphic to sl l +1 ( A ) for A an associative torus of type Λ . In this case, L is an invariant Lie torus.Proof. This is a special case of the Coordinatization Theorem of A l -graded Liealgebras ([BerM, Recognition Theorem 0.7]): A centreless Lie algebra L is A l -graded ( l = N −
1) if and only if L is Q (A l )-graded-isomorphic to sl N ( A ) /Z ( sl N ( A ))for some associative F -algebra A . If L is a Lie torus, it follows as in the Exercise 4.21above that A is an associative torus. But Z ( sl N ( A )) = { } for an associative torus([NY, (3.3.2)] and Exercise 4.19). (cid:3) Besides [BerM], related results are proven in [BGK, Th. 2.65] (see Cor. 4.27below), [GN, 2.11 and 3.4] and [Y2, Prop. 2.13].
Review 4.23 (Associative tori versus twisted group algebras) . In view of Th. 4.22it is of interest to know more about associative tori. First of all, the identity 1 A of an associative torus A satisfies 1 A ∈ A . Hence a − ∈ A − λ for every invertible ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 47 a ∈ A λ . Moreover, since the product of two invertible elements in an associativealgebra is again invertible, it follows that supp Λ A is a subgroup of Λ, whence (AT3)is equivalent to (AT3) ′ : supp Λ A = Λ.Next, choose a family ( u λ : λ ∈ Λ) of invertible elements u λ ∈ A λ . This is thenin particular an F -basis of A so that the algebra structure of A is completelydetermined by the equations(4.19) u λ u µ = c ( λ, µ ) u λ + µ for λ, µ ∈ Λ and suitable non-zero scalars c ( λ, µ ) ∈ F . It is not necessary that c ( λ, µ ) = 1, see for example Ex. 4.26. Rather, given an F -vector space with basis( u λ : λ ∈ Λ) one can define a multiplication on A by (4.19), and this multiplicationis associative if and only if(4.20) c ( λ, µ ) c ( λ + µ, ν ) = c ( µ, ν ) c ( λ, µ + ν )In this case, the algebra is an associative torus of type Λ. It is clear from theconstruction that, conversely, any associative torus is obtained in this way froma family ( c ( λ, µ )) λ,µ of non-zero scalars. The algebras constructed in this way arecalled twisted group algebras . The reader with some knowledge in group cohomologywill recognize that the families ( c ( λ, µ )) satisfying (4.20) are precisely the 2-cocyclesof Λ with values in F \{ } . One can show that two families define graded-isomorphictori if and only if their cohomology classes coincide. Example 4.24 (Group algebra) . Although, as we have pointed out, the c ( λ, ν )need not equal 1 in general, the family for which all c ( λ, µ ) = 1 satisfies (4.20) andso yields an example of a Λ-torus, called the group algebra of Λ and denoted F [Λ].In particular, this implies that associative tori exist for all Λ, and hence Lie toriexist for all types (A l , Λ), l ≥ S, Λ) where Λ is a freeabelian group of finite rank. This condition on Λ follows from the axiom (EA6) or,equivalently from the axiom (EARS7). We therefore discuss this special case now.
Definition 4.25.
Let q = ( q ij ) be an n × n matrix such that the entries q ij ∈ F satisfy q ii = 1 = q ij q ji for all 1 ≤ i, j ≤ n . The quantum torus associated to q isthe associative algebra F q presented by the generators t i , t − i , 1 ≤ i ≤ n subject tothe relations t i t − i = t − i t i , and t i t j = q ij t j t i for all 1 ≤ i, j ≤ n. For example, if all q ij = 1, then F q = F [ t ± , . . . , t ± n ] is the Laurent polynomial ringin n variables. Thus, a general F q is a non-commutative version of F [ t ± , . . . , t ± n ],the coordinate ring of the n -dimensional algebraic torus ( F \ { } ) n , which explainsthe name “quantum torus”. Exercise 4.26.
Let F q be a quantum torus. Show:(a) F q = L λ ∈ Z n F t λ for t λ = t λ · · · t λ n n .(b) The t λ satisfy the multiplication rule t λ t µ = c ( λ, µ ) t λ + µ with c ( λ, µ ) = Q ≤ j
We describe some more easy exam-ples, where easy means that they do not require some knowledge of non-associativealgebras, like Jordan algebras, alternative or structurable algebras.
Example 4.28 (Λ = { } ) . Let g be a finite-dimensional split simple Lie algebrawith splitting Cartan subalgebra h . Then g has a root space decomposition g = L ξ ∈ S g ξ where g = h and S is the root system of ( g , h ), a finite reduced rootsystem. Since g is simple, S is also irreducible. Using standard properties of finite-dimensional split simple Lie algebras, it is easy to check that g = L ξ ∈ S g ξ is a Lietorus of type ( S, { } ) . Conversely, if L is a Lie torus of type ( S, { } ), then L is a finite-dimensional splitsimple Lie algebra. Indeed, L = g in the notation of Prop. 4.8.Note that this fits nicely the picture of EALAs of nullity 0, which we havecharacterized in Prop. 3.24 as finite-dimensional split simple Lie algebras. Example 4.29.
As in the previous Example 4.28 let g be a finite-dimensional splitsimple Lie algebra with splitting Cartan subalgebra h and root system S . We would ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 49 like to consider a Lie algebra of the form g ⊗ A where A is an associative algebra.For g of type A we could take non-commutative “coordinates” A to get a Lie torus,see § g not of type A the algebra A must be commutative in orderto get a Lie algebra.Therefore, we let A = L λ ∈ Λ A λ be a commutative associative torus of typeΛ and consider g ⊗ A , which becomes a Lie algebra (over F ) by [ u ⊗ a , u ⊗ a ] = [ u , u ] ⊗ a a . It is a centreless Lie torus of type ( S, Λ) with respect to thehomogeneous subspaces ( g ⊗ A ) λξ = g ξ ⊗ A λ . Note that the support of the Q ( S ) ⊕ Λ-graded Lie algebra g ⊗ A is the setsupp Q ( S ) ⊕ Z n g ⊗ A = S × Λ , and that g ⊗ A is an invariant Lie torus with respect to the bilinear form( x ⊗ a λ | y ⊗ b µ ) = κ ( x, y ) ( a λ b µ ) where κ is the Killing form of g and c for c ∈ A is the 0-component of c . Thisexample works for any type of S . But it yields all examples only for special typesof S . Theorem 4.30.
Any centreless Lie torus of type ( S, Λ) for S of type D l , l ≥ or E l , l = 6 , , , is graded-isomorphic to an example as in for g of the correspondingtype and A a commutative associative torus of type Λ .Proof. The proof is analogous to the proof of Th. 4.22: One applies the Coordina-tization Theorem of [BerM] to get that L has the form g ⊗ A for some commutativeassociative F -algebra. One then has to discuss when such a Lie algebra is a Lietorus. This is the case exactly when A is a torus. (cid:3) Example 4.31 (Untwisted multiloop algebras) . For EALAs it is of interest todescribe the centreless Lie tori of type ( S, Λ) with Λ a free abelian group of finiterank, say of rank n . Hence Λ ∼ = Z n . It is immediate that g ⊗ A is a Lie torus of type( S, Z n ) if and only if A is a commutative quantum torus, i.e., a Laurent polynomialring in several, say n variables. In other words, these are the untwisted multiloopalgebra of (1.11), L ( g ) = g ⊗ F F [ t ± , . . . , t ± n ]Hence by Th. 4.11 the universal central extension of L ( g ), the toroidal Lie algebrasof § Corollary 4.32 ([BGK]) . Any centreless Lie torus of type ( S, Z n ) with S = D l , l ≥ or S = E l , l = 6 , , , is graded-isomorphic to an untwisted multiloop algebra L ( g ) as in Example . Perhaps the reader now expects that the next example will be the general multi-loop algebras L ( g , σ ) defined in (1.13). However, an arbitrary multiloop algebra isin general not a Lie torus, see [ABFP2, Th. 3.3.1] and [Na, Th. 5.1.4] for a charac-terization of centreless Lie tori which are multiloop algebras. But this phenomenondoes not occur in nullity 1. Exercise 4.33.
Verify that the loop algebra L ( g , σ ) of (1.3) is an invariant Lietorus of type ( S, Z ) where S is the root system of Table 3.7. The construction of all EALAs
Recall Th. 4.14: If (
E, H ) is an EALA, its core E c and its centreless core E cc are Lie tori, the latter being an invariant Lie torus. Moreover, if ( S, Λ) is the typeof E c and E cc then Λ is a free abelian group of finite rank. Thus:core E c (Lie torus) (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O EALA (
E, H ) o o o/ o/ o/ o/ o/ o/ o/ o/ o/ centreless core E cc (invariant Lie torus)In this chapter we will reverse the process, starting from an invariant Lie torus wewill construct an EALA.To motivate the construction it is useful to look again at the construction inExample 1.1 and 2.4 of an affine Kac-Moody Lie algebra. It can be summarized asfollows: We start with a twisted loop algebra L = L ( g , σ ), which as we now know isan invariant Lie torus (Ex. 4.33). We then take a central extension ˜ L , which in thisexample is the universal central extension and hence by Th. 4.11 again a Lie torus(of course, one can also verify this directly in this example). Finally, we add some(not all) derivations to ˜ L to get an affine Kac-Moody Lie algebra, and all affineKac-Moody Lie algebras are obtained in this way (Kac’s Realization Theorem 1.3).To summarize, using the EALA terminology:central extension of L (another Lie torus) addderivations / / /o/o/o/o/o/o/o/o/o/o/o EALA (
E, H )invariant Lie torus L O O O(cid:15)O(cid:15)O(cid:15)
To do something like this in general, one faces the following two problems.(A) An invariant Lie torus has in general many central extension. For example,the untwisted multiloop algebra L = g ⊗ F [ t ± , . . . , t ± n ] is an invariant Lie torusby Ex. 4.31. If n ≥
2, its universal central extension has an infinite-dimensionalcentre, a result we already mentioned in § affine extension (after all, the result will be anextended affine Lie algebra). In fact, affine extensions are a special case of so-called double extension , see for example [Bor].(5.1) central extension of L / / EALA (
E, H )invariant Lie torus L O O affineextension ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 51
The key idea is based on the construction of a 2-cocycle in Ex. 1.19: Any subspace D of skew-symmetric derivations will give rise to a 2-cocycle and hence to a centralextension. But not only do we get examples of central extensions. By Th. 1.25 andEx. 1.27, up to isomorphism all central coverings of L are of the form E( L, D, ψ D )for some graded subspace D of SDer( L ) with D ∩ IDer( L ) = { } . Observe that wecan indeed apply this theorem: The invariant Lie torus L (i) is perfect by Ex. 4.10 and is finitely generated as Lie algebra by Th. 4.13(recall that Λ is free of finite rank, where ( S, Λ) is the type of L ),(ii) has finite homogeneous dimension, even bounded homogeneous dimensionalso by Th. 4.13, and(iii) has an invariant nondegenerate Λ-graded symmetric bilinear form, by def-inition of an invariant Lie torus.But we need more than just a central covering. For example, the axiom (EA2)requires that we construct an ad-diagonalizable subalgebra H for which the sub-spaces L λξ , ξ = 0, are root spaces, as can be seen from Th. 4.14. By Lemma 4.4we can realize the subspaces L ξ as root spaces of some natural subalgebra h ⊂ L .But we do not have a result, which describes the subspaces L λ in a similar fashion,i.e., as root spaces of some toral subalgebra. There is in fact no natural choice of asubalgebra to do so. Rather, we will distinguish these subspaces “externally”, i.e.,via an action of some non-inner derivation algebra. The required formalism to dothis, is described in the next section. This has nothing to do with Lie algebras.Rather, it is a topic in the theory of graded vector spaces, and we will thereforedescribe it in this setting.5.1. Degree maps.
In this section, V is vector space over a field F , which could beof arbitrary characteristic until Prop. 5.3(b). Also, Λ denotes an arbitrary abeliangroup. We recall that a Λ-grading of V is simply a direct vector space decompositionof V by a family ( V λ : λ ∈ Λ) of subspaces V λ ⊂ V . Our goal in this section is topresent a method describing the homogeneous subspaces V λ of a given Λ-gradingof V as the joint eigenspaces of a subspace of diagonalizable endomorphisms.To motivate the construction, let us first look at the converse, namely inducinga grading of V via the action of endomorphisms. We will say that a subspace T ⊂ End F ( V ) is a subspace of simultaneously diagonalizable endomorphisms , if V = L λ ∈ T ∗ V λ for(5.2) V λ = { v ∈ V : t ( v ) = λ ( t ) v for all t ∈ T } . In this case, T obviously consists of pairwise commuting diagonalizable endomor-phisms. Conversely, it is well-known that a finite-dimensional subspace of pairwisecommuting diagonalizable endomorphisms is a subspace of simultaneously diagonal-izable endomorphisms (this is no longer true if T is infinite-dimensional). Observethat the decomposition (5.2) is a grading of V by the group span Z (supp T ∗ V ) wheresupp T ∗ V = { λ ∈ T ∗ : V λ = 0 } . Our goal is to realize a given grading of V in thisway.To do so, we will use the F -vector spaceD(Λ) = Hom Z (Λ , F )consisting of all maps θ : Λ → F which are Z -linear: θ ( λ + λ ) = θ ( λ ) + θ ( λ ) forall λ i ∈ Λ. This is an F -vector space by defining for θ, θ i ∈ D(Λ) and s ∈ F the sum θ + θ and the scalar multiplication sθ by ( θ + θ )( λ ) = θ ( λ ) + θ ( λ ) and( sθ )( λ ) = s ( θ ( λ )). Exercise 5.1. (a) Show D(Λ) ∼ = Hom F (Λ ⊗ Z F, F ) = (Λ ⊗ F F ) ∗ . Thus D(Λ) isnaturally a dual vector space.(b) If Λ is free of rank n , say with Z -basis ε , . . . , ε n , then D(Λ) = F ∂ ⊕· · ·⊕ F ∂ n where ∂ i ∈ D(Λ) is defined by ∂ i ( P j m j ε j ) = m i . In particular, dim F D(Λ) = n .We now suppose that V = L λ ∈ Λ V λ is a Λ-grading of the vector space V . Any θ ∈ D(Λ) defines an endomorphism ∂ θ ∈ End F ( V ) by ∂ θ ( v λ ) = θ ( λ ) v λ for v λ ∈ V λ . We put D ( V ) = { ∂ θ : θ ∈ D(Λ) } and call the elements of D ( V ) degree maps . If A = L λ ∈ Λ A λ is a Λ-grading ofan algebra A , the maps ∂ θ are derivations and D ( A ) is called the space of degreederivations .The map ∂ : D(Λ) → D ( V ) is clearly F -linear and surjective by definition. Itskernel is { θ ∈ D(Λ) : θ (supp Λ V ) = 0 } . To make ∂ an isomorphism we will(5.3) from now on assume span Z (supp Λ V ) = Λ . As we have pointed out at previous occasions, this is not a serious assumptions sinceone can always replace Λ by span Z (supp Λ V ) without changing the given grading.Since now ∂ is an isomorphism, we can define a linear form ev λ ∈ D ( V ) ∗ for every λ ∈ Λ: ev λ ( ∂ θ ) = θ ( λ )The F -linear map ev : Λ → D ( V ) ∗ , λ ev λ is called the evaluation map . By construction,(5.4) V λ ⊂ { v ∈ V : d ( v ) = ev λ ( d ) v for all d ∈ D ( V ) } since for d = ∂ θ and v ∈ V λ we have ∂ θ ( v λ ) = θ ( λ ) v λ = ev λ ( ∂ θ ) v λ . Definition 5.2.
In the setting of above, i.e., V = L λ ∈ Λ V λ is Λ-graded and (5.3)holds, we will say that a subspace T ⊂ D ( V ) induces the Λ -grading of V if V λ = { v ∈ V : t ( v ) = ev λ ( t ) v for all t ∈ T } holds for all λ ∈ Λ. Proposition 5.3.
Let V = L λ ∈ Λ V λ be a Λ -grading of the vector space V suchthat (5.3) holds. (a) A subspace T ⊂ D ( V ) induces the Λ -grading of V if the restricted evaluationmap ev T : Λ → T ∗ , ev T ( λ ) = ev λ | T is injective. (b) Suppose F has characteristic and Λ is torsion-free, i.e., nλ = 0 for some n ∈ Z implies λ = 0 . Then Λ embeds into the F -vector space U = Λ ⊗ Z F and forevery subspace S ⊂ D(Λ) separating the points of Λ in U the corresponding subspace T = ∂ ( S ) ⊂ D ( V ) induces the Λ -grading of V . In particular, this holds for D ( V ) itself. ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 53
The centroid of Lie algebras, in particular of Lie tori.
After the inter-mezzo on how to induce gradings of vector spaces in the previous section 5.1 wenow come back to Lie algebras, but not immediately to Lie tori and EALAs. Ofcourse, the topic of this section is motivated by the over-all goal of this chapter:The construction of EALAs from Lie tori using certain subspaces of derivations.The derivations, which in 5.4 will be used in the general construction, are productsof degree maps, studied in 5.1, and so-called centroidal transformations, to whichthis section is devoted.The basic idea of the centroid of a Lie algebra (or of any algebra for that matter)is that it identifies the largest ring over which the given algebra can be consideredas an algebra. For example, if one studies the real Lie algebra L which is sl n ( C )considered as a real Lie algebra by restricting the scalars to R , the centroid will be ∼ = C and will thus indicate that L can also be considered as a complex Lie algebra.In general, the centroid will not be a field but only a (commutative) ring. Hence,considering a Lie algebra as algebra over its centroid, necessitates that in the fol-lowing definition and the Lemma 5.6 after it we will deviate from our standardassumption and consider Lie algebras over rings. The definition of a Lie algebra L defined over a ring, say k , is not surprising: L is a k -module with a k -bilinear map[ ., . ] : L × L → L which is alternating, i.e., [ l, l ] = 0 for all l ∈ L , and satisfies theJacobi identity. Definition 5.4 ([J, Ch. X]) . The centroid
Cent k ( L ) of a Lie algebra L defined overa ring k is defined asCent k ( L ) = { χ ∈ End k ( L ) : χ ([ l , l ]) = [ l , χ ( l )] for all l , l ∈ L } . Of course, χ ∈ Cent k ( L ) ⇔ χ ([ l , l ]) = [ χ ( l ) , l ] for all l , l ∈ L . It is importantto indicate k in the notation Cent k ( L ) since the centroid depends on the base ring k . We have k Id L ⊂ Cent k ( L ) for every L . One calls L central if the map k → Cent F ( L ), s s Id L , is an isomorphism, and one says that L is central-simple if L is just that: central and simple.Let L = L λ ∈ Λ L λ be a Lie algebra graded by an abelian group Λ. We can thenalso define the Λ -graded centroid asgrCent k ( L ) = grEnd k ( L ) ∩ Cent k ( L ) = L λ ∈ Λ Cent k ( L ) λ where Cent k ( L ) λ consists of the centroidal transformations which have degree λ : χ ( L µ ) ⊂ L λ + µ for all µ ∈ Λ. Example 5.5.
As an immediate example we calculate the centroid of the Liealgebra L = sl ( C ), considered as real Lie algebra by restricting the scalars to R . We let ( e, h, f ) be an sl -triple in sl ( C ). Then the relations [ h, ce ] = 2 ce and [ h, cf ] = − cf for c ∈ C show that χ ( ce ) and χ ( cf ) are uniquely determinedby χ ( h ). For example, 2 χ ( ce ) = χ ([ h, ce ]) = [ χ ( h ) , ce ]. Moreover, χ ( h ) ∈ C h because [ χ ( h ) , ch ] = χ ([ h, ch ]) = 0. Hence dim R Cent R ( L ) ≤
2. On the other side, C Id L ⊂ Cent R ( L ) is clear, whence C Id L = Cent R ( L ).We leave it to the reader to show Cent R ( L ) = C Id L for L = sl n ( C ) consideredas real Lie algebra, without using any of the results mentioned below!The following lemma gives a mathematical meaning to the claims made beforethe Def. 5.4, and lists the most important properties of the centroid of Lie algebraswhich are not necessarily Lie tori. Lemma 5.6 (Folklore) . Let L be a Lie algebra defined over a ring k . (a) The centroid of L is always a unital associative subalgebra of the endo-morphism algebra End k ( L ) of L . Hence Cent k ( L ) is a k -algebra and L becomesa Cent k ( L ) -module by defining the action of Cent k ( L ) on L by χ · l = χ ( l ) for χ ∈ Cent k ( L ) and l ∈ L . (b) If the centroid of L is commutative, then with respect to the action of Cent k ( L ) on L defined in (a) , L is a Lie algebra over the ring Cent k ( L ) . Moreover, L is central as a Lie algebra over its centroid. (c) If L is perfect, its centroid is commutative and does not depend on the basering k : Cent k ( L ) = Cent Z ( Z L ) where Z L is the Lie algebra L with scalars restrictedto Z . (d) If L is simple, its centroid is a field and L as a Lie algebra over the field Cent F ( L ) is central-simple. In particular: (i) a finite-dimensional simple Lie algebra over an algebraically closed field F is central-simple, and (ii) the centroid of a simple real Lie algebra L is either ∼ = R Id , in which case L is central-simple, or is ∼ = C Id , in which case L is a simple complex Liealgebra, considered as a real Lie algebra. (e) Suppose L is Λ -graded. Then grCent k ( L ) is a Λ -graded subalgebra of the fullcentroid Cent k ( L ) . Moreover, grCent k ( L ) = Cent k ( L ) if L is finitely generated asan ideal, i.e., there exist l , . . . , l n ∈ L such that the ideal generated by l , . . . , l n isall of L . (f) If χ ∈ Cent k ( L ) and d ∈ Der k ( L ) , then χ ◦ d ∈ Der k ( L ) . With respect tothis operation, Der k ( L ) is a Cent k ( L ) -module and IDer( L ) is a submodule of the Cent k ( L ) -module Der k ( L ) . The proof of this lemma is a straightforward exercise, which the reader will beasked to do now. The exercise also lists some interesting additional facts on thecentroid.
Exercise 5.7. (a) For any χ ∈ Cent k ( L ) the kernel Ker χ and the image Im χ areideals of L satisfying [Ker χ, Im χ ] = 0.(b) Prove Lemma 5.6. For part (c) of the Lemma use (a) above.(c) If L is perfect, any χ ∈ Cent k ( L ) is symmetric with respect to any invariantbilinear form on L .Here is the result, which describes the centroid of the Lie algebras of interest inthis chapter. We will use the notion of an associative torus, introduced in 4.20 andfurther discussed in 4.23–4.26. Proposition 5.8 ([BN, Prop. 3.13]) . Let L = L ξ ∈ S, λ ∈ Λ L λξ be a centreless Lietorus of type ( S, Λ) . (a) With respect to the ( Q ( S ) ⊕ Λ) -grading of L we have (5.5) Cent F ( L ) = L λ ∈ Λ Cent F ( L ) λ = grCent F ( L ) . In particular, χ ( L ξ ) ⊂ L ξ for any χ ∈ Cent k ( L ) and ξ ∈ S . (b) Moreover, with respect to the decomposition (5.5) the centroid
Cent F ( L ) is anassociative commutative torus of type Γ , where Γ = supp Λ Cent F ( L ) is a subgroupof Λ , hence a twisted group algebra over Γ . ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 55 (c)
In particular, if Λ is free abelian of finite rank n , the centroid Cent F ( L ) isgraded-isomorphic to F [Γ] , the group algebra of Γ as defined in , and is thusisomorphic to a Laurent polynomial ring in ν variables, ≤ ν ≤ n . Moreover, L isa free module over its centroid.Proof. Parts (a) and (b) of this proposition are proven in [BN, Prop. 3.13]. Part(c) is ([Ne3, Th. 7]). The first part of (c) follows from (b): A twisted group algebraover a free group is a group algebra. The second part is a special case of a generalfact: Any graded module over an associative torus is free. (cid:3)
Example 5.9.
Let L = sl N ( A ) for A an associative F -algebra, see 4.4, and let Z ( A ) = { z ∈ A : [ z, A ] = 0 } be the centre of the associative algebra A . Any z ∈ Z ( A ) induces a centroidal transformation χ z defined by mapping x = ( x ij ) ∈ sl N ( A ) to χ z ( x ) = ( zx ij ). It is easily seen ([Ne5, 7.9]) that Z ( A ) → Cent F ( sl N ( A )) , z χ z is an isomorphism of F -algebras (the only non-obvious part is surjectivity).Let us now specialize to the case of a Lie torus sl N ( A ) of type (A N − , Z n ). Thus,by Cor. 4.27, A = F q is a quantum torus. A description of the centre Z ( F q ) is givenin Ex. 4.26(e) (see [BGK, Prop. 2.44] for a proof): Z ( F q ) = L γ ∈ Γ F t γ where Γ isthe subgroup Γ = { γ ∈ Z n : Q nj =1 q γ j ij = 1 for 1 ≤ i ≤ n } . of Z n . The centre of F q is therefore isomorphic to a Laurent polynomial ring in,say, ν variables, as claimed in Prop. 5.8(c). To see that the inequalities 0 ≤ ν ≤ n stated there are sharp, we consider the quantum torus associated to the matrix q = (cid:20) qq − (cid:21) . Specializing the description of Γ above we getΓ = ( { } , q not a root of unity ,m Z ⊕ m Z , q an m th root of unity . Hence ν = 0 in the first case and ν = 2 = n in the second case.However, the following result says that this is the only case in which the cen-troidal grading group Γ has smaller rank than Λ. Theorem 5.10 ([Ne3, Th. 7]) . Let L be a centreless Lie torus of type ( S, Z n ) with S not of type A . Then [ Z n : Γ] < ∞ and L is a free Cent F ( L ) -module of finiterank. This result, together with the Realization Theorem of [ABFP1] implies that aninvariant Lie torus of type ( S, Z n ), S = A l , is graded-isomorphic to a multiloopalgebra as defined in (1.13). A characterization of which multiloop algebras are Lietori is the main result of [ABFP2]. A more general approach to realizing Lie torias multiloop algebras is developed in [Na].It is easy to verify Th. 5.10 in case L is a Lie torus of type ( S, Z n ) and S of typeD or E. As we have seen in Th. 4.30, in this case L = g ⊗ F [ t ± , . . . , t ± n ]. Thecentroids of these types of Lie algebras are described in the next example. Example 5.11.
Let g be a finite-dimensional central simple Lie algebra. Forexample, by [BN, Remark 3.6] any finite-dimensional split simple Lie algebra is central and thus central-simple. (Over algebraically closed fields, this also followsfrom Lemma 5.6(d).) Also, let A be an associative commutative F -algebra.A straightforward verification shows that for s ∈ F and a ∈ A the map χ s,a ,defined by u ⊗ b su ⊗ ab , is a centroidal transformation of the Lie algebra g ⊗ A .It follows from [ABP3, Lemma 2.3(a)] or [Az2, Lemma 1.2] or [BN, Cor. 2.23] thatthese are all the maps in the centroid of g ⊗ A : F Id g ⊗ A ∼ = Cent F ( g ⊗ A ) , via s ⊗ a χ s,a . Although this will not be needed in the following, we mention that the centroidof an EALA is known too.
Proposition 5.12.
Let E be an EALA, let K = E c be its core and put D = E/K .Then K is a central Lie algebra, and Cent F ( E ) = F Id E ⊕ V ( K ) , V ( K ) = { χ ∈ Cent F ( E ) : χ ( K ) = 0 } . As a vector space, the ideal V ( K ) of Cent F ( E ) is canonically isomorphic to the D -module homomorphisms D → Z ( K ) : V ( K ) ∼ = Hom D ( D, Z ( K )) . This is proven in [BN, Cor. 4.13]. Observe that the reference to [Ne4, Th.6]in the proof of [BN] can now be replaced by the combination of Th. 4.13(c) andEx. 1.27.5.3.
Centroidal derivations of Lie tori.
In this section L is a centreless Lietorus of type ( S, Λ). Regarding L as a Λ-graded Lie algebra, the results of section5.1 apply and provide us with the subspace D = D ( L ) = { ∂ θ : θ ∈ D(Λ) } of degree derivations of L . Moreover, we can apply Lemma 5.6(f) and get that χ ◦ ∂ θ ≡ χ∂ θ is a derivation for any χ ∈ Cent F ( L ). We call the elements ofCDer F ( L ) = Cent F ( L ) D centroidal derivations . (A notion of centroidal derivations for arbitrary Λ-gradedLie algebra is developed in [Ne5, 4.9].) Recall from Prop. 5.8 that Cent F ( L ) = L γ ∈ Γ Cent F ( L ) γ is a commutative associative torus of type Γ, where Γ is a sub-group of Λ. Since D consist of degree 0 endomorphisms, CDer( L ) is Γ-graded,CDer F ( L ) = L γ ∈ Γ CDer F ( L ) γ for(5.6) CDer F ( L ) γ = Cent F ( L ) γ D = Cent F ( L ) ∩ End F ( L ) γ . It is then easily seen that CDer F ( L ) is a Γ-graded subalgebra of Der F ( L ). For χ γ ∈ Cent F ( L ) γ , χ δ ∈ Cent F ( L ) δ and θ, ψ ∈ D(Λ) the Lie algebra product ofCDer F ( L ) is given by the formula(5.7) [ χ γ ∂ θ , χ δ ∂ ψ ] = χ γ χ δ (cid:0) θ ( δ ) ∂ ψ − ψ ( γ ) ∂ θ (cid:1) . Thus, CDer F ( L ) is a generalized Witt algebra, see for example [NY, 1.9].Suppose now that L is an invariant Lie torus, say with respect to the invariantbilinear from ( ·|· ). We can then consider the skew centroidal derivations SCDer F ( L ) = SDer F ( L ) ∩ CDer F ( L ) , ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 57 defined as the centroidal derivations which are skew-symmetric with respect to( ·|· ). This is a Γ-graded subalgebra of CDer F ( L ) whose homogenous componentsare given by(5.8) SCDer F ( L ) γ = { χ γ ∂ θ : χ γ ∈ Cent F ( L ) γ , θ ( γ ) = 0 } . In particular, SCDer F ( L ) = D is a toral subalgebra of SCDer F ( L ) since [ ∂ θ , χ δ ∂ ψ ] = θ ( δ ) ∂ ψ by (5.7). It is alsoof interest to point out that [SCDer F ( L ) γ , SCDer F ( L ) − γ ] = 0, which implies thatSCDer F ( L ) is a semidirect product,SCDer F ( L ) = D ⋉ (cid:0) L γ =0 SCDer F ( L ) γ (cid:1) of the toral subalgebra D and the ideal spanned by the homogeneous subspaces ofnon-zero degree. For the construction of EALAs, the following theorem is funda-mental. Theorem 5.13 ([Ne3, Th. 9]) . Let L be an invariant Lie torus of type ( S, Λ) with Λ free of finite rank. Then Der F ( L ) is a semi-direct product, Der F ( L ) = IDer F ( L ) ⋊ CDer F ( L ) , hence (5.9) SDer F ( L ) = IDer F ( L ) ⋊ SCDer F ( L ) , where IDer F ( L ) denotes the ideal of all inner derivations. Some remarks on the proof of this theorem follow. By Prop. 5.8 the centroid of L is a Laurent polynomial ring. Let K be its field of fractions, a field of rationalfunctions. As a Cent F ( L )-module, L is torsion-free and hence L embeds into theLie K -algebra ˜ L = L ⊗ Cent F ( L ) K, the so-called central closure of L . If the Cent F ( L )-module L is finitely generated,its central closure is a finite-dimensional central-simple Lie algebra. Hence, in thiscase Der K ( ˜ L ) = IDer( ˜ L ), from which the theorem easily follows. If however L isnot finitely generated as a Cent F ( L )-module, then we know from Th. 5.10 that L is a Lie torus of type A. More precisely, as a consequence of the results in [Y1] and[NY] for type A , [BGKN] for type A and Cor. 4.27 for type A l , l ≥
3, such aLie torus is graded-isomorphic to sl n ( F q ). But in this case the result follows from[BGK, 2.17, 2.53], [BGKN, Th. 1.40] and [NY, Th. 4.11]. We will discuss the specialcase L = g ⊗ F [ t ± , . . . , t ± nn ] in Ex. 5.14. To avoid any confusion, we note that thesplitting (5.9) is not the one proven in [Be, Th. 3.12] for arbitrary root-graded Liealgebras.The importance of the theorem stems from Th. 1.25: It identifies a naturalcomplement of IDer( L ) in SDer F ( L ). Hence, up to graded-isomorphism, any gradedcovering of L has the form E( L, D gr ∗ , ψ D ) for a graded subspace D ⊂ SCDer F ( L ).Moreover, since D ⊂ SCDer F ( L ), we can require that D ⊂ D be not too smalland use it to distinguish the homogeneous spaces L λ by applying Prop. 5.3. Thiswill be our approach in section 5.4. But first some examples. Example 5.14.
Let L = g ⊗ A where g is a split simple finite-dimensional Lie alge-bra with root system S and where A = F [ t ± , . . . , t ± n ] is a Laurent polynomial ring in n variables. This is an invariant Lie torus of type ( S, Z n ), see the Examples 4.29and 4.31. We have seen in (1.18) thatDer F ( g ⊗ A ) = IDer( g ⊗ A ) ⊕ (Id g ⊗ Der F ( A )) . The reader has (or should have) determined Der F ( A ) in Ex. 1.7: Der F ( A ) = A D where D = span Z ( { ∂ i : 1 ≤ i ≤ n } ) in the notation of the quoted exercise. But byEx. 5.1, D = D ( A ) is also the space of degree derivations of A . Since the Λ-gradingof L = g ⊗ A is concentrated in the factor A , it follows that Id ⊗ D is the space ofdegree derivations of L , whence, by Example 5.11, we haveCDer F ( g ⊗ A ) = Id g ⊗ A D = Id g ⊗ Der F ( A ) . Thus, for the invariant Lie torus g ⊗ A the decomposition (1.18) is the same as thedecomposition (5.9)!We have seen in Ex. 4.29 that L is an invariant Lie torus with respect to the tensorproduct form ( ·|· ) = κ ⊗ β where κ is the Killing form of g and where β is the bilinearform on A defined by β ( t λ , t µ ) = δ λ, − µ . It is then easy to identify SCDer F ( L ) using(5.8). In particular, for n = 1 we see that SCDer( g ⊗ F [ t ± ]) = F d , where d is thedegree derivation of (1.8). In particular, this together with Th. 1.25 gives a newproof of the theorem, mentioned in 1.1, that the Lie algebra ˜ L ( g , σ ) of (1.7) is theuniversal central extension of the twisted loop algebra L ( g , σ ).5.4. The general construction.
Finally, we can describe the ingredients (
L, D, τ )of the general construction: • L = L ξ ∈ S,λ ∈ Λ L λξ is an invariant Lie torus of type ( S, Λ) with Λ a freeabelian group of finite rank; we put Γ = supp Λ Cent( L ), see Prop. 5.8. • D = L γ ∈ Γ D γ ⊂ SCDer F ( L ) is a graded subalgebra such that the evalua-tion map(5.10) ev D : Λ → D ∗ , λ → ev λ | D is injective. • τ : D × D → D gr ∗ is an affine cocycle , i.e., τ is a bilinear map satisfyingfor all d, d i ∈ Dτ ( d, d ) = 0 and P (cid:9) d · τ ( d , d ) = P (cid:9) τ ([ d , d ] , d ) , (5.11) τ ( D , D ) = 0 , and τ ( d , d )( d ) = τ ( d , d )( d )(5.12)Recall from Prop. 5.3 that the condition (5.10) implies that D induces theΛ-grading of L , i.e.,(5.13) L λ = { l ∈ L : d ( l ) = ev λ ( d ) l, for all d ∈ D } . For example, (5.10) holds for D = D = SCDer F ( L ) or D any graded subalgebrawith D = D . In (5.11), the symbol P (cid:9) denotes the cyclic sum: P (cid:9) d · τ ( d , d ) = d · τ ( d , d ) + d · τ ( d , d ) + d · τ ( d , d ) and analogously for P (cid:9) τ ([ d , d ] , d ).Moreover, d · c for c ∈ D gr ∗ is the contragradient action of D on the graded dualspace D gr ∗ . The condition (5.11) says that τ is an abelian -cocycle , meaning that D gr ∗ ⊕ D is a Lie algebra with respect to the product formula(5.14) [ c ⊕ d , c ⊕ d ] = (cid:0) d · c − d · c + τ ( d , d ) (cid:1) ⊕ [ d , d ]for c i ∈ D gr ∗ and d i ∈ D . Thus,0 / / D gr ∗ inc / / D gr ∗ ⊕ D pr D / / D / / ECTURES ON EXTENDED AFFINE LIE ALGEBRAS 59 is an abelian extension: D gr ∗ is an abelian ideal, not necessarily contained in thecentre. The conditions in (5.12) will allow us to define a toral subalgebra H andan invariant bilinear form ( ·|· ) below. We note that an affine cocycle is necessarilygraded of degree 0: τ ( D γ , D δ ) ⊂ ( D gr ∗ ) γ + δ for γ, δ ∈ Γ. There do exist non-trivial affine cocycles, see [BGK, Rem. 3.71] and[ERM].To data (
L, D, τ ) as above we associate a Lie algebra E = L ⊕ D gr ∗ ⊕ D with product ( l i ∈ L , c i ∈ D gr ∗ and d i ∈ D )[ l ⊕ c ⊕ d , l ⊕ c ⊕ d ] = (cid:0) [ l , l ] L + d ( l ) − d ( l ) (cid:1)(cid:0) ψ D ( l , l ) + d · c − d · c + τ ( d , d ) (cid:1) ⊕ [ d , d ] . (5.15)Here [ ., . ] L is the Lie algebra product of L , d i ( l j ) is the natural action of D on L ,and ψ D is the central 2-cocycle of (1.24). It is immediate from the product formulathat(i) L ⊕ D gr ∗ is an ideal of E , and the canonical projection L ⊕ D gr ∗ → L is acentral extension.(ii) The Lie algebra D gr ∗ ⊕ D of (5.14) is a subalgebra of E .The Lie algebra E has a a subalgebra H = h ⊕ D ∗ ⊕ D where h = span F { h λξ : ξ ∈ S × , λ ∈ Λ } = span F { h ξ : 0 = ξ ∈ S ind } . We embed S into the dual space h ∗ , using the evaluation map of (5.10), and extend ξ ∈ S ⊂ h ∗ toa linear form of H by ξ ( D ∗ ⊕ D ) = 0. We embed Λ ⊂ D ∗ , using the evaluationmap of Prop. 5.3, and then extend λ ∈ Λ ⊂ D ∗ to a linear form of H by putting λ ( h ⊕ D ∗ ) = 0. Then H is a toral subalgebra of E with root spaces E ξ ⊕ λ = ( L λξ , ξ = 0 ,L λ ⊕ ( D − λ ) ∗ ⊕ D λ , ξ = 0 . Observe H = E since h = L by Ex. 4.10. The symmetric bilinear form ( ·|· ) on E ,defined by (cid:0) l ⊕ c ⊕ d | l ⊕ c ⊕ d (cid:1) = ( l | l ) L + c ( d ) + c ( d )where ( ·|· ) L is the given bilinear form of the invariant Lie torus L , is nondegenerateand invariant. With respect to this bilinear form the set of roots of ( E, H ) is R = R ∪ R an , where R = { λ ∈ Λ ⊂ H ∗ : L λ = 0 } and R an = { ξ ⊕ λ : ξ = 0 and L λξ = 0 } . We have now indicated that the axioms (EA1) and (EA2) of an extended affine Liealgebra holds for the pair (
E, H ). The verification of the remaining axioms can beeasily be done by the reader, or can be looked up in [Na, Prop. 5.2.4]. This thenshows part (a) of the following theorem.
Theorem 5.15 ([Ne4, Th. 6]) . (a) The pair ( E, H ) constructed above is an extendedaffine Lie algebra, denoted E = E(
L, D, τ ) . Its core is L ⊕ D gr ∗ and its centrelesscore is L . (b) Conversely, let ( E, H ) be an extended affine Lie algebra, and let L = E c /Z ( E c ) be its centreless core, which by Cor. 4.16 is an invariant Lie torus, say of type ( S, Λ) ,with Λ free of finite rank.Then there exists a subalgebra D ⊂ SCDer F ( L ) and an abelian -cocycle τ sat-isfying the conditions (5.10)–(5.12) on ( D, τ ) such that E ∼ = E( L, D, τ ) . We have defined discrete EALAs in 2.1 as a special class of EALAs over the basefield F = C . They can now be characterized as follows. Corollary 5.16 ([Ne4, Th. 8]) . Let F = C . (a) Let L be an invariant Lie torus oftype ( S, Λ) with Λ free of finite rank and let D ⊂ SCDer C ( L ) be a graded subalgebrasuch that the evaluation map ev : Λ → D ∗ is injective with discrete image. Then,for any affine -cocycle τ the extended affine Lie algebra E( L, D, τ ) is a discreteEALA. Conversely, any discrete EALA arises in this way. References [AABGP] Allison, B., Azam, S., Berman, S., Gao, Y. and Pianzola, A.
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