Descent Constructions for Central Extensions of Infinite Dimensional Lie Algebras
by
Jie Sun
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics
Department of Mathematical and Statistical Sciences
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1*1 Canada To my parents on their 30th wedding anniversary Abstract
The purpose of this thesis is to give a new construction for central extensions of cer tain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between in finite dimensional Lie theory and descent theory and is crucial to the construction contained in this thesis, where central extensions of twisted forms of split simple Lie algebras over rings are constructed by using descent theory. This descent construc tion gives new insight to solve important problems in the structure theory of infinite dimensional Lie algebras such as the structure of automorphism groups of extended affine Lie algebras and universal central extensions of multiloop Lie algebras. There are four main results in this thesis. First, by studying the automorphism groups of infinite dimensional Lie algebras a full description of which automorphisms can be lifted to central extensions is given. Second, by using results from lifting auto morphisms to central extensions and techniques in descent theory a new construction of central extensions is created for twisted forms given by faithfully flat descent. A good understanding of the centre is also provided. Third, the descent construction gives information about the structure of the automorphism groups of central exten sions of twisted forms. Finally, a sufficient condition for the descent construction to be universal is given and the universal central extension of a multiloop Lie torus is obtained by the descent construction. Acknowledgements
When I started my PhD program I felt the urgency to quickly find accessible research problems. But I couldn't see them and I didn't know where to look. Professor Pianzola put to rest my fears and worries by gently suggesting that I focus on my course work. Starting from my second semester Arturo patiently worked together with me on a research problem. I sincerely appreciate all the long discussions we had, where I also finally saw mathematics come alive. Arturo always encouraged me by saying that "it is the mystery that makes mathematics interesting". I am grateful to him for introducing me to the beautiful bridge between finite and infinite dimensional Lie theory afforded by descent theory. Through it I experienced the excitement of seeing connections between two different and apparently unrelated areas of mathematics. Writing my first research paper with Arturo was a wonderful experience, and it helped me to gradually build up confidence in my own research abilities. From there, several research problems opened naturally before my eyes. Arturo encouraged me to pursue my own research interests and challenged me to think independently and to develop my own writing style. At the same time Arturo was a wonderful counselor. He gave me valuable advice for my future mathematical career. Arturo's hard work and dedication to his research inspires me always to strive for excellence. I shall always hold him in the highest regard.
I would like to thank my supervisory committee members, Professor Vladimir Chernousov and Professor Jochen Kuttler, and my external examiners, Professor Alexander Penin and Professor Yun Gao, for their valuable comments on my thesis. I would also like to thank Professor Erhard Neher at the University of Ottawa for his insightful comments on the part of my thesis concerning universal central extensions. I am grateful to all the professors who taught me mathematics at the University of Alberta including Professors Robert Moody, Alfred Weiss, Gerald Cliff, Terry Gannon and James Lewis. I owe a great debt of gratitude to my wonderful office-mates Alexander Ondrus and Serhan Tuncer who have been a constant source of support and encouragement. They patiently proofread my work and listened to my practise presentations, and I also fondly remember the many "algebra seminars" we had in our office discussing each other's research problems. I would also like to thank all my friends at the University of Alberta for their friendship and the staff members in the department for providing a nice environment to study and work. Finally, I would like to acknowledge the Queen Elizabeth II Graduate Scholarship Program for their generous financial assistance. Contents
1 Introduction 1
2 Infinite Dimensional Lie Theory 12 2.1 From Finite to Infinite 12 2.2 Affine Kac-Moody Algebras 14 2.3 Kac's Loop Construction 16 2.4 Extended Affine Lie Algebras 18 2.5 Multiloop Realization of Centreless Lie Tori 20
3 Descent Theory 24 3.1 The Descent Theorem 24 3.2 Twisted Forms and Cohomology 26 3.3 Applications to Lie Theory 32
4 Central Extensions of Twisted Forms 37 4.1 Some Generalities on Central Extensions 38 4.2 Lifting Automorphisms to Central Extensions 42 4.3 Descent Constructions for Central Extensions 52 4.4 Automorphism Groups of Central Extensions 58
4.5 Universal Central Extensions 60
5 Conclusions 71
Bibliography 73 Chapter 1
Introduction
Central extensions play a crucial role in physics as they can reduce the study of projective representations to the study of true representations. The trick of replacing a physical symmetry group G with its universal covering group (which is a central extension of G by its fundamental group) is so well known in the physics community that the universal covering group is considered as the true physical symmetry group. Sophus Lie (1842-1899) developed his notion of continuous transformation groups and their role in the theory of differential equations. Lie groups (topological groups with compatible manifold structures) and their universal covering groups have turned out to be of fundamental significance in our understanding of symmetries in physics. By Noether's theorem whenever you have a conservation law, then you have a Lie group symmetry which underlies it. For example, invariance of physical systems with respect to rotation gives the law of conservation of angular momentum, and invariance of physical systems with respect to time translation gives the law of conservation of energy. In quantum mechanics the probability of finding a particle in a certain region at a specific time is determined by the square of the wave function. The laws of quantum mechanics describe how the wave function evolves over time. The wave function of an electron is a vector in L2(R4) © L2(R4) which describes the states of the electron. In physics the state v and the state \v{\ € Cx) are identical. Thus the wave function
1 transforms according to projective representations (continuous group homomorphisms x from G to GLn(C)/C ) of the rotation group SO(3). The study of projective representations can be reduced to the study of true rep resentations (continuous group homomorphisms from G to GLn(C)) by enlarging the physical group of symmetries. The universal covering group of SO(3) is the special unitary group SU(2). Every rotation in SO(3) has a "double cover" in SU(2). Under a rotation R in SO(3) the wave function of an electron is multiplied by a matrix U in SU(2) which covers R. Thus a projective representation of SO(3) is reduced to a true representation of SU(2). In this sense, the universal covering group SU(2) is the "quantum mechanical rotation group" (see Chapter 1 §4 in [SW]). Matrix groups provide basic examples for finite dimensional Lie groups. There are also infinite dimensional Lie groups, such as loop groups which appear directly in Wess-Zumino-Witten string theory. A loop group L(G) of a compact Lie group G is the group of continuous maps from a circle S1 into G with pointwise composition in G. The central extension of L{G) by 51 play an important role in the representation theory of loop groups. The group of diffeomorphisms of S1, denoted by Diff(S1), is another infinite dimensional Lie group which acts naturally on L(G) by changing the parametrization of the loop. The entire local structure of a Lie group is codified in its Lie algebra which is much simpler and has a purely algebraic structure. The manifold structure of a Lie group G makes it possible to talk about the tangent space at a point of G, in particular the tangent space at the identity. This tangent space is called the Lie algebra g(G) of G. The Lie algebra Q(G) is a vector space over a filed k, but because of the group structure on the manifold it inherits a rich algebraic structure. In general, a Lie algebra over k is a A;-vector space g with bilinear multiplication [•, •] : g x g —> g,
2 called the Lie bracket, satisfying:
• (LI) Skew Symmetry: [x, x] = 0 for all x in g;
• (L2) Jacobi Identity: [x, [y, z]] + [y, [z, x]} + [z, [x, y}} = 0 for all x, y, z in g.
The Lie algebra of a matrix group can be obtained by differentiating curves through the identity. For example g(SO(3)) is all the matrices in M3(R) satisfying the condi T tions X = -X and trX = 0, while g(SU(2)) is all the matrices in M2(C) satisfying —T the conditions X — —X and trX = 0. It can be shown that these two Lie algebras are isomorphic over K. Matrix algebras provide basic examples for finite dimensional Lie algebras. The building blocks of finite dimensional Lie algebras are finite dimen sional simple Lie algebras, which have no nontrivial central extensions. Infinite dimensional Lie algebras, by contrast, have many interesting central ex tensions. For example, the Lie algebra of Diff(51) is called the Witt algebra which is the algebra of complex vector fields on S1. The Witt algebra is infinite dimensional and has a one dimensional central extension. This one dimensional central exten sion, called the Virasoro algebra, is in fact the universal central extension of the Witt algebra. The Witt and Virasoro algebras often appear in problems with conformal symmetry where the essential spacetime is one or two dimensional and space is peri odic, i.e. compactified to a circle. An example of such a setting is string theory where the string worldsheet is two dimensional and cylindrical in the case of closed strings (see §4.3 in [Gl]). Such worldsheets are Riemann surfaces which are invariant under conformal transformations. The algebra of infinitesimal conformal transformations is the direct sum of two copies of the Witt algebra. The role of central extension here is to reduce the study of projective representations of the Witt algebra to the study of true representations of the Virasoro algebra. The representations of the Virasoro
3 algebra that are of interest in most physical applications are the unitary irreducible highest weight representations. These are completely characterized by the central charge and the conformal weight corresponding to the highest weight vector (see §3.2 in [KR]). Another example of infinite dimensional Lie algebras is the Lie algebra of a loop group which is called a (non-twisted) loop algebra. When G is a compact Lie group over C, the Lie algebra of the loop group L(G) is g(G) <8>c C^1]. This loop algebra is infinite dimensional and its universal central extension (which is one dimensional in this case) gives an affine Kac-Moody algebra. The natural action of Diff(51) on a loop group is manifested locally as the action of the Witt algebra on a loop algebra. At the universal central extension level, to each affine Kac-Moody algebra there is an associated Virasoro algebra by Sugawara's construction (see §3.2.3 in [Gl]). A given unitary representation of an affine Kac-Moody algebra then naturally transforms into a unitary representation of the associated Virasoro algebra. The theory of affine Kac-Moody algebras, developed in the 1960s, not only gen eralizes essentially all of the well-developed theory of finite dimensional simple Lie algebras, but also draws together many different areas of mathematics and physics. Kac's loop construction reveals the key structure of affine Kac-Moody algebras as the universal central extensions of loop algebras based on finite dimensional simple Lie algebras over C (see Chapter 7 and Chapter 8 in [K]). Here the concept of loop algebras has been enlarged to include twisted cases. Given a finite dimensional simple Lie algebra g over C and a finite order diagram automorphism a of Q with am = 1, the loop algebra of (Q,CT) is defined by
i/m ±1 m HO,*) •= ©(0J®* ) CB0cC[t / ], jez
4 where : Z —»• Z/mZ is the canonical map and gy := {x € 0 | cr(x) = Cmx} *s th the eigenspace corresponding to the eigenvalue (m (the primitive m root of unity). Kac's realization works for both non-twisted and twisted affine Kac-Moody algebras and the universal central extensions in the affine case are all one dimensional. The journey from finite to infinite dimensional Lie theory began by V. Kac and R. Moody is far from complete. Several directions have been pursued to generalize the theory of affine Kac-Moody algebras during the last forty years, for example symmetrizable Kac-Moody algebras, Borcherds algebras, toroidal algebras and root- graded algebras. The generalization that is of interest for this thesis is extended affine Lie algebras which arose in the work of K. Saito and P. Slodowy on elliptic singularities and in the paper in 1990 by the physicists R. H0egh-Krohn and B. Torresani ([H-KT]) on Lie algebras of interest to quantum gauge field theory. A mathematical foundation of the theory of extended affine Lie algebras was provided in 1997 by B. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola in their AMS memoirs ([AABGP]). In this memoirs they established some basic structure theory and described the type of root systems appearing in these algebras. In the decade following their memoirs, both the structure theory and the rep resentation theory of extended affine Lie algebras were explored. To untangle the structure of an extended affine Lie algebra c, Kac's loop construction inspired a two- step philosophy. First, one needs to understand the centreless core of e. Second, one needs to recover e from its centreless core by central extensions. The centreless cores of extended affine Lie algebras were characterized axiomatically as centreless Lie tori by Y. Yoshii and E. Neher ([Yl], [Y2], [Nl] and [N2]). A Lie torus is a special graded Lie algebra. The connection between central extensions of graded Lie algebras and skew derivations was explored in the 1980s by R. Farnsteiner ([Fal] and [Fa2]). This
5 connection appeared again in E. Neher's constructions for central extensions of cen- treless Lie tori. In [N2] by using centroidal derivations E. Neher constructed central extensions of centreless Lie tori and stated that the graded dual of the algebra of skew centroidal derivations gives the universal central extension of a centreless Lie torus. The axiomatization of centreless Lie tori and Neher's construction for central extensions of centreless Lie tori paved the way for further development of the structure theory of extended affine Lie algebras. The loop structures of centreless Lie tori were investigated by B. Allison, S. Berman, J. Faulkner and A. Pianzola ([ABP1], [ABP2], [ABP3], [ABFP1] and [ABFP2]). The centreless core of an affine Kac-Moody algebra is a loop algebra based on a finite dimensional simple Lie algebra over C. It is natural to ask whether all centreless Lie tori over C can be realized by applying the loop construction iteratively to finite dimensional simple Lie algebras. This question gave birth to the concept of iterated loop algebras which are algebras obtained by a sequence of loop constructions, each based on the algebra obtained at the previous step (see Definition 5.1 in [ABP3]). Multiloop Lie algebras are special examples of iterated loop algebras and are crucial to understand the loop structures of centreless Lie tori. In [ABFP2] it was shown that over C (or any algebraically closed field of characteristic zero) a centreless Lie torus has a multiloop realization based on a finite dimensional simple Lie algebra if and only if it is finitely generated as a module over its centroid. All but one family of centreless Lie tori are finitely generated over their centroids (see Remark 1.4.3 in [ABFP2]). Thus in some sense "almost all" extended afRne Lie algebras have multiloop structures.
Carrying essential structures of extended affine Lie algebras, multiloop Lie alge bras became an important class of infinite dimensional Lie algebras. From a different
6 point of view, multiloop Lie algebras can also be thought as twisted forms, thus con nect infinite dimensional Lie theory and descent theory. This connection inspired a new philosophy to further develop the theory of extended afhne Lie algebras. Given a finite dimensional simple Lie algebra g over C and commuting finite order automor mi phisms (h...,in)€Z^ where ~ : Z —* Z/mjZ is the canonical map for 1 < j < n and flii,...,i„ := ix e 0 I ai(x) = Cmjx for l - •?' - n} is the simultaneous eigenspace corresponding to the eigenvalues (mj (the primitive mf roots of unity) for 1 < j < n. A multiloop Lie algebra is infinite dimensional over the given base field C, but is free of finite rank over its centroid C[tf1,..., t^1] and can be viewed as a twisted form of a much simpler Lie algebra g <8>c C[tf1,..., t^1]. This perspective of viewing multiloop Lie algebras as twisted forms of g (8>c C[tfx,..., i*1] presents a beautiful bridge between infinite dimensional Lie theory and descent theory. Grothendieck's descent theory was developed in 1950s and 1960s. The concept of twisted forms in descent theory captures objects which are globally different but locally the same. A good example of twisted forms is the Mobius strip as a twisted form of a cylinder. To be more precise, the Mobius strip L is a topological line bundle over the manifold X = S1 (a circle). The trivial line bundle A^- over the same manifold X gives a cylinder. The manifold X can be covered by two open subsets {Ui}i=i_2 where Ui ~ R. Over each open subset Ui both the Mobius strip L and the 2 cylinder A^ look like M , thus locally LVi ~ A^.. 7 The algebraic analogue of a manifold is a scheme. Just as a manifold looks locally like afflne space W1 or Cn, a scheme looks locally like an affine scheme SpecR, where R is a ring. Here R plays the role of the ring of analytic functions. The (analytic) geometry of a manifold can be recovered from its sheaf of analytic functions. Similarly, the (algebraic) geometry of an affine scheme can be recovered from its sheaf of regular functions. For an affine scheme Speci? the ring extension R —> S yields a scheme morphism U : Spec S —» Spec R. This map U can be thought as an "open cover" of the affine scheme X = Speci? in the Grothendieck topology on X. The open cover U : SpecS -> SpecR, where R = C[tf\ ... ,t^} and S = C[tf1/mi,... ,tn1/mn], sets the stage for multiloop Lie algebras to enter descent theory. In this case the map U is faithfully flat and etale (in fact finite Galois) and U is an open cover of X in the etale topology on X, where X is the complex plane with n punctures. A multiloop Lie algebra L(g, crl5..., an) and the much simpler Lie algebra QR := g L(g, tells that £(g, a1}..., an) is a twisted form of QR ([GP1], [GP2] and [P2]). This perspective of viewing multiloop Lie algebras as twisted forms of QR provides a new way to look at their structure through the lens of descent theory. Since the affine group scheme Aut(g^) is smooth and finitely presented, Grothendieck's descent formulism classifies all isomorphism classes of twisted forms of gR split by S by means of the pointed set H}t(S/R, Aut(g#)). There is a natural bijective map Isomorphism classes of twisted forms of QR split by S <—> H\t (S/R, Aut(g#)). 8 When R = C[t±x], as g runs over the nine Cartan-Killing types At, B(, C(, Dg, EQ, E7, Eg, F4 and G2, the 16 resulting classes in H\t correspond precisely to the isomorphism classes of affine Kac-Moody algebras (see Remark 3 in [P2]). For mul tiloop Lie algebras, the ring extension R —> S is finite Galois with the Galois group G = Z/TOIZ x ... x Z/m„Z. In this case, Hjt(S/R,Aut(gjj)) is the usual non- abelian Galois cohomology Hl(G, Aut(gi?)(5)), where the S-points of the affine group scheme Aut(g#) are given by Aut(flB)(S) = Aut5(gii ®R S) ~ Auts(g ®c S). X A cocycle in Z (G, Aut(gfi)(5)) is a map u = (ug)geG • G —> Auts(g®c5') satisfying 92 9 _1 ug2gi = ug2 ugi, where G acts on Auts(g <8>c S) by 6 = (1 <8> g) o 6 o (1 ® p ). Two cocycles u and it' are cohomologous if there exists some A in Auts(g <8>c 5) such that ff _1 l u'g = Ait5( A) . Each cohomology class [it] e H {G, Aut(g^)(S')) corresponds to one isomorphism class [£u] of twisted forms of QR. The descended algebra Lu given by Galois descent can be expressed as A. = {x e 0 ®c s I v* = x for a115 e Gl- Thus a multiloop Lie algebra L(g,ai,... ,an) as a twisted form of g# must be iso l morphic to an i?-Lie algebra Cu for some cocycle u in Z (G, Aut(g#)(S)). This perspective of viewing multiloop Lie algebras as twisted forms of QR gives new insight to study both the structure theory and the representation theory of extended affine Lie algebras. This thesis is devoted to study central extensions of twisted forms of split simple Lie algebras over rings. The general setting is twisted forms of g 0k R, where A; is a field of characteristic zero, g a finite dimensional split simple Lie algebra over k, and R a commutative, associative, unital A;-algebra. Multiloop Lie algebras are special 9 examples of, but usually do not exhaust all, twisted forms of g (g>c C^f1,... ,t^1]. A twisted form of QR :— g (E>fc R is an i?-Lie algebra descended from Qs := g ®fe S, where 5 is a commutative, associative, unital fc-algebra and R —> S is a faithfully flat ring extension. As an J?-Lie algebra, a twisted form of QR is centrally closed, but it is not as a k-Lie algebra, thus it has central extensions over k. In 1984 C. Kassel constructed the universal central extension of Qs by using Kahler differentials, namely Q~S = Qs © £ls/dS. In general Qs/dS is infinite dimensional. The purpose of this thesis is to construct central extensions of twisted forms of QR by using their defining descent data to construct k-Lie subalgebras of §5 and study the universality of this new construction. There are four main results in this thesis. First, by studying automorphism groups of infinite dimensional Lie algebras a full description of which automorphisms can be lifted to central extensions is given. In particular, for QR it is shown that the unique lift of an .R-linear automorphism of QR to its universal central extension g]J fixes the centre QR/CIR pointwise and all i?-linear automorphisms of QR lift to every central extension of QR. Second, by using results from lifting automorphisms to central extensions and techniques in descent theory a new construction (called the descent construction) of central extensions is created for twisted forms of QR given by faithfully flat descent. A good understanding of the centre is also provided. The idea of the descent construc tion is illustrated in the following for twisted forms given by Galois descent which include multiloop Lie algebras. Let Cu be a twisted form of g# corresponding to a 1 cocycle u = (ug)geo in Z (G, Aut(fli?)(5)), where ug G Auts(gs). Recall that Cu has the following expression 9 Cu = {X € Qs I ug X = X for all g G G}. 10 Each ug lifts uniquely to ug 6 Autfc(gs). The Galois group G acts naturally both on Q.s and on the quotient &-space Qs/dS, thus acts naturally on gg. The A;-Lie subalgebra of gs defined by a £a--={Xeos\ ug X = X for allp € G} G gives a central extension of Cu over k and the centre is i{C%) cz (Cls/dS) ~ Vtn/dR. Third, the descent construction gives information about the structure of automor phism groups of central extensions of twisted forms of QR. A sufficient condition is found under which every i?-linear automorphism of a twisted form of QR lifts to its central extension obtained by the descent construction and the lift fixes the centre pointwise. Finally, a sufficient condition for the descent construction to be universal is given. In particular, the universal central extension of a multiloop Lie torus is obtained by the descent construction. The structure of this thesis is as follows. The second and third chapters provide theoretical backgrounds for this thesis, with the second chapter focusing on infinite dimensional Lie theory and the third chapter concentrating on descent theory. The fourth chapter contains the main results of this thesis. Some future research directions related to this thesis are pointed out in the final chapter. 11 Chapter 2 Infinite Dimensional Lie Theory This chapter starts with a brief account of the motivation to develop infinite dimen sional Lie theory. This is followed by a review of the definition of afflne Kac-Moody algebras and their classification through Coxeter-Dynkin diagrams. Next, Kac's loop construction is presented in details as it not only reveals the loop structure of affine Kac-Moody algebras but also inspires the development of the structure theory of extended affine Lie algebras which is discussed in the last two sections. 2.1 From Finite to Infinite One of the early accomplishments of Lie theory was the classification by W. Killing (1847-1923) and E. Cartan (1869-1951) of the compact connected Lie groups via the classification of their Lie algebras. For any connected Lie group, there exists a unique simply connected universal covering group, and they have the same Lie algebra. Ev ery finite dimensional real or complex Lie algebra arises as the Lie algebra of an unique real or complex simply connected Lie group (Ado's theorem). As the building blocks of the integers are the prime numbers, the building blocks of finite dimen sional Lie algebras are the simple Lie algebras. There is a complete classification of finite dimensional simple Lie algebras over C At the heart of this classification lie some combinatorial objects, finite root systems and finite Weyl groups, which yield 12 Cartan matrices and their corresponding Coxeter-Dynkin diagrams. Up to isomor phism, there exists a one-to-one correspondence between finite dimensional simple Lie algebras over C and the Coxeter-Dynkin diagrams of finite type (Figure 1). A £ Sx «2"""Vl«ll 1 2 uJl-l"Jl al a2 a3 Ci4 a5 a6 "7 -z?«7h G2 •>=$• al a2 Figure 1: Coxeter-Dynkin Diagrams of Finite Type All of the combinatorial objects in the classification of finite dimensional simple Lie algebras admit natural infinite dimensional generalizations and it is possible to develop from them an infinite dimensional generalization of the finite dimensional simple Lie theory that parallels the finite case to a large degree. This generalization is far from complete. At the present time there is no general theory of infinite di mensional Lie groups and Lie algebras and their representations. There are, however, 13 four classes of generalizations that have undergone a more or less intensive study (see §0.2 in [K]). These are, first of all, Lie algebras of vector fields and the corresponding groups of diffeomorphisms of a manifold. The second class consists of Lie groups (resp. Lie algebras) of smooth mappings of a given manifold into a finite dimensional Lie group (resp. Lie algebra). The third class consists of the classical Lie groups and algebras of operators in a Hilbert or Banach space. Finally, the fourth class is called Kac-Moody algebras for which finite dimensional simple Lie algebras and affine Kac-Moody algebras are the basic examples. This last class of infinite dimensional Lie algebras, especially affine Kac-Moody algebras and their natural generalization extended affine Lie algebras, is of interest for this thesis. 2.2 Affine Kac-Moody Algebras V. Kac and R. Moody generalized the theory of finite dimensional simple Lie algebras to the infinite dimensional setting where the resulting Lie theory has proven to be of great structural beauty and of wide applicability in many areas of mathematics and physics. One example of these applications is conformal field theory where the Witt and Virasoro algebras and their representations appear naturally. To each affine Kac- Moody algebra there is an associated Virasoro algebra by Sugawara's construction (see §3.2.3 in [Gl]). A given unitary representation of an affine Kac-Moody algebra then naturally transforms into a unitary representation of the associated Virasoro algebra. Kac-Moody algebras can be defined in terms of generators and relations as follows. Definition 2.1 ([K]) Let A = (ay)"J=1 be a generalized Carton matrix, i.e., an n x n matrix with only integer coefficients such that a& = 2, a^ < 0 for i ^ j, and ay = 0 14 implies a,-* = 0. The associated Kac-Moody algebra is a complex Lie algebra on 3n generators e*, /*, /ij (i = 1,..., n) and the following defining relations (i,j — 1,..., n): • (KM1) [hi, hj] = 0, fe, /J = /ii, [Ci, /,-] = 0 if i ^ j; • (KM2) [/ij,ej] = a^e-j, [hi, fa] = -%/.,•; • (KM3) (ad eif-^ej = 0, {ad /i)1_^/j = 0 if i ^ j. An indecomposable generalized Cartan matrix is of finite type if and only if all its principle minors are positive. The associated Lie algebra is finite dimensional and simple. An indecomposable generalized Cartan matrix is of affine type if and only if all its proper principle minors are positive and detA = 0. The associated Lie algebra, called an affine Kac-Moody algebra, is infinite dimensional but has a Z-grading into finite dimensional subspaces where dimension grows at most polynomially (see §3.2.1 in [Gl]). . The theory of affine Kac-Moody algebras, developed in the 1960s, generalizes es sentially all of the well-developed theory of finite dimensional simple Lie algebras over complex numbers. In particular, there is a complete classification of affine Kac- Moody algebras. Up to isomorphism, there exists a one-to-one correspondence be tween affine Kac-Moody algebras and the Coxeter-Dynkin diagrams of affine type (Figure 2). There are 16 isomorphism classes of affine Kac-Moody algebras, where the ten classes corresponding to diagrams on the left column in Figure 2 are called non-twisted affine Kac-Moody algebras and the other six classes are called twisted affine Kac-Moody algebras. Both non-twisted and twisted affine Kac-Moody alge bras can be realized as central extensions of loop algebras based on finite dimensional simple Lie algebras over complex numbers. 15 A(1) A.U) (», > 2) 1 1 B(1>(4 > 3) 0-6-0-...-o-to i 12 2 2 2 ci11 (£ > 2) OHO ... 0*4 12 2 1 (2) 1 A D^ ' U > 4) 0-0-0 . . .- 0-0 A2 A(2) G'11 0-0^3 > 2) 2 (i 12 3 11 A(2) > 3) 21-1 U 'i 12 3 4 2 D*J. + 1' U > 2) «—0— ... —0=0 12 321 11 11 2 E'1' o-o-o-6-o-o-o E,<2) ' 12 34321 " 12321 E' O-O-O-O-O-O-O-O /,, 8 12345 642 D!J; 0 0>-0 12 1 Figure 2: Coxeter-Dynkin Diagrams of Affine Type 2.3 Kac's Loop Construction Kac's loop construction reveals the key structure of affine Kac-Moody algebras as the universal central extensions of loop algebras based on finite dimensional simple Lie algebras over C (see Chapter 7 and Chapter 8 in [K]). Given a finite dimensional simple Lie algebra g over C and a finite order diagram automorphism a of g with am = 1, V. Kac built a Lie algebra from the pair (g, a) as follows. First from a we have the eigenspace decomposition TO—1 j=0 16 where : Z —• Z/mZ is the canonical map and 0j := {x G 0 | HS, *) := 0(0J ® ti/m) C 0 ®c C^1""]. When a = id, we get the non-twisted loop algebra L(g) := 0 0c C[£±]-]. Finally we construct a one dimensional central extension L(g,a) := L(g,a) © Cc with the following Lie bracket p q p+,? [a 1 L(sl2(C), id) = sl2(C) ®c C^ ] © Cc and the twisted affine Kac-Moody algebra of type A2 is isomorphic to tr j 2 L(sh(C), a : X .- -X ) = 0jeZ(flj ® t / ) © Cc. In order to have a non-degenerate bilinear form on affine Kac-Moody algebras, V. Kac constructed "full" affine Kac-Moody algebras by adding derivations L(g,a) := L(g, a) ®Cd with d acts on L(g, a) by tj^ and kills the centre Cc. Using this beautiful construction, V. Kac showed that any (derived modulo its centre) affine Kac-Moody algebra can be obtained as a loop algebra of a finite dimensional simple Lie algebra over C. This fact is of great importance in the study of affine Kac-Moody algebras and also inspires the development of the structure theory of extended affine Lie algebras. 17 2.4 Extended Affine Lie Algebras Extended affine Lie algebras, as natural generalizations of affine Kac-Moody algebras, arose in the work of K. Saito and P. Slodowy on elliptic singularities and in the paper by the physicists R. H0egh-Krohn and B. Torresani ([H-KT]) on Lie algebras of interest to quantum gauge field theory. A mathematical foundation of the theory of extended affine Lie algebras was provided in 1997 by B. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola in their AMS memoirs ([AABGP]). In this memoirs they established some basic structure theory and described the type of root systems appearing in these algebras. The basic definition of an extended affine Lie algebra is broken down into a sequence of axioms EA1-EA4, EA5a and EA5b. Let e be a Lie algebra over C We assume first of all that c satisfies the following axioms EA1 and EA2. EAl. e has a non-degenerate invariant symmetric bilinear form denoted by (v):exe-»C. EA2. e has a nonzero finite dimensional abelian subalgebra f) such that adc(h) is diagonalizable for all h E f) and such that f) equals its own centralizer Cc(h) in e. One lets h* denote the dual space of f) and for a E h* we let za = {x E c | [h, x\ = a(h)x for all h E h}. e an Then we have that c = ©asi)* a d so since Ce(h) = eo by EA2, we have f) = Co- We define the root system R of e relative to f) by saying R = {a E h* | ta + 0}. Notice that (ea, zp) = 0 unless a + (3 = 0. In particular, the form is non-degenerate when restricted to h x h. This allows us to transfer the form to h* as follows. For each 18 a G h* we let ta be the unique element in f) satisfying (ta,h) — a(h) for all h G h. Then for a,/3 G h* we define (a,/?) := (ta,t0). We let i?° := {a e R \ (a, a) = 0} be the set of isotropic roots and let Rx := {a G R | (a, a) =£ 0} be the set of non-isotropic roots. Then we have the disjoint union R = R° U Rx. Now we can state the remaining axioms. x EA3. For any a G R and any x € ta the transformation adtx is a locally nilpotent on e. EA4. i? is a discrete subspace of h*. x x EA5a. R cannot be decomposed into a union R = Ri U i?2 where i?! and R2 are nonempty orthogonal subsets of Rx. EA5b. For any 5 e R° there is some a G Rx such that a + 5 £ R. Definition 2.2 ([ABPlj) A triple (e, h, (•, •)) consisting of a Lie algebra e, a subal- gebra I) and a bilinear form (•, •) satisfying EA1-EA4, EA5a, and EA5b is called an extended affine Lie algebra or EALA for short. The core of e, denoted by ec, is the x subalgebra of e generated by the root spaces eQ for non-isotropic roots a G R . The rank of the free abelian group generated by the isotropic roots of e is called the nullity of e. We say the EALA e is tame if the centralizer Cc(ec) of ec in e is contained in cc. Two EALAs are isomorphic if there is a Lie algebra isomorphism from one to the other preserving the given forms up to a nonzero scalar and preserving the given ad-diagonalizable subalgebras. The defining axioms for extended affine Lie algebras are modeled after the proper ties of affine Kac-Moody algebras and it turns out that EALAs are generalizations of affine Kac-Moody algebras in the following sense. Nullity 0 tame EALAs are precisely finite dimensional simple Lie algebras and affine Kac-Moody algebras are precisely 19 the tame EALAs of nullity 1 (see §2 in [ABGP]). Toroidal Lie algebras give examples of tame EALAs of arbitrarily high nullity. To untangle the structure of an extended afflne Lie algebra e, Kac's loop construc tion inspired a two-step philosophy. First, one needs to understand the centreless core C := ec/3(ec). Second, one needs to recover e from its centreless core by central ex tensions. This situation can be summarized by the following diagram cc ^e / / " / where in the afRne case ec is the derived algebra and C a loop algebra. The first step leads to the topic of multiloop realization of centreless Lie tori which will be presented in the next section. The second step gives rise to the theme of central extensions which will be discussed in the fourth chapter. 2.5 Multiloop Realization of Centreless Lie Tori The centreless cores of extended affine Lie algebras have been characterized axiomati- cally as centreless Lie tori ([Yl], [Y2], [Nl] and [N2]). Let A; be a field of characteristic 0. Let A be a finite irreducible root system over k. Recall that A is said to be reduced x x if 2a g A for a € A . If A is reduced then A has type At(l > 1), Bt(l > 2), C,(Z > 3),Di(l > 4), E6, E7, E8, F4 or G2, whereas if A is not reduced, A has type BQ(l > 1) ([B], Chapter VI, §4.14). Let Q = Q(A) := spanz( A algebra, then C = ©AGA£ is A-graded and C = ®aeQ^a is Q-graded with A £ := ©asQ^a for A € A and Ca := ©AeA£« for a € Q. 20 Definition 2.3 ([Nl]) A Lie K-torus (or a Lie torus for short) is a Q x A-graded Lie over algebra £ — ®(a,\)eQx\^a & which satisfies: • (LT1) £a = 0 for a € Q \ A; • (LT2) (i)UO^aeA, then £° ^ 0; ^ If 0 ^ a G Q, A e A and L\ ^ 0, then there exist elements e£ e ££ and /* € £l£ such that £* = fee*, £l„ = kf* and [[ea»/«].^] = (Aav)^ for x^ £^,/3£ Q; x o (LT3) £ is generated as an algebra by the spaces Ca, a € A ; • (LT4) A is generated as a group by supp^{C) := {A G A | Cx ^ 0}. The type of the root system A is then called the absolute type of C and the rank of A is called the nullity of C. A centreless Lie torus is a Lie torus with trivial centre. The loop structures of centreless Lie tori over k were investigated by B. Allison, S. Berman, J. Faulkner and A. Pianzola ([ABP1], [ABP2], [ABP3], [ABFP1] and [ABFP2]). The centreless core of an affine Kac-Moody algebra is a loop algebra based on a finite dimensional simple Lie algebra over C. It is natural to ask whether all centreless Lie tori over k can be realized by applying the loop construction iteratively to finite dimensional split simple Lie algebras. This question gave birth to the concept of iterated loop algebras. In general, if A is an algebra over k, an n-step iterated loop algebra based on A is an algebra that can be obtained starting from A by a sequence of n loop constructions, each based on the algebra obtained at the previous step (see Definition 5.1 in [ABP3]). Multiloop Lie algebras defined as below are special examples of iterated loop algebras. . 21 Definition 2.4 (fABPSj) Let n be a positive integer. Fix a sequence m\, 7712,..., mn of positive integers. Let k be a field of characteristic 0. Suppose that k contains the 1 primitive mf root of unity (mj for 1 < j < n and A is a /c-algebra. Let cr1;..., Then the n-step multiloop algebra of (A, tfi,..., cn), or the n-step multiloop algebra of G\,..., an based on A, is defined by n (h...,in)ez where ~ : Z —> Z/m^Z is the canonical map for 1 < j < n and A,...,^ = (^ € 0 I ^(z) = Qi.x for 1 < j < n} is the simultaneous eigenspace corresponding to the eigenvalues (mj for 1 < j < n. Let R = k[tf1,..., t^1] be the algebra of Laurent polynomials in n variables over k and let S = k^ ,... ,tn ]. Then L(A,c?i,... ,an) is an i?-subalgebra of A When ^l is a finite dimensional split simple Lie algebra g over k, then L(g, ci,..., an) is called a multiloop Lie algebra based on Q. Multiloop Lie algebras based on finite dimensional simple Lie algebras over C are crucial to understand the structure of extended affine Lie algebras. In [ABFP2] it was showed that over C (or any algebraically closed field of characteristic zero) a centreless Lie torus has a multiloop realization based on a finite dimensional simple Lie algebra if and only if it is finitely generated as a module over its centroid (f.g.c. for short). Recall that the centroid of £,, denoted by Ctd/C(£), is the set of all \ € Endfc(£) satisfying [X, adx] = 0 for all x G C. The centroid is a A;-subalgebra of the endomorphism algebra Endfc(£). Almost all centreless Lie tori are f.g.c. in the sense that only one family of centreless Lie tori are not f.g.c, namely the family of Lie algebras of the 22 form sl/+i(A;q), where kq is the quantum torus associated with a quantum matrix q containing an entry that is not a root of unity (see Remark 1.4.3 in [ABFP2]). The relationship between centreless Lie tori and multiloop Lie algebras can be illustrated by Figure 3. \ I I | f.g.c. Lie Tori Multiloop . = | Lie *•' | Multiloop Lie Tori Algebras | Centreless . • Lie Tori I L_ _ 2. _ _" Figure 3: Multiloop Realization of Centreless Lie Tori Carrying essential structures of extended affine Lie algebras, multiloop Lie alge bras became an important class of infinite dimensional Lie algebras. From a different point of view, multiloop Lie algebras can also be thought as twisted forms, thus con nect infinite dimensional Lie theory and descent theory. This connection, presented in the next chapter, inspired a new philosophy to further develop the theory of extended affine Lie algebras. 23 Chapter 3 Descent Theory This chapter provides backgrounds needed in descent theory to present the perspective of viewing multiloop Lie algebras as twisted forms. Two basic questions are answered in the first two sections respectively: When can a module descend? If a module can descend, how many different ways can it descend? The first question leads to the concept of descent data and is answered by the descent theorem. The second question gives rise to the concept of twisted forms and finds its answer in the classification theorem of twisted forms by cohomology. The last section presents the connection between infinite dimensional Lie theory and descent theory by viewing multiloop Lie algebras as twisted forms. 3.1 The Descent Theorem Let R and S be commutative rings with identity. A ring homomorphism R —> S is called flat if, whenever M —• N is an injection of .R-modules, then M®RS —> N®RS is also an injection. For example, any localization R —• Rp is flat, where P is a prime ideal of R. What is really needed in descent theory, however, is a condition stronger than flatness and not satisfied by localization. Theorem 3.1 ([Wa]) Let R —> S be flat. Then the following are equivalent: (1) M —> M ®R S (sending m to m ® 1) is injective for all M. 24 (2) M ®R S = 0 implies M = 0. (3) If M —> N is an R-module map and M ®R S —> N A ring homomorphism R —>• 5 with these properties is called faithfully flat, in particular i? maps injectively onto a subring of S. Note that R —> 5 is faithfully flat if 5 is a free i?-module. The following refined version of condition (1) is crucial in descent theory. Theorem 3.2 ([Wa]) Let R -> S be faithfully flat. Then the image ofMinM®RS consists of those elements having the same image under the two maps M ®# S —> M SR S S>R S sending m®b to m Let i? -> 5 be a faithfully flat ring extension. If an 5-module M can be con structed explicitly as N 6 : M #°(m 25 9l (m ® u #2(m ® a Such a 0 is precisely what is needed to "go down" from the S-module to the R- module, recapturing the descended module AT from M, namely JV={meM #(m Theorem 3.3 ([WaJ) Let R —> S be faithfully flat. Then the category of R-modules 9yiod N h-f &(N) = (N®RS, e-.N®RS®RS^N®RS®RS) where 6(n ®a®b) = n f:N->N' •-> • &(f) = f®id:N®RS^N'®RS with («£•(/) ® id)9 = 9'(&{f) ® id). 3.2 Twisted Forms and Cohomology If an 5-module M can descend, the next natural question to ask is "how many different ways can M descend?" In other words, how to classify .R-modules which become isomorphic when tensored with 5? s M R N N' N" ... 26 Suppose N is a given i?-module, possibly with some additional algebraic structure. An S/R-form of N, or twisted form of N split by S, is another i?-module with the same type of structure which becomes isomorphic when tensored with S. The concept of twisted forms in descent theory captures objects which are globally different but locally the same. A good example of twisted forms is the Mobius strip as a twisted form of a cylinder. To be more precise, the Mobius strip L is a topological line bundle over the manifold X = S1 (a circle). The trivial line bundle A^ over the same manifold X gives a cylinder. The manifold X can be covered by two open subsets {Ui}i=i,2 where [7* ~ R. Over each open subset Ui both the Mobius strip L and the 2 cylinder A^ look like M. , thus locally LVi ~ A^.. Figure 4: The Mobius Strip and the Cylinder The algebraic analogue of a manifold is a scheme. Just as a manifold looks locally like affme space M.n or C", a scheme looks locally like an affine scheme Speci?, where R is a ring. Here R plays the role of the ring of analytic functions. The (analytic) geometry of a manifold can be recovered from its sheaf of analytic functions. Similarly, 27 the (algebraic) geometry of an affine scheme can be recovered from its sheaf of regular functions. For an affine scheme Speci? the ring extension R —> S yields a scheme morphism U : Spec S —> Speci?. This map U can be thought as an "open cover" of the affine scheme X = Speci? in the fppf topology (Definition 3.4). A twisted form of N split by S and N as objects over Spec R are different (not isomorphic as i?-modules), but locally over the open cover U : Spec S —> Spec R they are the same (isomorphic as 5-modules after base change). Definition 3.4 ([WaJ) A finite set of maps {R —> Si} is called a fppf (fidelement plat de presentation finie) covering if R —> St are flat and finitely presented maps and R —* USi is faithfully flat. These maps define the fppf topology. Definition 3.5 ([WaJ) A ring homomorphism R —+ S is etale if S as an i?-module is flat and unramified (i.e., VLS/R = 0 where QS/R is the Kahler differentials of the i?-algebra S) and 5 as an i?-algebra is finitely presented. A ring homomorphism R —> 5 is finite etale if R —> S is etale and S as an .R-module is finitely generated. A finite set of maps {R —> Si} is called a (finite) etale covering if R —> Si are (finite) etale maps and R —> 115, is faithfully flat. These maps define the etale topology. Consider the Zariski coverings where each Si is a localization Rft. All of these are flat and the faithful flatness means that the ideal generated by all the /j's is all of R. Rft corresponds to the basic open set in Speci? where fi does not vanish. Faithful flatness says that these sets cover Spec R. Furthermore, Rfi 0 Rfj = R^fj corresponds to the intersection where both /, and /_,- do not vanish. There is a wide range of Grothendieck topologies which allow us to understand that a twisted form of A^ is locally isomorphic to N. The fppf topology and the etale topology will be considered in this thesis. 28 We want to classify twisted forms. Notice that different twisted forms correspond to different descent data on N ®R S. Suppose that we have some descent data ip : N ®R S N The automorphism p can be extended to automorphisms of N (dV)(n 0 a 0 u 0 6) = S n, 0 a* 0 u 0 bi, (d2p)(n 0 a 0 b 0 ii) = E n* 0 a* 0 6, 0 it- It is shown that ^V = ^ iff d°ip d2 This then is the condition for descent data in terms of the automorphism p. The twisted form of N corresponding to the descent data ip, which is the descended module from N0# S consisting of the elements m = E n* 0 a* satisfying m01 = ^(m0l) = 9p(m 0 1), can be expressed in terms of p> as {E rii® a,i | ip(E rii 0 a* 0 1) = E n* 0 1 0 a*}. Now different 29 descent data. Explicitly, let ip and ip' be descent data. For an 5-automorphism A of N SIR S, it is shown that ip'(X ® id) = (A d°ip d2ip = dV- They are called 1-cocycles. Two 1-cocycles Theorem 3.6 ([Wa]) The isomorphism classes of S/R-forms of N correspond to Hl(S/R, Aut(iV)). 30 This theorem can be read either way. Cohomology can be used to classify twisted forms and information about twisted forms can be used to compute cohomology. When R —> S is a finite Galois ring extension (Definition 3.7), the above cohomology is the usual Galois cohomology. Definition 3.7 (fKOJ) A ring homomorphism R —> S is finite Galois with the Galois group G if (i) R -f S is faithfully fiat; (ii) The Galois group G is a subgroup of Aut^(5); (iii) S ®R S ~ UQS is an isomorphism as 5-algebras (where S acts on the second component of S ®R S) under the map sending a 9 For convenience the elements ( ab)geG G ric.? can be written as the functions / : G —> S with 7(5) = 9a6. Then d°(a) = 1 92 (d°f)(9i,92)= ab = f(g2), 1 9l (.d f)(g1,92)= ab = f(9l), 2 9l 9 1 (d f)(9i,92)= a *b= *(* Let G be any group functor satisfying G(A x B) = G(A) x G(B). Consider G(S)=$G(S ®R S)E$G(S ®R S ®R S). We can break these up as follows. G(S ®R S) = G(nG5) = nGG(5), 31 Q(S ®R S®RS) = G(nGxG5) = UGxGG(S). UcG(S) can be identified with functions / : G —>• G(S) and IIGXGG(5) can be identified with functions h : G x G —• G(5). These functions are keeping track of which coordinate is which in the product. The G-action on G(S) is the one induced by functoriality by its action on S. Now / : G —> G(5) is a 1-cocycle iff /(52) ^/(p^Pi) = /(#i)- Setting g! := #25i, we can rewrite the equation as /GfcSi) = /(<&) 92/(<7i)- Two 1-cocycles / and /' are cohomologous if there exists some A in G(5) such that for all g £ G. This is an equivalence relation. The set of equivalence classes (coho- mology classes) is denoted by Hl(G, G(S)). It is a set with a distinguished element, the class of the trivial cocycle f(g) — id for all jeG. Thus twisted forms split by a finite Galois ring extension can be classified by using Galois cohomology. Theorem 3.8 (fKOJ) Let i? —> S be finite Galois with the Galois group G. Then the isomorphism classes of S/R-forms of N correspond to Hl(G, Aut(N)(S)). 3.3 Applications to Lie Theory As we have seen in the second chapter, multiloop Lie algebras based on finite di mensional simple Lie algebras over C play an important role in the structure theory of extended affine Lie algebras. A multiloop Lie algebra L{$,(j\,... ,an) is infinite dimensional over the given base field C, but is free of finite rank over its centroid C[tf\ ..., t^1}. The open cover U : SpecS -> SpecR, where R = C[tf \ ..., t^1} and 32 /mi S = C[t1 ,... ,tn ], sets the stage for multiloop Lie algebras to enter descent theory. In this case the map U is faithfully flat and etale (in fact finite Galois) and U is an open cover of X in the etale topology on X, where X is the complex plane with n punctures. A multiloop Lie algebra L(g, ax,..., an) and the much simpler Lie algebra gR := g <8>c R as objects over X = Spec R are different (not isomorphic as Lie algebras over R), but locally over the open cover U : Spec S —>• Speci? they are same (isomorphic as Lie algebras over S after base change). In the language of descent theory, the 5-Lie algebra isomorphism L(g, CTI,... , an) ®R S ~ gR ®R S tells that L(g, au ..., an) is a twisted form of gR ([GP1], [GP2] and [P2]). This perspective of viewing multiloop Lie algebras as twisted forms of gR provides a new way to look at their structure through the lens of descent theory. Since the affine group scheme Aut(g#) is smooth and finitely presented, Grothendieck's descent formulism classifies all isomorphism classes of twisted forms of QR split by S by means of the pointed set Hjt[S/R,Aut(gR)). There is a natural bijective map Isomorphism classes of twisted forms of gR split by 5 <—> H}t(S/R,Aut(gR)). 1 When R = C^ ], as g runs over the nine Cartan-Killing types A(, Be, Cg, D(, E6, E7, E$, F\ and G2, the 16 resulting classes in H\t correspond precisely to the isomorphism classes of affine Kac-Moody algebras (see Remark 3 in [P2]). For multiloop Lie algebras, the ring extension R —-> 5 is finite Galois with the Galois group G ~ TLjrtiiTL x ... x Z/m„Z. In this case, H.\t{SjR,Aut(gi?)) is the usual non-abelian 1 Galois cohomology H (G, Aut(gR){S)), where the S point of the affine group scheme Aut(gn) is given by Aut(gij)(5) = Auts(gfl ®R S) ~ Auts(g 0C S). 33 l A cocycle in Z (G, A.ut(QR)(S)) is a map u = (ug)geG : G —» Auts(Q®cS) satisfying a2 9 -1 ug29l = u92 ugi, where G acts on Auts(g ®c S) by 9 = (1 ® 5) o 9 o (1 ® g ). Each cohomology class [u] G Hl{G, Aut(g^)(5)) corresponds to one isomorphism class [Cu] of twisted forms of g#. The descended algebra Cu given by Galois descent can be expressed as 9 £U = {X £g®cS \ ug X = X for all g e G}. Thus a multiloop Lie algebra L(g, O\,..., Theorem 3.9 Let k be a field of characteristic 0 and g a finite dimensional split simple Lie algebra over k. Let R = k[tfr,... ,t^1] be the algebra of Laurent polyno mials in n variables over k and let C be a Lie algebra over k with centroid R. Then the following conditions are equivalent. (1) C ®R S ~ (9 ®fe R) ®R S ~ g <8>fc 5 for some fppf ring extension R—>S. (2) C ®R S ~ (g (3) C®RS ~ (g®k R) QRS ~ g®k S for some finite etale covering R —> S. (4) C®RS ~ {g®kR)®R,S — g®kS for some finite Galois ring extension R —>• 5. In particular, if k contains all roots of unity, then multiloop Lie algebras based on g satisfy all the above conditions. Ifk is algebraically closed, then centreless f.g.c. Lie tori of absolute type g are multiloop Lie algebras, thus satisfy all the above conditions. If k = C and n = 1, all the above conditions are equivalent to that C is an affine Kac-Moody algebra. 34 Multiloop Lie algebras are special examples of twisted forms of g f.g.c. Lie Tori Multiloop Twisted Lie *• Forms *• Multiloop Lie Tori Algebras (Descent Theory) -'I— Centreless • Lie Tori 1 1 t Figure 5: Multiloop Lie Algebras as Twisted Forms The general setting studied in this thesis is twisted forms of QR := g Aut(gB)(5 ®i? S) = A\its>(6R ®H S') ~ AutS/(gS')- All twisted forms of QR split by S are classified by Hl[S/R, Aut(flfl)). Each coho- 1 mology class [u] € H (S/R, Aut(QR)) corresponds to one isomorphism class [£u] of 35 twisted forms of QR. The descended algebra Cu given by faithfully flat descent can be expressed as jCu = {X€gs\uPl(X)=p2(X)}, where px and p2 are A;-Lie algebra homomorphisms defined by Pi : 0s ->• Ss', ^-iXi ®a,i^ SjXj P2 : 0s -• 05", Si^i ® a« ^ SjXj 36 Chapter 4 Central Extensions of Twisted Forms This chapter is devoted to study central extensions of twisted forms of split simple Lie algebras over rings. The general setting is twisted forms of g# := 0 37 the unique lift of an i?-linear automorphism of QR to its universal central extension OR fixes the centre QR/CIR pointwise and all i?-linear automorphisms of QR lift to every central extension of QR. Second, by using results from lifting automorphisms to central extensions and techniques in descent theory a new construction (called the descent construction) of central extensions is created for twisted forms of QR given by faithfully flat descent. A good understanding of the centre is also provided. Third, the descent construction gives information about the structure of automorphism groups of central extensions of twisted forms of QR. A sufficient condition is found under which every i?-linear automorphism of a twisted form of QR lifts to its central extension obtained by the descent construction and the lift fixes the centre pointwise. Finally, a sufficient condition for the descent construction to be universal is given. In particular, the universal central extension of a multiloop Lie torus is obtained by the descent construction. 4.1 Some Generalities on Central Extensions This section reviews the definition and some basic properties of central extensions. In particular, the classification of central extensions by Lie algebra cohomology is provided. Let C and V be Lie algebras over k. A central extension of C by V is a short exact sequence of Lie algebras 0—>V ^Z-^ C—>0 such that V is in the centre of C In other words, we have a surjective Lie algebra homomorphism ir : C —> L and an injective homomorphism % : V —> C such that V = ker{ir) is in the centre of C. The central extension C is said to be a covering of C in case C is perfect. 38 Given another central extension 0 —>V -^ £' X £ —>0 of £, then a morphism of the central extension £ to the central extension £' is a pair ( V —^-> £ —^— £ y —!—• £' —^ £ is commutative. The central extensions of £ with their morphisms form a category. Proposition 4.1 ([MP]) If the central extension £ of £ is a covering, then there exists at most one morphism of £ to a second central extension of £. A covering of £ is said to be universal if for every central extension of £ there exists a unique morphism from the covering to the central extension. To verify that a covering is universal, it suffices to show the existence of a morphism from the covering to any central extension. The uniqueness of each such morphism follows by the above proposition. It is clear from the definition that any two universal central extensions are isomorphic (in the sense of central extensions) and this isomorphism is unique. The following proposition decides which Lie algebras admit a universal central extension. Proposition 4.2 For Lie algebra £ to admit a (unique up to isomorphism) universal central extension, it is necessary and sufficient that £ is perfect. Proof. This result was proved in [vdK] Proposition 1.3 (ii) and (iii). The existence of an initial object in the category of central extensions of £ is due to Garland [Grl] 39 §5 Remark 5.11 and Appendix III. (See also Theorem 1.14 in [Nl], §1.9 Proposition 2 in [MP] and §7.9 Theorem 7.9.2 in [We] for details). D While finite dimensional split simple Lie algebras do not have nontrivial central extensions, both affine Kac-Moody algebras and their EALA descendants have non- trivial central extensions which play a crucial role in physics. In the case of affine Kac-Moody algebras, the universal central extension is one dimensional. For higher nullity extended affine Lie algebras, the universal central extensions are infinite di mensional, and one knows many interesting central extensions which are not universal (see [MRY] and [EF]). Lie algebra cohomology can be used to parameterize equivalence classes of central extensions of Lie algebras. Let £ be a Lie algebra over k. An C-module is a k- vector space V with a bilinear action C x V —> V where (x,v) i—»• x.v such that [x,y].v = x.y.v — y.x.v for all x,y 6 C and v € V. Let CP{C,V) (p > 1) be the A>vector space of alternating multilinear mappings £x£x ••• x£ —>V v ' p-times and let C°(C, V) = V, CP(£, V) - {0} (p < 0). Define a fc-linear map dp:Cp(C,V)—>Cv+l{C,V) by v 1 H d (f)(x1,x2, •.., xp+i) = Y^ (- ) "V(N, Xj],xi, ...,Xi,...,Xj,..., xp+1) P+i + Yl Xi-f(X^ • • • ' ^' • • • ' XP+0 fOT P - 1 40 and d°(m)(x) = x.m, dp = 0 for p < 0. It is clear that dp+ldv = 0 and +oo C*(£,V) = Q)CP(£,V) is a cochain complex of £ with values in V. Then Zp(£, V) = ker(dp) is the set of p-cocydes and BP(£,V) = im(dp~1) is the set of p-coboundaries. The sets of equivalence classes H*(£,V) = Z*(£,V)/B*(£,V) are called the cohomology groups of £ with coefficients in V. By definition we have H°(£, V) = Vc = {v € V | x.v = 0, V x G £}; ^(AV) = Derk(£,V)/IrmDerk(£,V). Let V be a trivial £-module, i.e., x.u = 0 for all x G £ and i> G V. Then we can write down Z2(£, V) and J32(£, V) explicitly. 2 2 Z (£,V) = {fe C (£,V) I f([xux2},x3) - f([xux3],x2) + f([x2,x3},Xl) = f([x1,x2},x3)+f([x3,x1},x2)+f([x2,x3},x1) = 0,Vxl,x2,x3 G £}; 2 £ (£,F) = {d\f) : £ x £ - ^(n.u) ^ -/([xi,x2]) | / € Hom,(£,F)}. The following proposition shows that #2(£, V) = Z2(£, V)/B2(£, V) can be used to classify all central extensions of £ by V up to equivalence. Two central extensions 0 —> V —> £, —> £ -^ 0 are equivalent if there is a Lie algebra isomorphism 4>: £\ ~ £2 such that the following diagram is commutative. V - £1 £ V £2 £ Proposition 4.3 ([We]) Let V be a trivial £-module. The set of equivalence classes of central extensions of £ by V is in 1-1 correspondence with H2(£,V). 41 The second Lie algebra cohomology group H2(C, V) thus gives a parameterization of all equivalence classes of central extensions of £ by V. Any cocycle P G Z2(£, V) leads to a central extension 0 —>V —>£P^£ —• 0 of £ by V as follows: As a fc-vector space £p = £ © V, and the bracket [, ]p on £p is given by [x + u,y + v]p = [x,y] + P(x,y) for x,y € £ and u,t; G V. The equivalence class of this central extension depends only on the class of P in H2(C,V). We will naturally identify C and V with subspaces of Cp. Assume £ is perfect. We fix once and for all a universal central extension of £ 0—>V—>£-^£—>0, referred to as the universal central extension of £. We can think of this universal central extension as being given by a "universal" cocycle P, i.e., £ = £p = £ © V. This cocycle is not unique, but we fix one such "universal" cocycle in our discussion. 4.2 Lifting Automorphisms to Central Extensions This section presents new results about lifting automorphisms to central extensions which will be used to construct central extensions of twisted forms in the next section. An automorphism 6 G Aut^(£) is said to lift to £p, if there exists an element Op G Autfc(£p) for which the following diagram commutes. £p • £ eP e £P • £ 42 We then say that Op is a lift of 9. By definition, Op stabilizes V, thereby inducing an element of GLk(V). In general, Op need not stabilize the subspace C of Cp. Note that 6p(x) — 0(x) € V for all x € C. Since V lies inside the centre i(Cp) of Cp, we get the useful equality 0P([x, y}P) = [0(x), 0(y)}P for all x, y E C. (4.1) The following lemma gives a complete description of which automorphisms of C can be lifted to its central extensions. Lemma 4.4 Let 5 : Homfc(£, V) —> Z2(C,V) be the coboundary map, i.e., S(/y)(x,y) = —j([x,y]). For 0 € Autfc(£) and P € Z2(C, V), the following conditions are equiva lent. (1) 0 lifts to CP. (2) There exists 7 € Homfc(£, V) and \i e GLfc(V) such that fxoP-Po(Ox0) = 8(j). In particular, for a lift 9p of 0 to exist, it is necessary and sufficient that there exists./J, € GLk(V) for which, under the natural right action of GLk(V) x Autfc(£) on H2(C,V), the element (n,6) fixes the class [P]. If this is the case, the lift Op can be chosen so that its restriction to V coincides with fi. Proof. (1)=^(2) Let us denote the restriction of Op to V by //. Define 7 : C —> V by 7 : x !->• 0p(x) — 6{x). Then 8P(x + v) = 0(x) + 7(2) + n(v) for all x 6 C and v G V. For all x and y in £, we have 43 (^oP-Po(9x 6))(x,y) = n(P(x,y)) - P(9(x),9{y)) = H([x,y]p - [x,y]) - ([9(x),9(y)}P - [9(x),9(y)}) = eP([x,y]P - [x,y]) - [9(x),9(y)}P + [9(x),9(y)} = 9P([x,y]P)-9P([x,y}) - [9(x),6(y)}P + {9(x),9(y)} = eP([x,y}P) - (9([x,y}) + j([x,y})) - [9(x),9(y)}P+ [9(x),9(y)} = [9(x),9(y)]P - 7([x,y\) - [9(x),9(y)]P (by 4.19) = -j([x,y}) = 5(i)(x,y). (2)=>(1) Define 9P € Endfe(£P) by 6P{x + v) = 9(x) + 7(x) + n(y) (4 for all x € C and v eV. Then #p is bijective; its inverse being given by l -1 e- (x + v) = e~\x) - tr^ie-Hx)) + M ^)- That #p is a Lie algebra homomorphism is straightforward. Indeed, 9P[x + u, y + v]P = 9P([x, y] + P(x, y)) = 9([x,y}) + -f([x,y}) + fioP(x,y) = 9([x,y}) + (voP-5(1))(x,y) = [9(x),9(y)] + P(9(x),9(y)) = [9(x), 9(y)}P = [0P{x + u), 9P{y + u)]P. Since the action of Autfc(£) x GLjt(V) on Z2(C, V) in question is given by 1 pW)=/i- oPo(«xS)1 the final assertion is clear. 44 V The final assertion in the above lemma is a Lie algebra version of a well known result in group theory. For a much more general discussion of the automorphism group of non-abelian extensions, one can consult [Nb] (specially Theorem B2 and its corollary). For future use, we recall the following fundamental fact. Proposition 4.5 Let £ be a perfect and centreless Lie algebra over k. Let 0—>V —>£-^£—>0 be its universal central extension. Then the centre %(£) of £ is precisely the kernel V of the projection homomorphism n : £ —> £ above. Furthermore, the canonical map Autfc(jC) —» Autfc(£) is an isomorphism. Proof. This result goes back to van der Kallen (see §11 in [vdK]). Other proofs can be found in [Nl] Theorem 2.2 and in [PI] Proposition 2.2, Proposition 2.3 and Corollary 2.1. • The above proposition tells that a perfect and centreless Lie algebra £ and its universal central extension £ have the same automorphism group, thus every auto morphism of £ can be lifted uniquely to its universal central extension £. It is also important to know informations about automorphism groups of other central exten sions of £. The following lemma gives equivalent conditions for an automorphism of £ to be lifted to every central extension of £. Lemma 4.6 Assume £ is perfect and centreless, and let 0 —> %{£.) —>£—>£ —> 0 be its universal central extension. For an automorphism 6 6 Autfc(£), the following conditions are equivalent. (1) The lift 6 of 6 to £ acts on the centre of £ by scalar multiplication, i.e., 0\ IQ = A Id for some A G kx. 45 (2) 6 lifts to every central quotient of C. (3) 9 lifts uniquely to every central quotient of C. (4) 9 lifts to every central extension of C. Proof. (1)=>(2) The lift 6 exists by Proposition 4.5. Let Cp be a central quotient of £. Then CP ~ C/J = (£ ffi l{C))/J ~ £ ffi i{C)/J. Since 9\i{£) = Aid for some Aefcx,we have 9(J) C J. So # induces an automorphism of Cp. (2) =4> (3) The point is that 0—> J —> £ —>• £ ffi l(fi)lJ ~* 0 is a universal central extension of £ © %{£.)/J. By Proposition 4.5 then, any two lifts of 0 to £ © l(C)/ J must coincide since they both yield 0 when lifted to £. (3)=»(4) There is no loss of generality in assuming that the central extension of £ is given by a cocycle, i.e., that it is of the form 0 —> V —> Cp —• £ —*• 0 for some P € Z2(C, V). There exists then a unique Lie algebra homomorphisms 0 • 3(£) > £ • C • 0 *l3(£) 0 • V • £P • £ • 0 Let J = ker. Then J C 3(£), and the central quotient £ ffi i(C)/J corresponds to the cocycle Q obtained by reducing modulo J the universal cocycle P chosen in modeling £. We have Cp = C © V = (/>(£ ffi %{C)/J) ffi V for some suitable subspace V' of V. Let #Q be the unique lift of 9 to £ 0 3(£)/J, which we then transfer, via 0, to an automorphism 9'Q of the subalgebra 4>(C®%(C)/J) of £p. Then 9p — 9'Q + ldvi is a lift of 9 to £p. (4)=>(1) Assume 0\ ,K ^ Aid for all A e fcx. Then, there exists a line J C j(£) such that 0(J) ^ J. Consider the central quotient £ ®i(C)/J. Let #p be a lift of 46 6 to C © l{£)/J. As pointed out before, 0 -+ J -> £ -> C ® l{C)/J -> 0 is the universal central extension of £ ® i(C)/J. So 0 is also the unique lift of Op. This forces 0(J) C J, contrary to our assumption. • The rest of this section will focus on studying explicitly lifting automorphisms of QR to its central extensions. We view QR = g [x®a,y®b] = [x,y]®ab (4.3) for all x,y E g and a,b G R. Of course QR can also be viewed naturally as an f2-Lie algebra (which is free of finite rank). It will be at all times clear which of the two structures is being considered. Let (VLn/k, dp) be the i?-module of Kahler differentials of the fc-algebra R. When no confusion is possible, we will simply write (£IR, d). Following Kassel [Ka], we consider the A>subspace dR of QR, and the corresponding quotient map ~ : QR —• Clji/dR. We then have a unique cocycle P = PR G Z2(gR,Qp/dR) satisfying P(x0 a,y®b) = (x\y)a~db, (4.4) where (• | •) denotes the Killing form of g. Let gU be the unique Lie algebra over k with the underlying space QR @ Qp/dR, and the bracket satisfying [x 0 a, y As the notation suggests, 0 —+ £VcLR —f QR- -^ gR —> 0 47 is the universal central extension of QR. There are other different realizations of the universal central extension (see [N3], [MP] and [We] for details on three other different constructions), but Kassel's model is perfectly suited for our purposes. The following proposition proves an important fact about QR, namely the unique lift of an i?-linear automorphism of QR to its universal central extension QR fixes the centre pointwise. This fact will be crucial in the descent construction for central extensions of twisted forms of g#. Proposition 4.7 Let 9 G Autk(QR), and let 6 be the unique lift of 8 to QR (see Proposition 4-5). If 8 is R-linear, then 9 fixes the centre QR/OIR ofQR pointwise. In particular, every R-linear automorphism of QR lifts to every central extension of QR. Proof. For future use, we begin by observing that [x 0 = [x(g> ab, x = [9(x®ab),e(x®l)]p(by4.19) = [y] Xj 0 abaj, y^ Xj 0 aj]p i j = 2_\xii xj) ® abaiaj + / ^(xi\xj)abaidaj. Thus y^y(xi\xj)abaidaj = 0 = YJ^, Xj] 0 abataj. (4.6) Since 8 is .R-linear, it leaves invariant the Killing form of the R-Lie algebra QR. We thus have x x a a 4 7 (x\x)g = (x® l\x 48 We are now ready to prove the proposition. By Lemma 4.6, it will suffice to show that 9 fixes Q.R/CLR pointwise. Now, ${(x\x)adb) = 9{[x®a,x®b]p) = [9{x®a),9(x®b)]p (by 4.19) = [}j Xj ® adj, )y Xj ® baj]p i 3 = \J[xj,Xj] Remark 4.8 Each element 6 G Autfc(-R) can naturally be viewed as an automor phism of the Lie algebra QR by acting on the i?-coordinates, namely 6{x®r) = a;® 9r for all x 6 g and r € R. The group Autjt(i?) acts naturally as well on the space Qn/dR, so that (adb) — 8adeb. A straightforward calculation shows that the map e 6 e GL/C(0R) defined by 9{y + z) = 9{y) + z for all y & QR and 2 € Q,R./dR, is an automorphism of the Lie algebra Q~R. Thus 9 is the unique lift of 9 to JR prescribed by Proposition 4.5. Note that 9 stabilizes the subspace QR. We now make some general observations about the automorphisms of QR that lift to a given central quotient of QR. Without loss of generality, we assume that the central quotient at hand is of the form (QR)P for some P E Z2(QR,V). Let 9 6 Autfc(g#). Since Q~R is the universal central extension of its central quotients, the lift 9p, if it exists, is unique (Lemma 4.6). We have Autfc(gfl) = Ant R(QR) XI Autk(R) (see [ABP2] Lemma 4.4 or [BN] Corollary 2.28). By Proposition 4.7 all elements of 49 Autp(gp) do lift, so the problem reduces to understanding which 0 € Autk(R) admit a lift Op to (QR)P. Since 0 stabilizes QR, the linear map 7 of Lemma 4.4 vanishes. This holds for every central extension of QR, and not just central quotients of QR. We conclude that 9p exists if and only if there exists a linear automorphism \i G GLfc(V) such that P = p(^'e). We will end this section by a concrete example showing how results about lifting automorphisms to central extensions can be used to describe automorphism groups of Lie algebras obtained by central extensions. Example 4.9 Let k = C and R = Cftf1,^1]. Fix ( e C, and consider the one di 2 mensional central extension Cp( = QR®CC, with cocycle P^ given by P^xQt^t™ , y Each 0 £ Antc(R) is given by 6(h) = \itftf and 6(t2) = \2tftf for some P\ V2 \ x x x € GL2(Z), and Ai, A2 € C . The natural copy of the torus T-C xC GL2(Z)^ consisting of elements of GL2(Z) that admit a lift to Autc(£pc). Then x x Autc(£pc) ^ Autfi(gfl) xi ((C x C ) xi GL2(Z)C). Note also that since Pie(P) = 1, the structure of the group Autp(g#) is very well understood [PI]. The GL2(Z)^ form an interesting 1-parameter family of subgroups of GL2(Z) that we now describe. Let 6 € GL2(Z). Then by Lemma 4.4, 6 € GL2(Z)^ if and only if there exists 7 : QR —> Cc and fi £ GLc(Cc) ~ Cx, such that j([x 50 for all x, y € 0, and all a,b G R (in fact 7 = 0, as explained in Remark 4.8) . Choose 1 x,y E 0 with (x\y) ^ 0. Since in QR we have [x •(;)-(::)(;)-(;)• ••- The group GL2(Z)^ could thus be trivial, finite, or even infinite, depending on some 2 arithmetical properties of the number (. For example if £ = — 1, then GL2(Z)f is a cyclic group of order 4, generated by the element a for which u{t\) = £2 and The one-parameter family GL2(Z)^ has the following interesting geometric inter pretation. By the universal nature of g^ = QR ® Qn/dR, we can identify the space Z2(QR,C) of cocycles with Hom(Oi?/di?,C) = (QR/CIR)*. Furthermore, this identifi cation is compatible with the respective actions of the group Aut^ (R). The action of the torus T on QR/CIR is diagonalizable, and the fixed point space T 1 1 V — (QR/CIR) is two dimensional with basis {t^ dti,t2~ dt2}- The cocycle P$ is T-invariant, and corresponds to the linear function F$ E (Qn/dR)* which maps t~[ldt\ 1—> 1, tl^dti 1—>• £, and vanishes on all other weight spaces of T on QR/dR. We can thus identify F^ with an element of V*. The action of GL2(Z) c Autc(-R) on Vtn/dR stabilizes V, and is nothing but left multiplication with respect to the chosen basis above. e) x Let 9 E GL2(Z). Then 6 lifts to CP(. if and only if Pc = P^' for some /i G C (Remark 4.8). With the above interpretation, this is equivalent to 6 stabilizing the line CF{ C V*; which is precisely equation (4.9) after one identifies V with V* via our choice of basis. 51 4.3 Descent Constructions for Central Extensions This section presents a new construction for central extensions of twisted forms of QR by using results from lifting automorphisms to central extensions and techniques in descent theory. A twisted form of QR is an R-Lie algebra descended from 05, where S is a commutative, associative, unital fe-algebra and R —> S is a faithfully flat ring extension. The descent data corresponding to a twisted form of QR is an element u € Aut(QR)(S®RS) satisfying d°u d?u = d}u. Let S' := S®RS and Qs' '•= 9®kS', then Aut(flfi)(5 All twisted forms of QR split by S are classified by H1(S/R, Aut(g#)). Each coho- 1 mology class [u] 6 H [S/R, Aut(gR,)) corresponds to one isomorphism class [£u] of twisted forms of QR. The descended algebra Cu given by faithfully flat descent can be expressed as £u = {Xegs\uPl(X)=P2(X)}, where px and P2 are fc-Lie algebra homomorphisms defined by Pi • 9s -> ds', ^iXi <8) a,i >-> EjXj (g) at 0 1, P2 : 9s —> 0S', S^Xj <8> a* H-> S^XJ (g) 1 As an /?-Lie algebra, Cu is centrally closed, but it is not as a fc-Lie algebra, thus it has central extensions over k. The purpose of this section is to construct central extensions of Cu by using its defining descent data to construct a &-Lie subalgebra of 9~s ~ 9s © fls/dS, where (Qs, d) is the 5-module of Kahler differentials of the k- algebra S. We first define two maps from Qs/dS to Cis'/dS' and the following lemma verifies that they are indeed well denned. 52 Lemma 4.10 The following two maps are well defined k-linear maps. if! : ns/dS -> Qs'/dS', J[a% i-> (s: ip2 : Vis/dS -> tts'/dS', sxds2 i-> (1 (8) Si)d(l Proof. By the universality of (fig, d) the ring homomorphism on : 5 —>• 5' defined by cti(s) = s By the above lemma we can define two fc-linear maps to be used for our construc tions of central extensions. Let X e Qs and Z € Qs/dS, define Pi-Ss-^Ss', X + Z ^pt{X) + ipi(Z), P2-9s ~> 9s>, X + Z h^ p2(X) + f2(Z). The following lemma verifies that they are indeed A;-Lie algebra homomorphisms. Lemma 4.11 pi andp2 are homomorphisms of k-Lie algebras. Proof. Let X = X + Z and Y = Y + W be elements in §s, where X = ^2ixi = ^[XJ, J/J] 53 On the other hand, [pl(X),pi(y)]fl7, = [pi(X),PI(Y)]Q = J>< ® = ]P[£i,%] ® ((oj ® l)(6j <8> 1)) + (xi|j/j)(a4(8)l)d(6j<8il) = ^2[xi} yj] ® a,6j (8) 1 + (xj|yj)(aj <8> l)d(6j <8> 1). a So pi is a k-Lie algebra homomorphism. Similarly we can show that p2 is A;-Lie algebra homomorphism. • The following proposition is the main result in this section and gives a natural construction for central extensions of twisted forms of QR. 1 Proposition 4.12 Let u e Auts'(gs>) be a cocycle in Z (S/R, Aut(QR)) and let u e Autfc(gs"/) be the unique lift of u to Qs*. Then the k-Lie subalgebra ofQs defined by £u:={Xe^\uf1(X)=f2(X)}. is a central extension of the descended algebra Cu corresponding to u. Proof. First £% is a ft-subspace of gj. Then let X, Y G CQ, we have ufi({X,Y}) = u(\pi(X)MY)]) = [2(pl(X)),S(pi(y))] = \p2(X),p2(Y)}=p2([X,Y}). So [X, Y] € £% and £% is a fc-subalgebra of §5. Moreover if X € 0s and Z € Qs/dS are such that I + ^G £2, then up[(X + Z) = p2(X + Z)= p2{X) + ip2(Z). Since ufi(X + Z) = U(PI(X) + ^(Z)) = uPl(X) + (u - u)(pi(X)) + ^(Z) and (u — u)(pi(X)) + 54 extension of Qs- Since the kernel of 7r|£s : £s —> Cu is visibly central, the only delicate point is to show that 7r(£a) = Cu. Fix X E Cu, i.e., upi(X) — p2pO- We must show the existence of some Z E Qs/dS for which X + Z € C$, i.e., ufi(X + Z) = f2(X + Z). Thus we must show the existence of some Z E Qs/dS for which (u — u)(pi(X)) = ( y 7(PIW) = 7(Ebi(^i).PiW)]) = E^(^)'^( <)] = E-*W(PI(X«),PIW) = ^i>PlpQWi(^)) - £(PI(*0,PIM) = ^P(p2pQ),p2(Y<)) - P(PlpQ,Pl (}-,)) = 5ZP(^Xy<8)l(E>ay,^yifc X - XI ^($Z *J <8> % i j,k -^2^2(xij\yik){aij = X ^(^jbifc)(y2(aij^fc) - ^\{aijdbik)) i j,k 55 Thus l(Pi(X)) = {^-^(^^(Xijly^aijdbik). i j,k So Z := J2i Yjj,k(xii\yik)aijdhk € Qs/dS as desired. D Remark 4.13 This construction recovers the result of Proposition 4.22 (2) in [PPS] when R —• S is a finite Galois ring extension with the Galois group G. The descent l data in the Galois case is a cocycle u = (ug)geG in Z (G, Aut(g^)(5)), where ug G Auts(fls). The corresponding twisted form of QR has the following expression g £u = {X € Qs ] ug X = X for all g e G}. The Galois group G acts naturally both on Qg and on the quotient fc-space Qs/dS, in such way that 9{sdt) = 9sd9t. This leads to an action of G on §5 for which 9(x®S + z) =£ 9 9 [xi 0si + 2i,x2 <8>s2 + 22]g5 = ([x!,X2] <8)Sis2 + (xi\x2)s1ds2) S S = [Xl,X2] ® 5! S2 + (24I £2)^1*^2 9 9 S 9 = [Xi® Si+ Zi,X2® S2+ ^2]gS- Accordingly, we henceforth identify G with a subgroup of Autfc(gs), and let G act on 9 l Autfe(gs) by conjugation, i.e., 9 = g9g~ . Each ug lifts uniquely to ug E Autfc(gs), 1 then u = (ug)geG is a cocycle in Z (G, Autfe(jjs)). The &-Lie subalgebra of gs defined in Proposition 4.12 has the following expression in the Galois case 9 CQ = {XE^\ ug X = X for alls € G} which coincides with the construction in [PPS]. 56 The construction in Proposition 4.12 is called the descent construction for central extensions of twisted forms of QR. A good understanding of the centre a(£a) is provided by the following proposition. Proposition 4.14 The centre of Cq is %(£$,) = {Z E Vis/dS \ p[(Z) = f2(Z)}. Proof. By faithfully flat descent considerations, Cu is centreless. Indeed, the centre i(Cu) C Cu is an .R-submodule of Cu. Since S/R is faithfully flat, the map l{Cu) ®RS -* CU®RS~QS is injective. Clearly the image of i{Cu) ®R S under this map lies inside the centre of 0s, which is trivial (as one easily sees by considering a fc-basis of S, and using the fact that 3(0) = 0). Thus i{Cu) ®R 5 = 0, and therefore i{Cu) = 0 again by faithfull flatness. Thus the centre of £s lies inside Qs/dS. Indeed, if X € 0s and Z E Qs/dS are such that X + Z e 3(£s), then we know X e Cu. Now we must show that X € i{Cu) = 0. For any Y € Cu there exists W € Cis/dS such that Y + W G £s since TT|£O is surjective. So [X + Z,Y + W\L~U = [X,Y]Ca = 0. Thus [X,Y]Cu = 0. This shows X e l(Cu) = 0, thus i{Cu) C Qs/dS. By Proposition 4.7 u fixes Qs>/dS' pointwise, thus 3(£s) = {Z € Qs/^5 | fi{Z) = f2(Z)}. D Remark 4.15 In the case of R —• 5 is a finite Galois ring extension we have 3(£s) = G (CLs/dS) ~ Ofi/di?. In general, however, fifl/dfl C 3(£s) (see [WG] §3 Example 3.1 for details). The descent construction provides a new way to construct central extensions of multiloop Lie algebras as they are special examples of twisted forms of QR, thus the descent construction gives new insight to solve important problems in the structure 57 theory of infinite dimensional Lie algebras such as the structure of automorphism groups of extended affine Lie algebras and universal central extensions of multiloop Lie algebras which will be discussed in the next two sections respectively. 4.4 Automorphism Groups of Central Extensions This section presents how the descent construction gives information about the struc ture of automorphism groups of central extensions of twisted forms of QR. For the Lie algebra QR we already know that every .R-linear automorphism of QR lifts to ev ery central extension of QR. For twisted forms of QR, a sufficient condition is found under which every i?-linear automorphism of a twisted form of QR lifts to its central extension obtained by the descent construction and the lift fixes the centre pointwise. We keep the same notation as before. Let Cu be a twisted form of QR descended from Qs for some faithfully flat ring extensions R —> S and let C% be the central extension of Cu obtained by the descent construction. The following proposition gives equivalent conditions for Cu — Cu® %(£,%)• Proposition 4.16 The following conditions are equivalent. == (1) MI Cu © 3(.MJ)- (2) Cu C (3) uPl{X) = p2(X) for any X e Cu. (4) upxiCu) (5) upi(Cu) =p2(£«)- Proof. (2)=^(1) Cz D Cu © i(Cu) is dear since Cu C £a and i(Cy) C £s- On the other hand, if X € 0s and Z € Qs/dS are such that X + Z e Cu, then we have 58 Ie£„c£i. So Z = (X + Z) - X € £u- Thus px{Z) = upi(Z) = f2(Z). This shows that Z E 3(£a). Thus £„ C Cu ® 3(£a). (1)=>(2) This is clear. (2)=^(3) For any I e £„ we have up\{X) = p2{X) and Pi(X) = pi(X) for i = 1,2. If £„ C Cu, then upi(X) = upl(X) = p2(X) = p2{X). (3)=^(2) This is clear. (3)=K4) This is clear. (4)=3-(3) For any X G Cu we have upi(X) = up\(X) + (u — u){pi{X)), where upi(X) E Qs> and (u - u)(pi(X)) € Qs'/dS'. If upi(Cu) C p2(Cu), then upi(X) e V2{CU) C gS: Thus %(X) = uPl(X)=p2(X). (4)=>(5) We have shown (4)=>(3). It is clear(3) implies upi{Cu) = p2(Cu). (5)=>(4) This is clear. • It turns out that the equivalent conditions described as above provides a sufficient condition for i?-linear automorphisms of Cu to lift to CQ. Proposition 4.17 // Cu = Cu ® i(£n), then every 9 € Autfl(£u) lifts to an auto morphism 6 of Cu that ftxe$ the centre of CQ pointwise. Proof. Let 9s be the unique 5-Lie automorphism of Q$ whose restriction to Cu coincides with 9. Let 9s be the lift of 6s to §g. We claim that 9s stabilizes Cu = Cu® }>(Cu). By Proposition 4.7 §s fixes I(CQ) pointwise. Let X € Cu. Since Cu is perfect, we can write X = J2iiXi^Yi\es f°r some xu^% e £«• Thus X = ^pQ,Y^]g - Z for some Z e 3(A)- Then ^(X) = £4[0(Xi),0Q'i)]gs - 2 € [£„,£„]£ + a(A) C *-M ©3(."Mz) ~ Cu- 1—1 Remark 4.18 The i?-group Aut(£„) is a twisted form of Aut(gfl) (§4.4 of [GP2]). In particular the afflne group scheme Aut(£u) is smooth and finitely presented. We 59 have AutR(£U) = Aut(£u)(R). Every automorphism of Cu as a fc-Lie algebra induces an automorphism of its centroid. The centroid of £„, both as an R and &-Lie algebra, coincides with R (acting faithfully on Cu via the module structure). By identifying now the centroid of Cu with R, we obtain the following useful exact sequence of groups 1 -> Aut*(£u) -• Autfc(£„) -> Autfc(i2). (4.10) This sequence is a split short exact sequence if Cu is the twisted loop algebra of type A^n_x, thus one has a new concrete realization of the automorphism group of the corresponding twisted affine Kac-Moody algebra. For other types of loop alge bras or multiloop Lie algebras, whether the above sequence is a split short exact sequence remains an open problem. If moreover the descent data for Cu falls under the assumption of Proposition 4.17, then one also has a very good understanding of the automorphism group of £%• When Cu is a multiloop Lie tori, the automor phism group of Cy, are crucial to understand the structure of automorphism groups of extended afRne Lie algebras. 4.5 Universal Central Extensions This section presents a sufficient condition for the descent construction to be univer sal. In particular, the universal central extension of a multiloop Lie torus is given by the descent construction. For a non-twisted multiloop Lie algebra g^ where R is the algebra of Laurent polynomials in n variables, its universal central extension can be constructed by Kassel's model, namely QR = QR © CLR/CLR. It is much more complicated in the twisted case. Kassel's model was generalized in [BK] under certain conditions. Unfortunately twisted multiloop Lie tori do not satisfy these conditions. 60 In [N2] E. Neher constructed central extensions of centreless Lie tori by using cen- troidal derivations and stated that the graded dual of the algebra of skew centroidal derivations gives the universal central extension of a centreless Lie torus. Since the centroidal derivations are essentially given by the centroid, to calculate Neher's con struction of universal central extensions of centreless Lie tori depends on a good understanding of the centroid. In this section, it is proven that for a multiloop Lie torus £u, its universal central extension can be obtained by the descent construction and a good understanding of the centre is provided, namely £u ~ £Q = £u © QR/dR and the centre i{£u) ~ ClR/dR. Throughout this section R —• S is a finite Galois ring extension with the Galois G group G. We identify R with a subring of S and VtR/dR with {Q,s/dS) through a 1 chosen isomorphism. Let u = (ug)gSG £ Z (G, Auts(gs)) be a constant cocycle with ug~vg® id for all g e G. Then the descended Lie algebra corresponding to u is g CU = {X EQS\ ug X = X for all g € G} 9 = {EjXj x Let 0o = { £ 0 I vg(x) = x for all g G G}. Then g0 is a fc-Lie subalgebra of g. We write QoR — go <2)k R- Clearly goR is a A;-Lie subalgebra of Cu. Assume go is perfect and let go^ = go# © Qn/dR be the universal central extension of QoR. We first prove a useful, lemma and then generalize C. Kassel's proof in [Ka] that g~K is the universal central extension of g^. Lemma 4.19 Let £ be a Lie algebra over k and let V be a trivial C-module. Ifs C £ is a finite dimensional semisimple k-Lie subalgebra and £ is a locally finite s-module, then every cohomology class in H2{£, V) can be represented by an s-invariant cocycle. 61 Proof. For any cocycle P G Z2(£, V), our goal is to find another cocycle P' G Z2(£, V) such that [P] = [P'} and P'(£,s) = {0}. Note that Eomk(C,V) is a £-module given byy.p(x) = P(-[y,x}). Define a A;-linear map / : s —* Honifc(£, V) by f(y)(x) = P(x,y). We claim that / G ^(s.Homjfe^, V)). Indeed, since P G Z2(£, V), we have P(>,y) = -P(y,x) and P([x,y],z) + P([y,2:],£) + P([z,x],y) = 0 for all x,y,z G C. Then P(x, [y,z]) = P([x,y],z) + P([z,x],y), namely f([y,z])(x) = f(z)([x, y]) + f(y)([z, x]) for all x, y, z G £. Thus /(M) = !/./(*)-*./(y) implies / € Z^s.HonifcOC, V)). By our assumption that s is finite dimensional and semisimple, the Whitehead's first lemma (see §7.8 in [We]) yields H1[s,Hom.k(£,V)) = 0. Note that the stan dard Whitehead's first lemma holds for finite dimensional .s-modules. However, Homfc(£, V) is a direct sum of finite dimensional s-modules when £ is a locally finite s-module and V is a trivial £-module, so the result easily extends. So / = d°(r) for some r G Hom/c(>C,V), where d° is the coboundary map from Honifc(£, V) to C^s.HomfcOClO). Let P' = P+e^fV), where d1 is the coboundary map from Horm.(£, V) to C2(£, V). Then [P'] — [P]. For all x G £ and y G s we have P'(x,y) = Pfo^ + d^Xz.J/) = P(x,y)-T([x,y\) = P(x,y)-f(y)(x) = 0. Thus P' is an s-invariant cocycle. D 62 Remark 4.20 A version of this lemma was proved in [ABG] §3.3-3.5 for Lie algebras graded by finite root systems. Proposition 4.21 Let Cu be the descended algebra corresponding to a constant cocy- l cle u = (ug)geG 6 Z (G, Auts(0s)). Let Up be a central extension of CU with cocycle 2 P € Z (CU,V). Assume 0o is central simple, then there exist a k-Lie algebra homo- morphism ip : QOR —> Cp and a k-linear map tp : Qn/dR —• V such that the following diagram commutes. 0 • QR/dR 9OR QOR 0 v inclusion 0 • V cu —• o Proof. Since the first row of the diagram is a universal central extension of 0O_R and the second row is a central extension of Cu, the above proposition is a special case of [vdK] Proposition 1.3 (v). We give another proof here by constructing the maps ip and ip explicitly which are to be used in the proof of Proposition 4.26. 2 Our goal is to find PQ G Z (CU, V) with [P0] = [P] satisfying 4 PQ(x® a,y 0 1) = 0 for all x,y G 0O and a € R. ( -H) Applying Lemma 4.19 to L = QQR and s — 0o ®fc k, it is clear that £ is a locally finite s-module and thus we can find an s-invariant cocycle P' £ Z2(£,V), where P' = P \CxC + d}(T) for some r € Homfe(£, V). We can extend this r to get a fc-linear l map T0 : Cu = Q0R © 0o]? -* V by T0 , = r and T0 , L = 0. Let P0 = P + d (TQ), yu Ft yu R 1 2 where d is the coboundary map from H.om.k(£u,V) to C (CU,V). Then [PQ] = [P] 63 and it is easy to check that for all x, y E go and a E R we have 1 P0(x®a,y® 1) = P(x®a,y®l) + d (T0)(x®a,y®l) = P(x<8)a,y(8)l) + d1(T)(a;(8ia,y<8) 1) = P'(x 2 Replace P by Po. Since P G Z (CU, V), we have P(x®a,y®b) = -P(y®b,x®a), (4.12) P([x P([z,x] So a([z,x],y) = a(x,[y,z}). This tells us a([x,z],y) = a(x,[z,y]), namely a is an invariant bilinear form on go- Since go is central simple by our assumption, go has a unique invariant bilinear form up to scalars. It follows that there is a unique za}> E V such that for all x, y E go we have P(x (8) a, y <8> 6) = a(x, y) = (x\y)za>b, (4.14) where (• | •) denotes the Killing form of g. From (4.11), (4.12), (4.13) and (• | •) is symmetric we have (i) za,i = 0, (ii) zafi = -zbta, (in) zahfi + zhc,a + zCa,b = 0. (4.15) 64 Then by (ii) and (iii) the map p : QR/U — Hi(R, R) ~ R ®k Rj < ab Finally let a : Cu —• Cp be any section map satisfying [o{x X G 0Oi? and Z G Qn/dR. Clearly ip is a well-defined ^-linear map. We claim that ip is a Lie algebra homomorphism. Indeed, let x ^([a; [^(x (8) a), ?/>(y (8) 6)]£p = [CT(X ® a), cr(y By (4.14) this shows that ip is a Lie algebra homomorphism. It is easy to check the following diagram is commutative. 0 • QR/dR QOR 8OR 0 0 • V cu —• 0 D Remark 4.22 The above proposition generalizes C. Kassel's result in [Ka] over fields of characteristic zero. When u is a trivial cocycle, we have Cu = QR and Q0 = Q. The above proposition shows that §# is the universal central extension of QR. To understand the universal central extensions of twisted forms of QR, we need to construct a cocycle PQ which satisfies a stronger condition than (4.11). For each 65 9 a G SV^O} define ga = {x G g | vg(x) Lemma 4.23 (1) ga = go ffl6 i?\{0}. In particular, gi = g0. (%) [flo, 0b] C ga& /or any a, 6 6 5\{0}. [ga, g0] C ga /or any a G S^O}. f^ Honife(ga, V) is a g0-module for any k-vector space V, a G 5\{0}. 9 Proof. (1) If a G -R, then a = a for all g e G. Thus ga = {x G g : vg(x) <8> a = x (8> a for all g G G}. Clearly ga D go- On the other hand, let x G ga and let {ajj®aj}ie/jej be a fe-basis of g§. Assume x = SjAjXj, t>ff(x) = EjAfxi and a = E^-a,. Then fs(x) <8> a = x 0 a implies that EyAf/Zj(xj Af/ij = Aj/Uj for all i € I,j €. J and g £ G. Since a ^ 0, there exists /Xjo ^ 0. By Af//ja = \Hja we get Af = Aj for alH G / and g G G. Thus x G go, so ga = g0. 9 s (2) Let x e Qa and y G g&. Then us([x,y]) ® (a6) = [i>5(x),t;s(y)] (3) Let y G go and /3 G Homfc(g0,V). Define y./3(x) = /3(—[y,x]). We can check y.j3 is a well-defined g0 action. D Proposition 4.24 Lei £„ 6e the descended algebra corresponding to a constant co- l cycle u = (ug)geG G Z (G, Auts(gs)). Lei £p 6e a central extension of Cu with 2 cocycle P G Z (£U,V). Assume g0 is simple and g /ias a frasis consisting of simulta 2 neous eigenvectors of {vg}geG, then we can construct a cocycle PQ G Z (CU,V) with [Lb] = [P] satisfying P0(x 66 Proof. For each a G ^i G ga., thus Xi <8> % G Caj • Define r : Cu —> V to be the unique linear map such that r(xi ®aj) = Taj(xi). l Let P0 = P + d (T). Then [P0] = [P]. For each ^ (; G J) we have 1 Po(x®dj,y® 1) = P(x®aj,y = P(x®aj,y <8> 1) - r([x<8>aj,y <8> 1]) = P(x®a,j,y = P(x®a,-,y®l)-/^([x,y]) - P(x<8>aj,2/0l)-/^(y)(x) = O, for all x G goj-,2/ G go and a,j G 5. Note that our proof does not depend on the choice of f3aj because ker(d°) — Hom^g^., F)9 and different choices of f3ai become the same when restricted to [g^So]- Thus for any x Lemma 4.25 Let SS — {x, ® aj}ieijej be a k-basis of Cu with {xi}i&1 consisting of simultaneous eigenvectors of {vg}g^G- Take Xj (g> cij,xi (g> ak G Cu. If 0 ^ ajdak G Qn/dR, then a^ak G P and [x,,x;] G go- 67 Proof. Let vg(Xi) — Xgx^ where A* G k. If x* 1 1 l l 1 Mafc = flajdflofc = (A^)- aid(A^)- afc = {\g)~ {\g)- ~a~da~k = (\g\g)- a~dak'. 1 9 9 1 1 So if ~a~da~k + 0, then (A^)" = AgAj, = 1. Thus {ajak) = afak - (A^)- ^^)" ^ = ajCLk. So djdk G R and [XJ,XJ] G [gOj-,0aJ C 0o3% = 80 by Lemma 4.23. • Now we are ready to prove the main result of this section. l Proposition 4.26 Let u = (ug)geG G Z (G, Auts(gs)) be a constant cocycle with ug = vg® id. Let Cu be the descended algebra corresponding to u and let £a be the central extension of £u obtained by the descent construction. Assume go is central simple and Q has a basis consisting of simultaneous eigenvectors of {vg}geG- Assume AJ = £u © Slri/dR, then £5 is the universal central extension of Cu. Proof. First of all, Cu is perfect. Indeed, let X + Z G £2, where X G Cu and r son Z G Ctit/dR. Since Cu is perfect, we have X — ^[X^Y^Cu f° ie Xj,YJ G Cu. By the assumption £2 = £„ © Q,R/dR we have Cu c £2, then Xi,Yt G £2. Thus ^[X^ta = VilXi^Cu + W for some W G ftp/di?. So X + Z = ^[X^Y^ + (Z - W), where Z - W G ftR/dR C [QOR,QOR]CZ C [C$,Cu]cn- Thus £s is perfect. 2 Let £p be a central extension of Cu with cocycle P G Z (CU, V). By Proposition 4.24, we can assume that P(x ® a,y [a(x 68 for all x 0 a, y®b £ Cu. Define ip : £s -• £P by ^(X + Z) = CT(X) + (£) for all X E Cu and Z € Q#/a\R, where •0([x ®a,y® b]Cu) = tp{[x, y)®ab + (x\y)adb) = cr([x, y] 0 aft) + (x|y) [ip(x 0 a),tp(y 0 ft)]£p = [ By (4.14) we have P(x <8> a,y 0 ft) = (x|y)zaif, for all x,y G 90 and a, ft G P. If a, ft G 5\i?, we have two cases. Since ip is well-defined, we only need to consider basis elements in Cu. Let SB = {xt 0 aj}i6/j€j be a fc-basis of £„ with {xi}iej consisting of eigenvectors of the t>s's. Take Xi 0 a,, X; 0 a^ € £„. If 0 ^ ajdafc G Vt^/dR, then aj-afe G i? and [x,, X;] G go by Lemma 4.25. Thus [XJ 0 a,, x; 0 afc]£s C QoR © Q.R/(IR. By Proposition 4.21 ^ is a &-Lie algebra homomorphism in this case. If 0 = cijdak G Clft/dR, then [x, 0 CLJ, XJ 0 a^]/;- = [XJ 0 aja/c, x; 0 l]cu- By Proposition 4.24 we have P(xi 0 ajOfc, x; 0 1) = 0. So ^ is a fc-Lie algebra homomorphism as well in this case. It is easy to check the following diagram is commutative. 0 • QR/dR • Cu > Cu • 0 i> identity 0 > V • Cp • Cu • 0 D Remark 4.27 By a general fact about the nature of multiloop Lie algebras as twisted forms (see [P2] for loop algebras, and [GP2] §5 in general), the cocycle u is a group homomorphism u : G —»• Autfc(fl). In particular, u is constant (i.e., it has trivial Galois action). The multiloop Lie algebra Cu has then a basis consisting of eigenvectors of 69 the Ug's, and therefore ug(Cu) C Cu for all g Corollary 4.28 If Cu is a multiloop Lie torus over an algebraically closed field of characteristic zero, then CQ is the universal central extension of Cu and the centre of CQ is Qji/dR. Proof. If Cu is a multiloop Lie torus, by Remark 4.27 we have £% = Cu © flR/dR. By the definition of multiloop Lie algebras, {vg}gEG is a set of commuting finite order automorphisms of g, thus g has a basis consisting of simultaneous eigenvectors of {vg}geG- By [ABFP2] g0 is simple. Thus for mulitloop Lie torus Cu, our construction Cu gives the universal central extension. • Remark 4.29 Proposition 4.26 provides a good understanding of the universal cen tral extensions of twisted forms corresponding to constant cocycles. The assumption that go is central simple is crucial for our proof. As an important application, Corol lary 4.28 provides a good understanding of the universal central extensions of twisted multiloop Lie tori. Recently E. Neher calculated the universal central extensions of twisted multiloop Lie algebras by using a result on a particular explicit description of the algebra of derivations of multiloop Lie algebras in [A] or [P3]. Discovering more general conditions under which the descent construction gives the universal central extension remains an open problem. 70 Chapter 5 Conclusions This thesis gives a new construction for central extensions of certain class of infinite dimensional Lie algebras. The philosophy behind this construction is a connection be tween infinite dimensional Lie theory and descent theory. This connection, developed in [ABGP], [ABP1], [ABP2], [ABP3], [GP1], [GP2], [P2], [ABFP1] and [ABFP2], starts from viewing multiloop Lie algebras as twisted forms. On the side of infinite dimensional Lie theory, multiloop Lie algebras play a crucial role in the structure theory of extended affme Lie algebras. On the side of descent theory, multiloop Lie algebras provide important examples of twisted forms. Thus multiloop Lie algebras stand as a bridge between infinite dimensional Lie theory and descent theory. Grothendieck's descent formulism reveals a new way to look at the structure of a multiloop Lie algebra through its defining descent data. A multiloop Lie algebra is a twisted form of QR = Q 71 the universal central extension. The descent construction for central extensions of twisted forms of QR is to construct fc-Lie subalgebras of §s by using their defining descent data. As the descent data of a twisted form of QR is an element in Aut(g/0(S ®R S) = Auts-isR ®R S') ~ AutS/(flS/), where S' — S <8>_R S and Qs' — 9 ®k S', a good understanding of the affine group scheme Aut(g#) and lifting automorphisms of 0s/ to its universal central extension gs' is needed for doing the descent construction. The descent construction addresses an important difficult open problem in the structure theory of infinite dimensional Lie algebras. For a non-twisted multiloop Lie algebra QR, Kassel's model tells that its universal central extension is QR = QR © QR/CIR with Clji/dR as the centre. There have been many attempts to understand the universal central extensions in the twisted cases. For a multiloop Lie torus C, the descent construction gives its universal central extension and a good understanding of the centre is provided, namely C = C © QR/CIR with Q,R/dR as the centre. The descent construction for central extensions of twisted forms of QR is obtained solely from their defining descent data and this fact is surprising for the following reason: A multiloop Lie algebra is a twisted form of QR as an i?-Lie algebra, but it is centrally closed over R. By contrast, a multiloop Lie algebra is not a twisted form of QR as a fc-Lie algebra, but it has central extensions over k. This delicate duality suggests some difficulty when one tries to solve problems in infinite dimensional Lie theory through the perspective of viewing multiloop Lie algebras as twisted forms. 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Then d°X and dxA in G(5 ®^ 5) are precisely derived from A in G(5) by the functoriality of G; that is, d°X and d1X are the images of A induced by the algebra 1 1 2 maps d° and d . Similarly d°ip, d ^ and d ip are the results of taking if in G(5 k R) <8>R S ~ g k S for some etale covering R —• S. k R- In the case of n = 1, loop algebras exhaust all twisted forms of g <8>fc R. In the case of n > 1, the Margaux algebra (see Example 5.7 in [GP2]) provides an example of twisted forms of g k R, where k is a field of characteristic zero, g a finite dimensional split simple Lie algebra over k, and R a commutative, associative, unital fc-algebra. A twisted form of QR is an i?-Lie algebra descended from gs := g <8>fc S, where 5 is a commutative, associative, unital fc-algebra and R —> S is a faithfully flat ring extension. The descent data corresponding to a twisted form of g^ is an element u € Aut(gfl)(5 <8>R 5) satisfying d°u d?u = dlu. Let S' :— S ®RS and Qs' := g <8>fc 5", then k R, where A; is a field of characteristic zero, g a finite dimensional split simple Lie algebra over k, and R a commutative, associative, unital fc-algebra. A twisted form of g# is an R-Lie algebra descended from Qs '•= Q®kS, where 5 is a commutative, associative, unital fc- algebra and R —> S is a faithfully flat ring extension. As an R-Lie algebra, a twisted form of QR is centrally closed, but it is not as a fc-Lie algebra, thus it has central extensions over k. In 1984 C. Kassel constructed the universal central extension of 0s by using Katiler differentials, namely 0s = 05 © Qs/dS. In general Qs/dS is infinite dimensional. The purpose of this chapter is to construct central extensions of twisted forms of 0R by using their defining descent data to construct fc-Lie subalgebras of 0s" and study the universality of this new construction. This chapter contains four main results in this thesis. After a brief review of some generalities on central extensions in the first section, the four main results are presented in the next four sections respectively. First, by studying automorphism groups of infinite dimensional Lie algebras a full description of which automorphisms can be lifted to central extensions is given. In particular, for gR it is shown that t^1], a straightforward computation based on (4.8) yields i? S) = Aut5/(flH ®R S') ~ AutS/(gS/). 1), 1). Similarly, we can show that <^2 is a well defined A;-linear map. • a, y ® 1) = 0 /or all x G ga, y G go and a G S. a, G £u, then Xi G gar So 9 9 9 -1 vg(Xi) <8> % = Ag^j