CENTRAL EXTENSIONS of SOME LIE ALGEBRAS the Lie Algebra Of

Home , Lie algebra

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 9, September 1998, Pages 2569–2577 S 0002-9939(98)04348-2

CENTRAL EXTENSIONS OF SOME LIE ALGEBRAS

WANGLAI LI AND ROBERT L. WILSON

(Communicated by Roe Goodman)

Abstract. We consider three Lie algebras: Der C((t)), the Lie algebra of all derivations on the algebra C((t)) of formal Laurent series; the Lie algebra of all diﬀerential operators on C((t)); and the Lie algebra of all diﬀerential operators on C((t)) Cn. We prove that each of these Lie algebras has an essentially unique nontrivial⊗ central extension.

1 The Lie algebra of all derivations on the Laurent polynomial algebra C[t, t− ]can also be characterized as the Lie algebra of vector fields on the circle. The analogous object over a field F of characteristic p>0, Der F[t]/(tp), is called the Witt algebra 1 [C], and this name is sometimes applied to Der C[t, t− ] as well. It is known [Bl] that Der F[t]/(tp) has an essentially unique nontrivial one-dimensional central 1 extension, and also [GF] that Der C[t, t− ] has an essentially unique nontrivial one- dimensional central extension. The proofs of these facts are similar. The nontrivial 1 one-dimensional central extension of Der C[t, t− ] is called the Virasoro algebra. It is one of the fundamental objects in representation theory as well as in theoretical physics. 1 For a positive integer n, the Lie algebra of all differential operators on C[t, t− ] ⊗ Cn has a nontrivial one-dimensional central extension, and the extended Lie algebra is related to the representation theory of affine Lie algebras [KP]. It is proved in [L] that this extension is essentially unique (also see [F]). When n =1,the Lie 1 algebra of all differential operators on the Laurent polynomial ring C[t, t− ]can also be characterized as the Lie algebra of differential operators on the circle; the corresponding extension is referred to, particularly in the physics literature, as 1+ . Some representations of 1+ have been studied recently (see, e.g., [KR], W ∞ W ∞ [FKRW]). In [FKRW], it is shown that some representations of 1+ have natural structures of vertex operator algebras (see, e.g., [Bo] and [FLM]W for∞ definitions). 1 Each of these constructions involves the Laurent polynomial algebra C[t, t− ]. This algebra is, of course, contained in C((t)), the algebra of formal Laurent series. 1 In this paper, we consider the Lie algebras obtained by replacing C[t, t− ]byC((t)) in each of these constructions. We show that each of the resulting Lie algebras has an essentially unique nontrivial one-dimensional central extension. We work over the field of complex numbers, though all results hold over any field of characteristic zero.

Received by the editors September 13, 1996 and, in revised form, February 4, 1997. 1991 Mathematics Subject Classiﬁcation. Primary 17B65, 17B56; Secondary 17B66. Key words and phrases. Lie algebra, central extension, 2-cocycle. The second author was supported in part by NSF Grant DMS-9401851.

c 1998 American Mathematical Society

2569

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1. Some basic definitions

Let L and Lˆ be two Lie algebras over C.TheLiealgebraLˆis said to be a one-dimensional central extension of L if there is a Lie algebra exact sequence 0 Cc Lˆ L 0, where Cc is the one-dimensional trivial Lie algebra and → → → → the image of Cc is contained in the center of Lˆ. It is well-known that Lˆ is a one- dimensional central extension of L if and only if Lˆ is the direct sum of L and Cc as vector spaces and the Lie bracket [ , ] in Lˆ is given by · · 1 [x, y]1 =[x, y]+ϕ(x, y)c,

[x, c]1 =0 for all x, y L, where [ , ] is the Lie bracket in L and ϕ : L L C is a bilinear form on L satisfying∈ the· following· conditions: × → (1) (i) ϕ(x, y)= ϕ(y,x), − (2) (ii) ϕ([x, y],z)+ϕ([y,z],x)+ϕ([z,x],y)=0 for all x, y, z L. The bilinear form ϕ is called a 2-cocycle on L. A central extension ∈ is trivial if Lˆ is the direct sum of a subalgebra M and Cc as Lie algebras, where the subalgebra M is isomorphic to L. A 2-cocycle ϕ corresponding to a trivial central extension is called a 2-coboundary, or a trivial 2-cocycle, and is given by a linear function f from L to C: ϕ(x, y)=f([x, y]) for all x, y L. The 2-coboundary defined by f is denoted by α .Thesetof ∈ f all 2-cocycles on L is a vector space, denoted by Z2(L, C). The set of all 2- coboundaries is a subspace of Z2(L, C), denoted by B2(L, C). The quotient space Z2(L, C)/B2(L, C) is called the 2nd cohomology group of L with coefficients in C, and denoted by H2(L, C). If dimH2(L, C)=1, we say that L has an essentially unique nontrivial one-dimensional central extension. We say that 2-cocycles ϕ, ψ are equivalent if ϕ ψ is a 2-coboundary. The following two− lemmas will be used in the proofs of our main results. Lemma 1. Let L be a Lie algebra and S asubsetofLsuch that S spans L and for each x S, x =[yx,zx] for some yx,zx L. Ifa2-cocycleϕsatisfies ϕ(yx,zx)=0 for all x∈ S, then either ϕ =0or ϕ is∈ nontrivial. ∈ Proof. Suppose that ϕ is trivial, so that ϕ = αf for some linear function f. Then for each x S, ∈ f(x)=f([yx,zx]) = ϕ(yx,zx)=0.

Thus f =0sinceSspans L. This implies that ϕ = αf =0. Lemma 2. Let L be a Lie algebra and ϕ a 2-cocycle on L. Suppose there are linear endomorphisms E and F of L such that ϕ(Ex,y)=ϕ(x, F y) for all x, y L, E is surjective and F is locally nilpotent (i.e., for any y L, there is a positive∈ integer n such that F ny =0). Then the 2-cocycle ϕ is 0. ∈ Proof. For x, y L, let n be a positive integer such that F ny =0.Since E is ∈ n surjective, we have x0 L such that x = E x0. Thus, ∈ n n ϕ(x, y)=ϕ(E x0,y)=ϕ(x0,F y)=0.

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Let A be a (not necessarily associative) algebra. A linear map δ: A A is called a derivation, if δ(ab)=δ(a)b+aδ(b) for all a, b A. → Now consider the algebra of all formal Laurent series∈

i C((t)) = ait ai C,n Z .  ∈ ∈  i ,i n  ∈XZ ≥ 

It is known that the set of all derivations on C((t)) is  d = f(t) f(t) C((t)) , A dt ∈

d d i where is the formal derivation defined by : C((t)) C((t)), ait dt dt → 7−→ ia ti 1. For convenience, we denote d by D and denote d f(t)byf(t). For i − dt dt P 0 f(t)= ati ((t)), define Res f(t)=a .If Res f(t)=0,we can define the P i C 1 formal integral∈ of f(t)as − P a i ti+1, i+1 i= 1 X6 − denoted by f(t). Then is a Lie algebra under the bracket operation: A R [f(t)D, g(t)D] =(f(t)D) (g(t)D) (g(t)D) (f(t)D) ◦ − ◦ =f(t)g0(t)D g(t)f0(t)D − for f(t),g(t) C((t)), where the is the composition of operators. More generally,∈ consider the space◦ of all differential operators on the algebra of formal Laurent series C((t)) :

l =span f(t)D l N,f(t) C((t)) . B ∈ ∈

is a Lie algebra under the bracket operation: B [f(t)Dl,g(t)Dk]=(f(t)Dl) (g(t)Dk) (g(t)Dk) (f(t)Dl) ◦ − ◦ l k l l i k+i k k j l+j = f(t) D − g(t) D g(t) D − f(t) D . i − j i=0 j=0 X X Furthermore, we may consider = gln(C), the space of diﬀerential operators C B⊗ on C((t)) Cn.Notethat End (C((t)) Cn). Deﬁne the bracket operation on by linearity⊗ and the commutatorC⊂ ⊗ C [f(t)Dl A, g(t)Dk B] ⊗ ⊗ =(f(t)Dl A) (g(t)Dk B) (g(t)Dk B) (f(t)Dl A) ⊗ ◦ ⊗ − ⊗ ◦ ⊗ l k l l i k+i k k j l+j = f(t) D − g(t) D AB g(t) D − f(t) D BA. i ⊗ − j ⊗ i=0 j=0 X X Hence is a Lie algebra. C

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2. Main results and proofs In this section, we will give our main results and their proofs. In each case we exhibit (using Lemma 1) a nontrivial 2-cocycle on the Lie algebra under consider- ation. The 2-cocycle is analogous to the standard nontrivial 2-cocycle on the Lie algebra obtained from Laurent polynomial algebra. Then for any given 2-cocycle on the Lie algebra, we reduce the 2-cocycle to a 2-cocycle which is equivalent to the original one and takes value 0 whenever the standard 2-cocycle takes value 0. We use Lemma 2 to show that the reduced 2-cocycle is a multiple of the standard one.

Theorem 1. dimH2( , C)=1. A Proof. Let β be a 2-cocycle on . Deﬁne a linear function fβ : Cby A A→

fβ (g(t)D)=β D, g(t)D for g(t) C((t)),Resg(t)=0 ∈ Z and 1 f t 1D = β t 1D, tD . β − 2 − Then β1 = β αfβ is a 2-cocycle on which is equivalent to β. − i A For f (t)= ait C((t)), ∈ i=0 X6

β1(D, f(t)D) = β(D, f(t)D) f ([D, f(t)D]) − β (3) = β(D, f(t)D) f (f0(t)D)=0, − β and

1 β1 t− D, tD 1 1 = β t− D, tD f [t− D, tD] − β 1 1 (4) = β t− D, tD f 2t− D =0. − β Lemma 3. β (D, )=0and β (tD, )=0 . 1 A 1 A Proof of Lemma 3. From (3) and β (D, D)= β(D, D), we have β (D, )=0. 1 − 1 1 A For f (t) C((t)) and Res f(t)=0, ∈ β1(tD, f(t)D)

= β1 tD, D, f(t)D Z

= β1 [tD, D], f(t)D + β1 D, tD, f(t)D =0. Z Z

β (tD, ) = 0 follows from this and (4). 1 A Lemma 4. β (t2D, )=0. 1 A

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Proof of Lemma 4. For f (t) C((t)) and Res f(t)=0,we have ∈ 2 β1(t D, f(t)D)

2 = β1 t D, D, f(t)D Z 2 2 = β1 [t D, D], f(t)D + β1 D, t D, f(t)D =0. Z Z Also 2 1 β1 t D, t− D

2 1 1 = β t D, tD, t− D 1 −2 1 2 1 1 2 1 = β t D, tD ,t− D β tD, t D, t− D −2 1 − 2 1 1 2 1 1 2 1 = β t D, t− D = β t D, t− D . −2 1 − 2 1 2 1 This implies that β1 t D, t− D =0.

3 Lemma 5. If f(t) C((t)) andRes f(t)=0,then β1(t D, f(t)D)=0. ∈ Proof of Lemma 5. We have 3 β1 (t D, f(t)D)

3 = β1 t D, D, f(t)D Z 3 3 = β1 t D, D , f(t)D + β1 D, t D, f(t)D Z Z = β 3t2D, f(t)D =0. 1 − Z Deﬁne α : Cby A×A→

i+1 j+1 3 α ait D, bjt D = aib i(i i)   − − i j i X X X   for a ti+1D, b tj+1D .Note that the sum on the right-hand side is ﬁnite i j ∈A i j andXα is a 2-cocycleX on .LetSbe the subset of given by A A 1 S = t− D f(t)D Res f(t)=0 . ∪ ∈A Now for f(t) C((t)),Resf(t)=0,we have ∈ 1 t 1D = t 1D, tD , − 2 − f(t)D = D, f(t)D , Z 1 α t− D, tD =0,

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and

α D, f(t)D =0. Z

3 1 Since α is nonzero (in fact, α(t D, t− D) = 6), Lemma 1 shows that α is nontrivial. Also α = α1. Applying Lemma 3, Lemma 4 and Lemma 5 to α, we have

α(D, )=α(tD, )=α(t2D, )=0, A A A and

3 α t D, f(t)D =0 for f(t) C((t)),Resf(t)=0. ∈

3 1 Suppose that β1(t D, t− D)=6rfor some r C. Deﬁne β2 = β1 rα;thenwe have ∈ −

(5) β (D, )=β (tD, )=β (t2D, )=0 2 A 2 A 2 A and

(6) β (t3D, )=0. 2 A

We now show that β2 =0,completing the proof of Theorem 1. Let ad: ,ad(a)b =[a, b], be the adjoint operator; then A→A

β2 (adD(f(t)D),g(t)D)

=β2 ([D, f(t)D] ,g(t)D)

=β2 ([D, g(t)D] ,f(t)D)+β2(D, [f(t)D, g(t)D]) (7) = β (f(t)D, adD(g(t)D)) . − 2 Similarly,

(8) β (ad(tD)(f(t)D),g(t)D)= β (f(t)D, ad(tD)(g(t)D)) , 2 − 2 (9) β ad(t3D)(f(t)D),g(t)D = β f(t)D, ad(t3D)(g(t)D) . 2 − 2 Now we want to use formulas (7), (8) and (9) to construct two linear endomorphisms E and F on so that we can use Lemma 2. Set A E =(adD)2ad(t3D) (adtD)3 3(adtD)2 + 4adtD, − − F = ad(t3D)(adD)2 +(adtD)3 3(adtD)2 4adtD. − − − Then, using (7), (8) and (9), we have

(10) β2(E(f(t)D),g(t)D)=β2(f(t)D, F(g(t)D)).

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i+1 j+1 For f (t)= ait ,g(t)= bjt C((t)), we have ∈ i j X X (adD)2ad(t3D)(f(t)D) = D, D, t3D, f(t)D = D, D, (i 2)a ti+3D − i " " i ## X = D, (i 2)(i +3)ati+2D − i " i # X = (i 2)(i +3)(i+2)ati+1D − i i X = (i3 +3i2 4i 12)a ti+1D. − − i i X Also k k i+1 (adtD) (f(t)D)= i ait D i X for all k N. This implies that E(f(t)D)= 12f(t)D for all f(t) C((t)). Thus ∈ j+1− ∈ E is invertible. Similarly, for g(t)= bjt C((t)), we have ∈ j X ad(t3D)(ad D)2(g(t)D)= ( j3+3j2 +4j)b tj+1D. − − j j X Thus F (g(t)D) = 0 for all g(t) C((t)). From Lemma 2, we have β2 =0or ∈ β1=rα.

Theorem 2. dimH2( , C)=1. B Proof. For any 2-cocycle ψ on , deﬁne B 1 f g(t)Dl = ψ t, g(t)Dl+1 ψ −l +1 for g(t) C((t)). Then ψ1 = ψ αfψ is a 2-cocycle and equivalent to ψ.Wehave ∈ − l+1 ψ1 t, g(t)D =0 for all g(t) C((t)) and l N. ∈ ∈ Lemma 6. If g(t) C((t)) and Res g(t)=0,then ψ1 (t, g(t)) = 0. ∈ Proof of Lemma 6. For f (t) C((t)) and Res f(t)=0,we have ∈ ψ1 (t, f(t))

= ψ1 ([tD, t] ,f(t))

= ψ1 (tD, [t, f(t)]) + ψ1 ([tD, f(t)] ,t)

= ψ (t, tf 0(t)) . − 1 Therefore, ψ1(t, f(t)+tf 0(t)) = 0. Note that every element g(t)inC((t)) with Res g(t) = 0 can be written in the form f(t)+tf 0(t)forsomef(t) C((t)) with Res f(t)=0. ∈

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Deﬁne ϕ : C, B×B→ l+m l k+n k l m + l ϕ amt D, bnt D = amb m( 1) l!k! . − − l + k +1 m n ! m X X X Then ϕ is a 2-cocycle on .LetSbe the subset of given by B B l S = f(t)D l N,f(t) C((t)) . ∈ ∈

For any l N and f(t) C((t)), ∈ ∈ 1 f(t)Dl = t, f(t)Dl+1 −l +1 and ϕ t, f(t)Dl+1 =0. 1 From Lemma 1 and the fact that ϕ(t, t− )=1 ,we have that ϕ is nontrivial. If 1 ψ1(t, t− )=s, we deﬁne ψ2 = ψ1 sϕ. Then using Lemma 6, we have ψ2(t, )=0. Note that − B l k ψ2 adt(f(t)D ),g(t)D l k =ψ2 t, f(t)D ,g(t)D k l l k =ψ2 t, g(t)D ,f(t)D +ψ2 t, f(t)D ,g(t)D (11) = ψ f(t)Dl, adt(g(t)Dk) . − 2 l+m For f (t)= amt C((t)), ∈ m X l l 1 [t, f(t)D ]= lf(t)D − . − Therefore the operator adt is surjective and locally nilpotent. Let E =adtand F = ad t. From equation (11) and Lemma 2, we have ψ2 =0.This gives ψ1 = sϕ. − Remark. This method gives a simpliﬁed proof of Theorem 2.1 of [L].

Consider the Lie algebra = gln(C). Deﬁne a bilinear map φ : C by C B⊗ C×C→

φ a tl+mDl A, b tk+nDk B m ⊗ n ⊗ m n ! X X l m + l = amb m( 1) l!k! tr(AB) − − l + k +1 m X l+m k+n for amt , bnt C((t)),l,k Nand A, B gln(C). Then φ is a 2- ∈ ∈ ∈ m n cocycleX on . UsingX a method similar to the proof of Theorem 2.2 of [L], we have C Corollary. dimH2( , C)=1. C Acknowledgment The authors are grateful to James Lepowsky for his helpful comments. In addi- tion, the ﬁrst author is grateful to Professor Lepowsky for his continuous encour- agement as thesis advisor.

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References

[Bl] R. E. Block, On the Mills-Seligman axioms for Lie algebras of classical type, Trans. Amer. Math. Soc. 121 (1966), 378-392. MR 32:5795 [Bo] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068-3071. MR 87m:17033 [C] H. -J. Chang, Uber¨ Wittsche Lie-Ringe, Abh. Math. Sam. Hansischen 14 (1941), 151- 184. MR 3:101A [F] B. L. Feigin, The Lie algebra gl(λ) and the cohomology of the Lie algebra of differential operators, Usp. Math. Nauk. 43, No. 2 (1988), 157-158. MR 90b:17024 [FKRW] E. Frenkel, V. Kac, A. Radul, W. Wang, 1+ and (glN ) with central charge N, W ∞ W Commun. Math. Phys. 170 (1995), 337-357. MR 96i:17024 [FLM] I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988. MR 90h:17026 [GF] I. M. Gelfand, D. B. Fuchs, Cohomology of the Lie algebra of vector fields on a circle, Funct. Anal. Appl. 2 (1968), 92-93. English translation: Funct. Anal. Appl. 2 (1968), 342-343. MR 39:6348a [KP] V. Kac, D. H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA 78 (1981), 3308-3312. MR 82j:17019 [KR] V. Kac, A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 157 (1993), 429-457. MR 95f:81036 [L] W. Li, 2-cocycles on the algebra of differential operators, J. Algebra 122 (1989), 64-80. MR 90d:17018

Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 E-mail address: [email protected]

E-mail address: [email protected]

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