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A CHARACTERIZATION OF THE DIFFERENTIAL

arXiv:hep-th/9302141v1 27 Feb 1993A CHARACTERIZATION OF THE DIFFERENTIALIN SEMI-INFINITE COHOMOLOGYF¨usun AkmanDepartment of Mathematics, Yale University,New Haven, CT 06520akman@pascal.math.yale.edu(to appear in Journal of Algebra)1

A CHARACTERIZATION OF THE DIFFERENTIALIN SEMI-INFINITE COHOMOLOGYF¨usun AkmanDepartment of Mathematics, Yale University,New Haven, CT 06520ACKNOWLEDGEMENTIt is my pleasure to thank Gregg Zuckerman for reading the manuscript and forthe semi-infinitely many comments.His Fall 91 course on Representations of InfiniteDimensional Lie Algebras at Yale University restored my faith in mathematical physics,besides providing an extensive background.INTRODUCTIONSemi-infinite cohomology is by now firmly established as a means of describing certainaspects of string theory in modern mathematical physics [5]. For mathematicians, the firstrigorous formulation was given by Feigin [1], and further clarification and explicit calcu-lations came with [3].

Several articles ([8],...) have appeared on the subject (also studiedas BRST cohomology), and yet so far, existence arguments for the differential have beengiven solely via formulas, or “by analogy”. Moreover, one has to show that the square ofthe differential is zero each time, the length of the proof depending on the complexity ofthe author’s favorite module and on the square of the width of the formula.

Hence, com-plete proofs practically never get published and semi-infinite cohomology remains exotic2

and inaccessible to most mathematicians.The main goal of this article is to characterize a universal differential, d, which willencode the properties of a given Z-graded Lie algebraG = ⊕n∈ZGnwith dim Gn < ∞for all n; satisfy the minimal properties that any self-respecting differen-tial should possess; and induce the cohomology operator for a very large class of (projective)representations of G. The operators of semi-infinite cohomology theory will be realized asinner derivations of an associative algebra Y ∞G, which is (almost) the completed universalenveloping algebra of the Lie superalgebra˜s(G) = π( ˜G) ⊕ι(G) ⊕ǫ(G′) ⊕Kkwhere ˜G is a certain central extension of G, G′ is the restricted dual of G, and ι(G) ⊕ǫ(G′) ⊕Kk (K fixed field of characteristic zero, k central) is the super Heisenberg Liealgebra based on the underlying vector space of G. The two gradings ∗-deg= superdegreeand deg= secondary degree on ˜s(G), coming from ∗-deg ι(x) = −1 and ∗-deg ǫ(x′) = 1and the grading on G respectively, will induce gradings on Y ∞G and its derivations by thePoincar´e-Birkhoff-Witt Theorem (PBW). One group of such derivations will be {θ(x)}x∈G,corresponding to elements {θ(x)}x∈G of Y ∞G, which constitute a natural representationof G on Y ∞G.

All these inner derivations will now act on Y ∞G modules of the formM ⊗∧∞/2+∗G′,M a smooth ˜G module, ∧∞/2+∗G′ the module of semi-infinite forms on G. Modulo theprecise definitions in the following sections and the choice of a special homogeneous basis{ei}i∈Z for G, the main result can now be stated asTHEOREM. (a) There exists a unique superderivation d of Y ∞G with ∗-deg 1, deg 0,satisfyingdι(x) = θ(x)∀x ∈G.

(b) Moreover,dθ(x) = 0∀x ∈G3

andd2 = 0. (c) There is a unique element d of the same degrees in Y ∞G, namelyd =Xiπ(ei)ǫ(ei′) +Xi

(d) Hence, if M is any smooth ˜G module on which the central element acts by 1, thend induces a well-defined differential on the Y ∞G moduleM ⊗∧∞/2+∗G′.The cohomology H∗(M ⊗∧∞/2G′, d) is by definition the semi-infinite cohomology of G withcoefficients in M.We will show the existence, uniqueness, and nilpotency of a canonical differential donce and for all, starting with the only piece of data we need, namely G, and using onlythe language of derivations.In the end, we will obtain the familiar but generic and -algebraically speaking- well motivated formula that establishes d as an inner derivation.Note that by the Theorem (and by the definition of completion) d has to yield the classicaldifferential in case of an ungraded finite dimensional Lie algebra G. Hopefully some physicsterminology will be demystified in the process.PRELIMINARIESUnlike the classical theory, derivations of associative and Lie algebras have never beenutilized systematically in semi-infinite cohomology. If A,B are (super) Z-graded associative4

or Lie algebras, with a degree-preserving algebra map φ : A →B, a superderivationD : A →B of ∗-deg N with respect to φ is a linear map of ∗-deg N satisfyingD(u · v) = (Du) · (φv) + (−1)N(∗degu)(φu) · (Dv)for u, v ∈A, u homogeneous. In case of secondary Z-gradings also preserved by φ, wewill still talk about a superderivation of ∗-deg M and (secondary) deg N even though thisnew grading may not be compatible with the “super” structure.

Note that any associativealgebra derivation is a Lie algebra derivation in a natural way. The converse is not true.A reliable way of constructing a derivation is extending another derivation, or just alinear map, to an associative algebra generated in some way by the original space.

Forexample, any linear map f : E →A into an associative algebra extends to an algebraderivation f : TE →A from the tensor algebra, if we first define p-linear mapsE × · · · × E →Awith(u1, . .

., up) 7→pXk=1φ(u1) · · ·f(uk) · · · φ(up).Here φ : E →A is a given linear map and φ : TE →A the corresponding algebra map.Using this prototype as an intermediate step, a Lie algebra derivation f : ℓ→A into anassociative algebra can be extended to an associative algebra derivation f : Uℓ→A fromthe universal enveloping algebra (again φ : ℓ→A given Lie algebra map, with extensionφ : Uℓ→A by the universal property of Uℓ) : The derivation f : Tℓ→A factors throughTℓ→Uℓ= Tℓ/idealsince elements of the formu ⊗(x ⊗y −y ⊗x −[x, y]) ⊗v = u ⊗([x, y]T ℓ−[x, y]ℓ) ⊗vwith x, y ∈ℓ, u, v ∈Tℓare annihilated. The result also holds for super Lie algebras, sowe immediately get extensions of linear maps E →A to associative algebra derivationsSE →A and ∧E →A (here SE, ∧E are the symmetric and exterior algebras) as specialcases.

Once we define completions of certain enveloping algebras, our Lie derivations willextend uniquely to the completion and we will not distinguish between the generatingmap and the full one by virtue of this discussion. Derivations form a super Lie algebra5

themselves under the obvious superbracket. The map φ will be an inclusion most of thetime, and we will sometimes omit the prefix super-.Let us now define the completion of a tame Z-graded (super) Lie algebra ℓwith asecondary Z-gradingℓ= ⊕nℓn.“Tame” means each ℓn is finite dimensional (the terminology is Zuckerman’s).

Letℓ≤0 = ⊕n≤0ℓnandℓ+ = ⊕n>0ℓn.By PBW their universal enveloping algebras are related byUℓ∼= Uℓ≤0 ⊗K Uℓ+as vector spaces. The grading in Uℓ+ induces a subsidiary grading which we denote bysdeg, as opposed to deg, which comes from the total (secondary) grading on ℓ.

A typicalhomogeneous element of Uℓis of the formrXk=1ukvkwith uk ∈Uℓ≤0, vk ∈Uℓ+, and deg(ukvk) =deg(ulvl) ∀k, l. If infinite expressions∞Xk=1ukvkwith degvk =sdeg(ukvk) →∞as k →∞are also allowed, one gets a canonical completionU∞ℓof Uℓwith respect to deg [10]. Completions of universal enveloping algebras haveappeared in the work of Kac [6] and Zhelobenko [9], for example, but the construction issomewhat different and the grading used is the so-called primary grading for Kac-Moodyalgebras.

Probably the object closest to this one in literature is the “universal envelopingalgebra of a vertex operator algebra” in the Frenkel-Zhu paper [4], where infinite sums arebuilt out of modes of vertex operators. In general, it can be shown that products exist andare exactly what we think they ought to be [10].

U∞ℓis a topological associative algebra6

with Uℓas a dense subalgebra, and both the multiplication and the induced Lie bracketare continuous. Note thatU∞ℓ= ⊕n(Uℓn)∞by definition, that is, each homogeneous piece is completed separately.U∞ℓprovides a context for the puzzling infinite sums of mathematical physics in termsof elements of ℓ.

The normal ordering : : that appears in most sums merely (another alge-braist’s belittlement) ensures the validity of such expressions as elements of the completedalgebra. What we really want to do with them is to take their supercommutators, and: : is superfluous once inside the bracket.

Since U∞ℓacts on every smooth ℓmodule (aZ-graded ℓmodule where each Uℓ+ orbit is finite dimensional) by its very definition, Liealgebra mapsG →U∞H,which abound in physics (G = constraint algebra, H = current algebra), are used togenerate an action of G on smooth H modules. Such a map always extends toU∞G →U∞Hby abstract nonsense.

More about this later.At this point we reluctantly fix a homogeneous basis {ei}i∈Z for G such that wheneverei ∈Gn, either ei+1 ∈Gn or ei+1 ∈Gn+1, with the ultimate purpose of generating formulas.What precedes that will be basis independent. Following the notation of [1], [3], and [10],letG′ = ⊕nG′nbe the restricted dual of G, withG′n = HomK(G−n, K).Then {ei′} is the dual basis in G ′.

Lets = s(G) = G ⊕G ′ ⊕Kkbe the super Heisenberg Lie algebra with center Kk in which elements of G and G ′ will bedistinguished by the additional symbols ι and ǫ, emphasizing that G and G ′ are now the7

adjoint and coadjoint representations. The ∗-degrees for elements of G, G ′, and Kk are−1, 1, and 0 respectively, and both commutators and anticommutators will be denoted bythe superbracket [,].

Thus[ι(x), ι(y)] = [ǫ(x′), ǫ(y′)] = 0and[ǫ(x′), ι(y)] =< x′, y > kfor all x, y ∈G, x′, x′ ∈G ′. A secondary Z-grading is given bydeg ι(x) = n,x ∈Gn,deg(x′) = −n,x′ ∈Gn′deg k = 0.We will use the same letter to denote dual elements, e.g.

ι(x) and ǫ(x′).In the abstract bigraded algebra s, the two compatible gradings have different uses:The super identities[u, v] + (−1)(∗deg u)(∗deg v)[v, u] = 0and(−1)(∗deg u)(∗deg w)[u, [v, w]]+(−1)(∗deg v)(∗deg u)[v, [w, u]]+(−1)(∗deg w)(∗deg v)[w, [u, v]] = 0hold for ∗-deg, and we complete Us with respect to deg. LetC∞G = U∞s/(k = 1),the completed Clifford algebra on G. Then by PBWC∞G ∼= (∧G ⊗∧G′)∞as vector spaces.

(Another approach would be to define C∞G to be U∞s[1/k] and modifyall formulas by a factor of 1/k. )∧∞/2+∗G′ is the CG = Us(G)/(k = 1) module induced from the one dimensionaltrivial representation of the subalgebra∧(G≥0 ⊕G′−),8

whose elements are called annihilation operators. This module was first realized as thespan of the semi-infinite formse′i1 ∧e′i2 ∧· · · ∧e′ik · · ·where i1 > i2 > · · · and ik+1 = ik −1 from some point on.

The completed algebra alsoacts on ∧∞/2+∗G′.The following results, together with Lemma 3, an extension of the second one, essen-tially form the proof of the Theorem. As an unexpected bonus, they help one guess andverify formulas for derivations, as well as take commutators of normal ordered expressionseasily.LEMMA 1.

The center of C∞G is K.Proof. Any PBW monomial in C∞G is uniquely determined by its commutators withall the ι(x) and ǫ(y′): If there exists an ǫ(e′i) in the monomial, [ι(ei),] replaces it with±1.

Otherwise it kills the monomial. Similarly [ǫ(e′i),] replaces an existing ι(ei) by ±1or else kills the monomial.

A central element commutes with all ι(x) and ǫ(y′), hencecannot contain a nonconstant monomial as a summand. So Center C∞G = K, of ∗-degand deg 0.LEMMA 2.

Let D : s →C∞G (equivalently, C∞G →C∞G) be a superderivation of∗-degree N≥0. Then D is determined by its values on ι(G) ⊕Kk ⊂s(G).Proof.

A linear map D : s →C∞G of ∗-degree N is a derivation iffD[u, v]s = [Du, v]C∞G + (−1)N(∗deg u)[u, Dv]C∞Gfor all u, v ∈s, u homogeneous. If Dk = 0, this is reduced to[Du, v] = (−1)N(∗deg u)+1[u, Dv].Let D1, D2 be derivations of ∗-deg N such that D1 = D2 on G ⊕K.

Then for D = D1 −D2,we have[Dǫ(x′), ι(y)] = (−1)−N+1[ǫ(x′), Dι(y)] = 0∀x′ ∈G′, y ∈G.9

This shows that Dǫ(x′), which is of ∗-deg N+1 ≥0, does not have any ǫ’s. Constants arealso excluded by degree considerations so Dǫ(x′) = 0.

Therefore D1 = D2 on s.Remarks. (1) A similar result holds for ∗-degree N≤0 and G′ ⊕Kk.

(2) Since C∞G is generated by s, the extension of any derivation of s into C∞G carriesthe center to the center. Hence any derivation of nonzero ∗-degree is trivial on Kk.The new techniques can best be demonstrated by constructing an action of G (rather,of a central extension) on C∞G by inner derivations.

The actionι(y) 7→ι(ad(x) · y),ǫ(y′) 7→ǫ(ad′(x) · y′),1 7→0of x ∈G in s extends to a derivation of C∞G by previous arguments. If we can producean element, say ρ(x), such that ρ(x) = [ρ(x),]has the correct degrees and agrees withthe action of x on ι(G) ⊕Kk, then the extension is exactly ρ(x) by Lemma 2.

(We willconsistently underline elements corresponding to inner derivations.) Our candidate isρ(x) =Xi∈Z: ι(ad(x) · ei)ǫ(e′i) :∈C∞G(the formula looks slightly different than the traditional one, say in [3], but this order ismore natural and leads directly to the correct formula for d).

The normal ordering , : :,means a factor with positive degree should appear on the right, and if we have to changethe given order, we had better multiply that term by −1. If we take the commutator ofρ(x) with an element of s, we will be allowed to do it term by term by continuity, andthe central term arising from the possible change of order will vanish.

The sum over Z isshorthand for two different sums over nonnegative integers. Also recall thatdeg ǫ(e′i) = deg e′i = −deg eianddeg ι(ad(x) · ei) = deg[x, ei] = deg x + deg ei,so that ρ(x) is a valid expression.

From now on we will write such sums with a clearconscience and not offer explanations. We immediately check that[ρ(x), ι(y)] = ι(ad(x) · y)x, y ∈G,10

hence ρ(x) and the action of x are the same derivation by Lemma 2. Here is part of thecomputation:[: ι(ad(x) · ei)ǫ(e′i) :, ι(ej)] = [ι(ad(x) · ei)ǫ(e′i), ι(ej)]= ι(ad(x) · ei)[ǫ(e′i), ι(ej)]= ι(ad(x) · ei)δij.Of course, addition of central terms to ρ(x) will not change ρ(x), but we always imposethe condition that deg ρ(x) be the same as that of x.

By Lemma 1, ρ(x) is uniquelyrepresented by ρ(x) as long as deg x ̸= 0.The span {ρ(x)} ⊂DerC∞G closes as a Lie algebra, but {ρ(x)} ⊂C∞G does not ingeneral. We have[[ρ(x), ρ(y)], ι(z)] = [ρ([x, y]), ι(z)] = ι([[x, y], z])for all x, y, z ∈G.

For reasons to become clear later we would like to haveγ(x, y) =def [ρ(x), ρ(y)] −ρ([x, y]) = 0for x, y ∈G. It is easy to see that γ is a cocycle in the classical sense, i.eγ(x, [y, z]) + γ(y, [z, x]) + γ(z, [x, y]) = 0.The calculation makes use of the Jacobi identity as well as the ∗-deg of ρ(x) (zero).

Wecan manage to have γ ≡0 when H2G = 0 (e.g. G = V irasoro, or G = ℓ⊗K[t, t−1] ⊕Kc,affine Kac-Moody algebra) if we redefine ρ byρ(x) =Xi: ι(ad(x) · ei)ǫ(e′i) : +β(x),where β is a special element of G′0 ([3], [7] Chapter 7).

On the other hand, one can chooseto leave ρ(x) as it is, for example when G = Witt = DerK[t, t−1] or G = ℓ⊗K[t, t−1]might be more appropriate than their famous central extensions. We will adopt the secondmethod and make up for the inconvenience later.On a final note, one can always pretend that the formula for ρ(x) is inspired by Lemma2, where ǫ(e′i) deletes ι(ei), to be replaced by ι([x, ei]).11

FIRST STEPWe will start by characterizing the part of d that acts on C∞G, namely d0. It willbe the unique derivation of C∞G of ∗-deg 1, deg 0 mapping ι(x) to ρ(x).

It suffices todefine d0 on s(G), i.e. we have to figure out the image of ǫ(x′).

This differential only hasto satisfy[d0ι(x), ι(y)] = [ι(x), d0ι(y)](1)[d0ǫ(x′), ǫ(y′)] = [ǫ(x′), d0ǫ(y′)](2)[d0ǫ(x′), ι(y)] = [ǫ(x′), d0ι(y)](3)for all x, y ∈G, x′, y′ ∈G′. Condition (1) is satisfied since d0ι(x) = ρ(x).

Next, we assume[d0ǫ(x′), ǫ(y′)] = 0∀x′, y′ ∈G′(4)and hope that this works. After all, d0ǫ(x′) is of ∗-deg 2 and every term has two more ǫ’sthan ι’s.

(4) merely says that there are exactly two ǫ’s and no ι’s. Sod0ǫ(x′) =Xdeg ei+deg ej=deg xλij : ǫ(e′i)ǫ(e′j) :,λij ∈K,which can be completely determined if all [d0ǫ(x′), ι(y)] are known.

But then we simplydefine[d0ǫ(x′), ι(y)] = [ǫ(x′), d0ι(y)]= [ǫ(x′), ρ(y)]= −ǫ(ad′(y) · x′)via condition (3). Incidentally, this gives us the formulad0ǫ(x′) = −1/2Xi: ǫ(ad′(ei) · x′)ǫ(e′i) :.The factor 1/2 is there because there are two ǫ’s for an ι to cross.

We have−1/2[Xi: ǫ(ad′(ei) · x′)ǫ(e′i) :, ι(ej)] = −ǫ(ad′(ej) · x′)as the result of some mild linear algebra, which verifies our guess.12

Now d20 = 1/2[d0, d0] is a superderivation of C∞G of ∗-deg 2, deg 0, and is determinedby its values on G. Thend20 = 0⇔d20ι(x) = 0∀x ∈G⇔d0ρ(x) = 0∀x ∈G,but d0ρ(x) is in turn an element of ∗-deg 1 and again Lemma 2 applies:[d0ρ(x), ι(y)] = d0[ρ(x), ι(y)] −[ρ(x), d0ι(y)]= d0ι([x, y]) −[ρ(x), ρ(y)]= ρ([x, y]) −[ρ(x), ρ(y)]= −γ(x, y).So d20 = 0 is equivalent to the closure of {ρ(x)}x∈G, which was remarked upon earlier. Thisapproach makes it unnecessary to square giant sums.A formula for d0?

Why not. We find an inner derivation of C∞G of ∗-deg 1 and deg0, which is the same as d0 on G. Letd0 = 1/2Xi: ρ(ei)ǫ(e′i) := 1/2Xi̸=j: ι([ei, ej])ǫ(e′j)ǫ(e′i) :=Xi

The cocycle −γ(x, y) determines a central extension of G, say ˜G. We extend the chosenbasis of G to {ei} ∪{c}, but now[ei, ej] ˜G = [ei, ej]G −γ(ei, ej)c.Form the Lie superalgebra ˜s as the direct sum˜s = ˜s(G) = ˜G ⊕s(G)where ˜G is of ∗-deg 0 and the rest is as before.

Elements of ˜G will be denoted by π(x)and will have secondary grading induced from G, with degπ(c) = 0.˜G will be “itself” in˜s unlike G ⊂s(G), that is ,[π(x), π(y)]˜s = π([x, y] ˜G).DefineY ∞G = U∞˜s(G)/(k = 1, π(c) = 1)which is isomorphic to(U ˜G ⊗∧G ⊗∧G′)∞by PBW. We will consider derivations D : ˜s →Y ∞G and imitate the former construction.LEMMA 3.

Let D : ˜s →Y ∞G be a superderivation of ∗-deg N≥1 (in particulark 7→0). Then D is determined by its values on G only.Proof.

Let D1 = D2 on G, and D = D1 −D2. We want to showDǫ(x′) = 0,Dπ(x) = 0∀x ∈G, x′ ∈G′.Again,[Dǫ(x′), ι(y)] = ±[ǫ(x′), Dι(y)] = 0.But Dǫ(x′) has at least two ǫ’s in each term, so Dǫ(x′) = 0.

Similarly, from [π(x), ι(y)] = 0,we get[Dπ(x), ι(y)] = −[π(x), Dι(y)] = 0,and Dπ(x), with ∗-deg at least 1, has no ǫ’s. Then Dπ(x) = 0 and D1 = D2.14

Defineθ(x) = π(x) + ρ(x)∀x ∈G,where ρ(x) has no β(x) term. This time[θ(x), θ(y)] = [π(x), π(y)] + [ρ(x), ρ(y)]= π([x, y] ˜G) + ρ([x, y]) + γ(x, y)= π([x, y]G) −γ(x, y)π(c) + ρ([x, y]) + γ(x, y)= θ([x, y]) + γ(x, y)(1 −π(c))= θ([x, y]),which is the phenomenon known as cancellation of anomalies.

The corresponding innerderivations induce a genuine action of the graded Lie algebra G on modules M ⊗∧∞/2+∗G′,and this will give us a square-zero differential.PROOF OF THE THEOREMAs before, we define d on ǫ(x′) and π(x) only, making sure it is a derivation. Againassume[dǫ(x′), ǫ(y′)] = 0, and define dǫ(x′) via[dǫ(x′), ι(y)] = [ǫ(x′), dι(y)] = −ǫ(ad′(y) · x′),and dπ(x) via[dπ(x), ι(y)] = −[π(x), dι(y)] = −[π(x), θ(y)]= −[π(x), π(y)] = −π([x, y] ˜G),thanks to Lemma 3.

Then dǫ(x′) is the same as d0ǫ(x′) anddπ(x) = −Xiπ([x, ei] ˜G)ǫ(e′i).It remains to check the relations[dπ(x), ǫ(y′)] + [π(x), dǫ(y′)] = 0(7)15

and[dπ(x), π(y)] + [π(x), dπ(y)] = d[π(x), π(y)]= dπ([x, y] ˜G)(8).Eq. (7) is just 0 + 0 = 0.

As for (8),−Xiπ([[x, ei] ˜G, y] ˜G)ǫ(e′i) −Xiπ([x, [y, ei] ˜G] ˜G)ǫ(e′i)= −Xiπ([[x, ei] ˜G, y] ˜G + [[ei, y] ˜G, x] ˜G)ǫ(e′i)= −Xiπ([[x, y] ˜G, ei] ˜G)ǫ(e′i)by the Jacobi identity.Once more,d2 = 0anddθ(x) = 0follows from:[dθ(x), ι(y)] = d[θ(x), ι(y)] −[θ(x), dι(y)]= d[ρ(x), ι(y)] −[θ(x), θ(y)]= dι([x, y]) −[θ(x), θ(y)]= θ([x, y]) −[θ(x), θ(y)]= 0.Finally,[d, ι(er)] = π(er) + [d0, ι(er)]= π(er) + ρ(er)= θ(er),which shows d is an inner derivation on Y ∞G; moreover d is unique subject to the degreeconditions. Since [D1, D2] = [D1, D2] in general, we have d2 = 0.AN EXAMPLE: THE SEMI-INFINITE WEIL COMPLEXThe semi-infinite Weil complex W ∞/2G associated to a Z-graded tame Lie algebraG is the tensor product of the semi-infinite symmetric and exterior modules S∞/2G′ and16

∧∞/2G′ together with the semi-infinite differential induced by d.The construction ofthese modules is as follows: Starting with the Heisenberg and super Heisenberg algebrash(G) = I(G)⊕E(G′)⊕Kk1 and s(G) = ι(G)⊕ǫ(G′)⊕Kk2 with the secondary Z-grading, wechoose two subalgebras S(G≥M ⊕G′

(M = N = 0 isusually referred to as the standard choice of vacuum. This convention will be in effect forthe rest of this example.

)An algebra more specific than Y ∞G, namelyA∞G = U∞(h(G) ⊕s(G))/(k1 = k2 = 1),will be home to our operators. It is well-known that the cocycles corresponding to the tworealizationsπ(x) =Xi: I([x, ei])E(e′i) :andρ(x) =Xi: ι([x, ei])ǫ(e′i) :of G cancel [2], hence θ(x) = π(x) + ρ(x) automatically induces a representation of G inthe Weil complex.

There is a degree derivation on A∞G represented bydeg =Xii : I(ei)E(e′i) : +Xii : ι(ei)ǫ(e′i) :which multiplies monomials by their (secondary) degrees. By Lemma 2 and its symmet-ric version, it suffices to check the formula only on I(G) ⊕ι(G) !

(Normal ordering forthe symmetric case does not require a factor of −1.) This operator commutes with thedifferential and hence the cohomology H∗(W ∞/2G, d) of the complex is the direct sum ofcohomologies of the eigenspaces of deg for nonpositive integer eigenvalues.

The calculationof H∗(W ∞/2G, d) for arbitrary G is in general an extremely difficult problem, but usingthe techniques developed in this paper it is not too hard to prove the following results:17

PROPOSITION. For any Z-graded tame Lie algebra G we have((W ∞/2G)deg=0, d) ∼= (WG0, d)andH∗((W ∞/2G)deg=0, d) ∼= H∗(WG0, d).COROLLARY.

Let G be the Witt algebra ⊕nKLn where [Lm, Ln] = (m −n)Lm+n.ThenH∗(W ∞/2G, d) ∼= W(KL0).Remark. We know that H∗(Wℓ, d) is exactly(Sℓ′)ℓ⊗(∧ℓ′)ℓfor a finite dimensional reductive Lie algebra ℓ, so the Proposition gives us a start forcohomological computations in a lot of interesting cases.

The Corollary follows immediatelyfrom the Cartan identitydι(L0) + ι(L0)d = θ(L0)which reduces the cohomology to the degree zero part since −θ(L0) is the degree operator.If we change the vacuum, the cohomology changes dramatically as shown in [2]. In case ofLie algebras that do not naturally possess an element that acts by degree, the remainingpart of the cohomology may have a very rich structure, even for the standard vacuum.Results concerning the d-cohomology of the Weil complex of the loop algebra ofsl(2, K) and more will be presented in the author’s Ph.D. Dissertation (Yale 1993).REFERENCES1.

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GREEN, J. H. SCHWARZ, AND E. WITTEN, “Superstring Theory”, Vol. I,Cambridge University Press, Cambridge, 1987.6.

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USA 81 (1984), 645-647.7. V. KAC, “Infinite Dimensional Lie Algebras”, Birkh¨auser, Boston, 1984.8.

B. H. LIAN AND G. ZUCKERMAN, BRST cohomology and highest weight vectors,Comm. Math.

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D. P. ZHELOBENKO, S-Algebras and Verma modules over reductive Lie algebras,Soviet Math. Dokl.

28 (1983), 696-700.10. G. ZUCKERMAN, “Representations of Infinite Dimensional Lie Algebras”, Lecturenotes (Yale University), Fall 1991.19

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