We prove that the Lie superalgebra of regular differential
operators on the superspace ℂM|N[t,t−1] has an essentially unique non-trivial
central extension.

1. Introduction

The W infinity algebras naturally arise in various physical systems, such as two-dimensional quantum gravity and the quantum Hall effects (see the review [1, 2] and references there in). The most fundamental one is the W1+∞ which is the central extension of the Lie algebra of regular differential operators on the circle [1–5], and it contains the W∞ algebra as a subalgebra. Various extensions where constructed: super extension (W∞1∣1) [6, 7], u(M) matrix version of W1+∞(W1+∞M) [8], and the most general super matrix generalization W1+∞M∣N presented in [1, 2, 9]. It seems difficult to decide where and when the first definition of a (version of) super-W algebra appeared, but a book by Guieu and Roger [10] has a good historical and bibliographic base, including the pioneering papers of Radul where the superanalogues of the Bott-Virasoro cocycles were introduced (see [11]). The original W1+∞ corresponds to M=1,N=0. The general study of representation theory of W infinity algebras started in the remarkable work [4] by Kac and Radul and continued in several works (some of them are [6, 12–14]). Matrix generalizations are deeply related to the main examples of infinite rank conformal algebras (see [15–17]).

The super matrix generalization W1+∞M∣N is defined as a specific central extension of the Lie superalgebra of regular differential operators on the superspace ℂM∣N[t,t-1]. Only in the special case of W1+∞ (i.e., M=1,N=0) was it proved that the 2-cocycle defining this central extension is unique up to coboundary [18]. The main goal of the present work is to extend this result to the super matrix generalization W1+∞M∣N. Similar studies of central extensions for q-analogs and other versions can be found in [19, 20].

2. Basic Definitions and Main Result

Let L and L̂ be two Lie superalgebras over ℂ. The Lie superalgebra L̂ is said to be a one-dimensional central extension of L if L̂ is the direct sum of L and ℂC as vector spaces and the Lie superbracket in L̂ is given by
[a,b]̂=[a,b]+Ψ(a,b)C,[a,C]̂=0,
for all a,b∈L, where [·,·] is the Lie bracket in L and Ψ:L×L→ℂ is a 2-cocycle on L, that is, a bilinear ℂ-valued form satisfying the following conditions for all homogeneous elements a,b,c∈L: (1)Ψ(a,b)=-(-1)|a||b|Ψ(b,a),(2)Ψ([a,b],c)=Ψ(a,[b,c])-(-1)|a||b|Ψ(b,[a,c]),
where |a| denote the parity of a. A central extension is trivial if L̂ is the direct sum of a subalgebra M and ℂC as Lie algebras, where M is isomorphic to L. A 2-cocycle corresponding to a trivial central extension is called a 2-coboundary, and it is given by an f∈L* as follows: αf(a,b)=f([a,b]),
for a,b∈L. It is easy to check that αf is a 2-cocycle. We say that the 2-cocycles Ψ,ϕ are equivalent if ϕ-Ψ is a 2-coboundary. The second cohomology group of L with coefficients in ℂ is the set of equivalent classes of 2-cocycles, and it will be denoted by H2(L,ℂ). If dim H2(L,ℂ)=1, we say that L has an essentially unique nontrivial one-dimensional central extension.

Now, we will introduce the Lie superalgebra that will be considered in this work. Let us denote by Mat(M∣N) the associative superalgebra of linear transformations on the complex (M∣N)-dimensional superspace ℂM∣N. Namely, we consider the set of all (M+N)×(M+N) matrices of the form A=(A0A+A-A1),
where A0,A+,A-,A1 are M×M,M×N,N×M,N×N matrices, respectively, with complex entries. The ℤ2-gradation is defined by declaring that matrices of the form (2.4) with A+=A-=0 are even, and those with A0=A1=0 are odd. We denote by |A| the degree of A with respect to this ℤ2-gradation. The supertrace is defined by Str(A)=tr(A0)-tr(A1),
and it satisfies Str(AB)=(-1)|A||B|Str(BA).

Let 𝒟as be the associative algebra of regular differential operators on the circle, that is, the operators on ℂ[t,t-1] of the form E=ek(t)∂tk+ek-1(t)∂tk-1+⋯+e0(t),whereei(t)∈C[t,t-1].
The elements Jkl=-tl+k(∂t)l(l∈Z+,k∈Z)
form its basis, where ∂t denotes d/dt. Another basis of 𝒟as is Lkl=-tkDl(l∈Z+,k∈Z),
where D=t∂t. It is easy to see that Jkl=-tk[D]l.
Here and further we use the notation [x]l=x(x-1)…(x-l+1).

Denote by S𝒟asM∣N the associative superalgebra of (M+N)×(M+N) (super)matrices with entries in 𝒟as. The ℤ2-gradation is the one inherited by the corresponding ℤ2-gradation in Mat(M∣N). By taking the usual superbracket we make 𝒮𝒟asM∣N into a Lie superalgebra, which is denoted by 𝒮𝒟M∣N. A set of generators is given by {tsf(D)A:s∈ℤ,f∈ℂ[x],A∈Mat(M∣N)}.

Let W1+∞M,N=𝒮𝒟M|N⊕ℂC be the central extension of 𝒮𝒟M∣N by a one-dimensional vector space with a specified generator C, whose commutation relation for homogeneous elements is given by
[trf(D)A,tsg(D)B]=tr+sf(D+s)g(D)AB-(-1)|A||B|tr+sf(D)g(D+r)BA+Ψ(trf(D)A,tsg(D)B)C,
where the 2-cocycle Ψ is given by
Ψ(trf(D)A,tsg(D)B)={(∑-r≤j≤-1f(j)g(j+r))Str(AB)ifr=-s≥0,0,ifr+s≠0.

Now, we are in condition to state our main result.

Theorem 2.1.

One has the following: dim H2(𝒮𝒟M∣N,ℂ)=1.

3. Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>

We will need the explicit expression of the bracket of basis elements of type (2.9) in 𝒮𝒟M∣N: [tm[D]lEij,tn[D]kErs]=tm+n([D+n]l[D]kδjrEis-(-1)|Eij||Ers|[D]l[D+m]kδisErj).
In particular, we have [t-1DEii,tm[D]lEii]=(l+m)tm-1[D]lEii,[t-l-1[D]lEii,DEii]=(l+1)t-l-1[D]lEii,[Eii,tm[D]lEij]=tm[D]lEij,i≠j.

Let β be a 2-cocycle on 𝒮𝒟M∣N. We consider the linear functional in 𝒮𝒟M∣N defined byfβ(tm-1[D]lEii)=1l+mβ(t-1DEii,tm[D]lEii),l≠-m,fβ(t-l-1[D]lEii)=1l+1β(t-l-1[D]lEii,DEii),fβ(tm[D]lEij)=β(Eii,tm[D]lEij),i≠j.
Then β1=β-αfβ is a 2-cocycle on 𝒮𝒟M∣N that is equivalent to β, and using (3.3), we obtain β1(t-1DEii,tm[D]lEii)=0,l≠-m,β1(t-l-1[D]lEii,DEii)=0,β1(Eii,tm[D]lEij)=0,i≠j.

In order to complete the proof we need to show that Ψ=aβ1 for some a∈ℂ. By observing the supertrace that appears in the expression of Ψ in (2.12), we immediately obtain that for any f,g∈DasΨ(fEij,gEsk)=0ifi≠korj≠s.
In Lemmas 3.1 and 3.2, we will show that β1 also satisfies (3.5).

Lemma 3.1.

For any f,g∈𝒟as, β1(fEii,gEsj)=0 if i≠j or i≠s.

Proof.

Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M120"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M121"><mml:mi>s</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula>.

Using that Eii is even, i≠s, and (2.2), we obtain that
β1(fEii,gEsi)=β1(fEii,[Ess,gEsi])=-β1([Ess,gEsi],fEii)=-β1(Ess,[gEsi,fEii])+β1(gEsi,[Ess,fEii])=-β1(Ess,(g∘f)Esi)=0,(usingi≠sand(3.4)),
where g∘f is the product in 𝒟as.

Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M127"><mml:mi>j</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M128"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula>.

In this case we have
β1(fEii,gEij)=β1(fEii,[gEij,Ejj])=-β1([gEij,Ejj],fEii)=-β1(gEij,[Ejj,fEii])+β1(Ejj,[gEij,fEii])(by(2.2))=β1(Ejj,(f∘g)Eij)=0(usingi≠jand(3.6)).

Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M131"><mml:mi>j</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M132"><mml:mi>s</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:math></inline-formula>.

By taking the usual bracket, we make the associative algebra 𝒟as into a Lie algebra which is denoted by 𝒟. Observe that
D=SD1∣0.
It is easy to show that [𝒟,𝒟]=𝒟; therefore, for any f∈𝒟, we have
f=∑l[fl,hl],fl,hl∈D.
Thus, if j≠i and s≠i, using (2.2),
β1(fEii,gEsj)=β1(∑l[flEii,hlEii],gEsj)=∑lβ1(flEii,[hlEii,gEsj])-∑lβ1(hlEii,[flEii,gEsj])=0.
The proof is finished.

Lemma 3.2.

For any f,g∈𝒟as and i≠j,s≠k, β1(fEij,gEsk)=0 when i≠k or j≠s.

Proof.

If i≠j and k≠i, we have
β1(fEij,gEsk)=β1([Eii,fEij],gEsk)=β1(Eii,[fEij,gEsk])-β1(fEij,[Eii,gEsk])=δj,sβ1(Eii,(f∘g)Eik)-δi,sβ1(fEij,gEik)=-δi,sβ1(fEij,gEik)(using (3.4)).
Hence we have β1(fEij,gEsk)=0.

Finally, using skew-symmetry and the previous case, if i≠j, s≠k, and s≠j, we have that β1(fEij,gEsk)=0.

Now, it remains to consider the expression β1(fEij,gEji). In order to do it, consider again the Lie algebra 𝒟=𝒮𝒟1∣0 (see (3.8)) and denote by ψ𝒟 the 2-cocycle Ψ defined in (2.12) with M=1 and N=0.

In fact, from the expression of Ψ, we have Ψ(fA,gB)=ψD(f,g)Str(AB).

Lemma 3.3.

There exist ai∈ℂ such that for all f,g∈𝒟asβ1(fEii,gEii)=aiψD(f,g).
Moreover, the constants ai satisfy ai=(-1)|Eij|aj for all i≠j.

Proof.

Let γi:𝒟×𝒟→ℂ be the bilinear map defined by (i=1,…,M+N)γi(f,g)=β1(fEii,gEii).
Since Eii is even, we have that γi is a 2-cocycle in 𝒟.

The following statement was proved in [18] (see Proof of Theorem 2.1 in page 74 and (3.2) and (3.3) in this work): if a 2-cocycle β1 in 𝒟 satisfies (l∈ℤ+,m∈ℤ)β1(tm[D]l,t-1D)=0,β1(t-1-l[D]l,D)=0.
Then β1=aψ𝒟 for some a∈ℂ. Now, using (3.4), we have that γi satisfies (3.15); thus, we get γi=aiψ𝒟 for some ai∈ℂ, proving the first part of this lemma.

In order to prove the second part, consider i≠j. Then
β1(tEii,t-1Eii)=β1(t(Eii-(-1)|Eij||Eji|Ejj),t-1Eii)(by Lemma3.1)=β1([Eij,tEji],t-1Eii)=β1(Eij,[tEji,t-1Eii])-(-1)|Eij||Eji|β1(tEji,[Eij,t-1Eii])=β1(Eij,Eji)+(-1)|Eij||Eji|β1(tEji,t-1Eij).
Similarly,
β1(tEjj,t-1Ejj)=β1(tEjj,t-1(Ejj-(-1)|Eij||Eji|Eii))(byLemma3.1)=β1(tEjj,[Eji,t-1Eij])=β1([tEjj,Eji],t-1Eij)+β1(Eji,[tEjj,t-1Eij])=β1(tEji,t-1Eij)-β1(Eji,Eij)=β1(tEji,t-1Eij)+(-1)|Eij||Eji|β1(Eij,Eji).
Therefore, β1(tEii,t-1Eii)=(-1)|Eij||Eji|β1(tEjj,t-1Ejj), which means that, ai=(-1)|Eij|aj for all i≠j, finishing the proof.

Lemma 3.4.

β1(Eij,gEji)=β1(gEij,Eji) for i≠j and g∈𝒟as.

Proof.

Since i≠j,
β1(Eij,gEji)=β1(Eij,[Eji,gEii])=β1([Eij,Eji],gEii)+(-1)|Eij||Eji|β1(Eji,[Eij,gEii])=β1(Eii,gEii)-(-1)|Eij||Eji|β1(Ejj,gEii)-(-1)|Eij||Eji|β1(Eji,gEij)=aiψD(1,g)+β1(gEij,Eji),(byLemmas3.3and3.1)=β1(gEij,Eji)(by definition of ψD)

Lemma 3.5.

β1(fEij,gEji)=β1(fEii,gEii) for i≠j and any f,g∈𝒟as.

Proof.

Observe that
β1(fEii,gEii)=β1([fEij,Eji],gEii)(byLemma3.1)=β1(fEij,[Eji,gEii])-(-1)|Eij||Eji|β1(Eji,[fEij,gEii])=β1(fEij,gEji)+(-1)|Eij||Eji|β1(Eji,(g∘f)Eij)=β1(fEij,gEji)-β1((g∘f)Eij,Eji).
Similarly,
β1(fEii,gEii)=(-1)|Eij||Eji|β1(fEjj,gEjj)(byLemma3.3)=(-1)|Eij||Eji|β1(fEjj,[gEji,Eij])(byLemma3.1)=(-1)|Eij||Eji|β1([fEjj,gEji],Eij)+(-1)|Eij||Eji|β1(gEji,[fEjj,Eij])=(-1)|Eij||Eji|β1((f∘g)Eji,Eij)+β1(fEij,gEji)=-β1((f∘g)Eij,Eji)+β1(fEij,gEji)(byLemma3.4).
Hence, from (3.19) and (3.20), we obtain
β1([f,g]Eij,Eji)=0.
Since [𝒟,𝒟]=𝒟, we have β1(𝒟Eij,Eji)=0. Therefore, (3.19) becomes the statement of this lemma.

Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>.

From the previous lemmas, one can easily see that β1=a1Ψ, by observing that the relation between the ai′s in Lemma 3.3 is essentially the supertrace term in expression (2.12) of Ψ.

Acknowledgments

C. Boyallian and J.L. Liberati were supported in part by grants of Conicet, ANPCyT, Fundación Antorchas, Agencia Cba Ciencia, Secyt-UNC, and Fomec (Argentina).

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