# The Central Extension Defining the Super Matrix Generalization ofW1+∞

Authors: Carina Boyallian; Jose I. Liberati
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

TheW 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 generalizationW1+∞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

LetL 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
(2.1)[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: (2.2)(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: (2.3)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 (2.4)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 (2.5)Str(A)=tr⁡(A0)-tr⁡(A1),
and it satisfies Str(AB)=(-1)|A||B|Str(BA).Let𝒟as be the associative algebra of regular differential operators on the circle, that is, the operators on ℂ[t,t-1] of the form (2.6)E=ek(t)∂tk+ek-1(t)∂tk-1+⋯+e0(t),whereei(t)∈C[t,t-1].
The elements (2.7)Jkl=-tl+k(∂t)l(l∈Z+,k∈Z)
form its basis, where ∂t denotes d/dt. Another basis of 𝒟as is (2.8)Lkl=-tkDl(l∈Z+,k∈Z),
where D=t∂t. It is easy to see that (2.9)Jkl=-tk[D]l.
Here and further we use the notation (2.10)[x]l=x(x-1)…(x-l+1).Denote byS𝒟asM∣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)}.LetW1+∞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
(2.11)[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
(2.12)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: dimH2(𝒮𝒟M∣N,ℂ)=1.


## 3. Proof of Theorem2.1

We will need the explicit expression of the bracket of basis elements of type (2.9) in 𝒮𝒟M∣N: (3.1)[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 (3.2)[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.Letβ be a 2-cocycle on 𝒮𝒟M∣N. We consider the linear functional in 𝒮𝒟M∣N defined by(3.3)fβ(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 (3.4)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 thatΨ=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∈Das(3.5)Ψ(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 anyf,g∈𝒟as, β1(fEii,gEsj)=0 if i≠j or i≠s.Proof.
Casej=i and s≠i.
Using thatEii is even, i≠s, and (2.2), we obtain that
(3.6)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.Casej≠i and s=i.
In this case we have(3.7)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)).Casej≠i and s≠i.
By taking the usual bracket, we make the associative algebra𝒟as into a Lie algebra which is denoted by 𝒟. Observe that
(3.8)D=SD1∣0.
It is easy to show that [𝒟,𝒟]=𝒟; therefore, for any f∈𝒟, we have
(3.9)f=∑l[fl,hl],fl,hl∈D.
Thus, if j≠i and s≠i, using (2.2),
(3.10)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 anyf,g∈𝒟as and i≠j,s≠k, β1(fEij,gEsk)=0 when i≠k or j≠s.Proof.
Ifi≠j and k≠i, we have
(3.11)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, ifi≠j, s≠k, and s≠j, we have that β1(fEij,gEsk)=0.Now, it remains to consider the expressionβ1(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 ofΨ, we have (3.12)Ψ(fA,gB)=ψD(f,g)Str(AB).Lemma 3.3.
There existai∈ℂ such that for all f,g∈𝒟as(3.13)β1(fEii,gEii)=aiψD(f,g).
Moreover, the constants ai satisfy ai=(-1)|Eij|aj for all i≠j.Proof.
Letγi:𝒟×𝒟→ℂ be the bilinear map defined by (i=1,…,M+N)(3.14)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)(3.15)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, consideri≠j. Then
(3.16)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,
(3.17)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.
Sincei≠j,
(3.18)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 ofψD)Lemma 3.5.
β1(fEij,gEji)=β1(fEii,gEii) for i≠j and any f,g∈𝒟as.Proof.
Observe that(3.19)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,
(3.20)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
(3.21)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 Theorem2.1.
From the previous lemmas, one can easily see thatβ1=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.



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*Source: 870613-2011-09-29.xml*