The Central Extension Defining the Super Matrix Generalization of 𝑊_1+∞

Abstract

We prove that the Lie superalgebra of regular differential operators on the superspace ℂ^𝑀|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 𝑊_1+∞ 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 𝑊_1+∞(𝑊^𝑀_1+∞) [8], and the most general super matrix generalization 𝑊^𝑀∣𝑁_1+∞ 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 𝑊_1+∞ 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 𝑊^𝑀∣𝑁_1+∞ is defined as a specific central extension of the Lie superalgebra of regular differential operators on the superspace ℂ^𝑀∣𝑁[𝑡,𝑡⁻¹]. Only in the special case of 𝑊_1+∞ (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 𝑊^𝑀∣𝑁_1+∞. 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 𝐿 be two Lie superalgebras over ℂ. The Lie superalgebra 𝐿 is said to be a one-dimensional central extension of 𝐿 if 𝐿 is the direct sum of 𝐿 and ℂ𝐶 as vector spaces and the Lie superbracket in 𝐿 is given by [𝑎,𝑏]̂=[𝑎,𝑏]+Ψ(𝑎,𝑏)𝐶,[𝑎,𝐶]̂=0, (2.1) 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(),(2)Ψ])([−()𝑏,𝑎([𝑎,𝑏,𝑐=Ψ𝑎,𝑏,𝑐])−1^|𝑎||𝑏|Ψ(𝑏,[𝑎,𝑐]), (2.2) where |𝑎| denote the parity of 𝑎. A central extension is trivial if 𝐿 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: 𝛼_𝑓(𝑎,𝑏)=𝑓([𝑎,𝑏]), (2.3) for 𝑎,𝑏∈𝐿. It is easy to check that 𝛼_𝑓 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 ℂ^𝑀∣𝑁. Namely, we consider the set of all (𝑀+𝑁)×(𝑀+𝑁) matrices of the form 𝐴𝐴=⁰𝐴⁺𝐴⁻𝐴¹, (2.4) 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)=tr𝐴⁰−tr𝐴¹, (2.5) and it satisfies Str(𝐴𝐵)=(−1)^|𝐴||𝐵|Str(𝐵𝐴).

Let 𝒟_as be the associative algebra of regular differential operators on the circle, that is, the operators on ℂ[𝑡,𝑡⁻¹] of the form 𝐸=𝑒_𝑘(𝑡)𝜕^𝑘_𝑡+𝑒_𝑘−1(𝑡)𝜕_𝑡^𝑘−1+⋯+𝑒₀(𝑡),where𝑒_𝑖(𝑡)∈ℂ𝑡,𝑡⁻¹. (2.6) The elements 𝐽^𝑙_𝑘=−𝑡^𝑙+𝑘𝜕_𝑡^𝑙𝑙∈ℤ₊,𝑘∈ℤ (2.7) form its basis, where 𝜕_𝑡 denotes 𝑑/𝑑𝑡. Another basis of 𝒟_𝑎𝑠 is 𝐿^𝑙_𝑘=−𝑡^𝑘𝐷^𝑙𝑙∈ℤ₊,𝑘∈ℤ, (2.8) where 𝐷=𝑡𝜕_𝑡. It is easy to see that 𝐽^𝑙_𝑘=−𝑡^𝑘[𝐷]_𝑙. (2.9) Here and further we use the notation [𝑥]_𝑙=𝑥(𝑥−1)…(𝑥−𝑙+1). (2.10)

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

Let 𝑊^𝑀,𝑁_1+∞=𝒮𝒟^𝑀|𝑁⊕ℂ𝐶 be the central extension of 𝒮𝒟^𝑀∣𝑁 by a one-dimensional vector space with a specified generator 𝐶, whose commutation relation for homogeneous elements is given by 𝑡^𝑟𝑓(𝐷)𝐴,𝑡^𝑠𝑔(𝐷)𝐵=𝑡^𝑟+𝑠𝑓(𝐷+𝑠)𝑔(𝐷)𝐴𝐵−(−1)^|𝐴||𝐵|𝑡^𝑟+𝑠𝑓(𝐷)𝑔()𝑡𝐷+𝑟𝐵𝐴+Ψ^𝑟𝑓(𝐷)𝐴,𝑡^𝑠𝑔(𝐷)𝐵𝐶, (2.11) where the 2-cocycle Ψ is given by Ψ𝑡^𝑟𝑓(𝐷)𝐴,𝑡^𝑠𝑔(𝐷)𝐵=_{−𝑟≤𝑗≤−1}𝑓𝑗𝑔𝑗+𝑟Str(𝐴𝐵)if𝑟=−𝑠≥0,0,if𝑟+𝑠≠0. (2.12)

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

Theorem 2.1. One has the following: dim 𝐻²(𝒮𝒟^𝑀∣𝑁,ℂ)=1.

3. Proof of Theorem 2.1

We will need the explicit expression of the bracket of basis elements of type (2.9) in 𝒮𝒟^𝑀∣𝑁: 𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗,𝑡^𝑛[𝐷]_𝑘𝐸_𝑟𝑠=𝑡^𝑚+𝑛[𝐷+𝑛]_𝑙[𝐷]_𝑘𝛿_𝑗𝑟𝐸_𝑖𝑠−(−1)^{|𝐸𝑖𝑗||𝐸𝑟𝑠|}[𝐷]_𝑙[𝐷+𝑚]_𝑘𝛿_𝑖𝑠𝐸_𝑟𝑗. (3.1) In particular, we have 𝑡⁻¹𝐷𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑖=(𝑙+𝑚)𝑡^𝑚−1[𝐷]_𝑙𝐸_𝑖𝑖,𝑡^−𝑙−1[𝐷]_𝑙𝐸_𝑖𝑖,𝐷𝐸_𝑖𝑖=(𝑙+1)𝑡^−𝑙−1[𝐷]_𝑙𝐸_𝑖𝑖,𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗=𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗,𝑖≠𝑗. (3.2)

Let 𝛽 be a 2-cocycle on 𝒮𝒟^𝑀∣𝑁. We consider the linear functional in 𝒮𝒟^𝑀∣𝑁 defined by𝑓_𝛽𝑡^𝑚−1[𝐷]_𝑙𝐸_𝑖𝑖=1𝛽𝑡𝑙+𝑚⁻¹𝐷𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑖𝑓,𝑙≠−𝑚,_𝛽𝑡^−𝑙−1[𝐷]_𝑙𝐸_𝑖𝑖=1𝛽𝑡𝑙+1^−𝑙−1[𝐷]_𝑙𝐸_𝑖𝑖,𝐷𝐸_𝑖𝑖,𝑓_𝛽𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗=𝛽𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗,𝑖≠𝑗. (3.3) Then 𝛽₁=𝛽−𝛼_𝑓𝛽 is a 2-cocycle on 𝒮𝒟^𝑀∣𝑁 that is equivalent to 𝛽, and using (3.3), we obtain 𝛽₁𝑡⁻¹𝐷𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑖𝛽=0,𝑙≠−𝑚,₁𝑡^−𝑙−1[𝐷]_𝑙𝐸_𝑖𝑖,𝐷𝐸_𝑖𝑖𝛽=0,₁𝐸_𝑖𝑖,𝑡^𝑚[𝐷]_𝑙𝐸_𝑖𝑗=0,𝑖≠𝑗. (3.4)

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∈D_asΨ𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘=0if𝑖≠𝑘or𝑗≠𝑠. (3.5) In Lemmas 3.1 and 3.2, we will show that 𝛽₁ also satisfies (3.5).

Lemma 3.1. For any 𝑓,𝑔∈𝒟_as, 𝛽₁(𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑗)=0 if 𝑖≠𝑗 or 𝑖≠𝑠.

Proof. Case 𝑗=𝑖 and 𝑠≠𝑖.
Using that 𝐸_𝑖𝑖 is even, 𝑖≠𝑠, and (2.2), we obtain that 𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑖=𝛽₁𝑓𝐸_𝑖𝑖,𝐸_𝑠𝑠,𝑔𝐸_𝑠𝑖=−𝛽₁𝐸_𝑠𝑠,𝑔𝐸_𝑠𝑖,𝑓𝐸_𝑖𝑖=−𝛽₁𝐸_𝑠𝑠,𝑔𝐸_𝑠𝑖,𝑓𝐸_𝑖𝑖+𝛽₁𝑔𝐸_𝑠𝑖,𝐸_𝑠𝑠,𝑓𝐸_𝑖𝑖=−𝛽₁𝐸_𝑠𝑠,𝑔∘𝑓𝐸_𝑠𝑖=0,using𝑖≠𝑠and(3.4), (3.6) where 𝑔∘𝑓 is the product in 𝒟_as. Case 𝑗≠𝑖 and 𝑠=𝑖.
In this case we have 𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑗=𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑗=−𝛽₁𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑗,𝑓𝐸_𝑖𝑖=−𝛽₁𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑗,𝑓𝐸_𝑖𝑖+𝛽₁𝐸_𝑗𝑗,𝑔𝐸_𝑖𝑗,𝑓𝐸_𝑖𝑖by(2.2)=𝛽₁𝐸_𝑗𝑗,𝑓∘𝑔𝐸_𝑖𝑗=0using𝑖≠𝑗and(3.6). (3.7)Case 𝑗≠𝑖 and 𝑠≠𝑖.
By taking the usual bracket, we make the associative algebra 𝒟_as into a Lie algebra which is denoted by 𝒟. Observe that 𝒟=𝒮𝒟^1∣0. (3.8) It is easy to show that [𝒟,𝒟]=𝒟; therefore, for any 𝑓∈𝒟, we have 𝑓=_𝑙𝑓_𝑙,ℎ_𝑙,𝑓_𝑙,ℎ_𝑙∈𝒟. (3.9) Thus, if 𝑗≠𝑖 and 𝑠≠𝑖, using (2.2), 𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑗=𝛽_1𝑙𝑓_𝑙𝐸_𝑖𝑖,ℎ_𝑙𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑗=_𝑙𝛽₁𝑓_𝑙𝐸_𝑖𝑖,ℎ_𝑙𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑗−_𝑙𝛽₁ℎ_𝑙𝐸_𝑖𝑖,𝑓_𝑙𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑗=0. (3.10) The proof is finished.

Lemma 3.2. For any 𝑓,𝑔∈𝒟_as and 𝑖≠𝑗,𝑠≠𝑘, 𝛽₁(𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘)=0 when 𝑖≠𝑘 or 𝑗≠𝑠.

Proof. If 𝑖≠𝑗 and 𝑘≠𝑖, we have 𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘=𝛽₁𝐸_𝑖𝑖,𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘=𝛽₁𝐸_𝑖𝑖,𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘−𝛽₁𝑓𝐸_𝑖𝑗,𝐸_𝑖𝑖,𝑔𝐸_𝑠𝑘=𝛿_𝑗,𝑠𝛽₁𝐸_𝑖𝑖,𝑓∘𝑔𝐸_𝑖𝑘−𝛿_𝑖,𝑠𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑖𝑘=−𝛿_𝑖,𝑠𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑖𝑘using(3.4). (3.11) Hence we have 𝛽₁(𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘)=0.
Finally, using skew-symmetry and the previous case, if 𝑖≠𝑗, 𝑠≠𝑘, and 𝑠≠𝑗, we have that 𝛽₁(𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑠𝑘)=0.

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

In fact, from the expression of Ψ, we have Ψ𝑓𝐴,𝑔𝐵=𝜓_𝒟𝑓,𝑔Str(𝐴𝐵). (3.12)

Lemma 3.3. There exist 𝑎_𝑖∈ℂ such that for all 𝑓,𝑔∈𝒟_as𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖=𝑎_𝑖𝜓_𝒟𝑓,𝑔. (3.13) Moreover, the constants 𝑎_𝑖 satisfy 𝑎_𝑖=(−1)^{|𝐸𝑖𝑗|}𝑎_𝑗 for all 𝑖≠𝑗.

Proof. Let 𝛾_𝑖∶𝒟×𝒟→ℂ be the bilinear map defined by (𝑖=1,…,𝑀+𝑁)𝛾_𝑖𝑓,𝑔=𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖. (3.14) Since 𝐸_𝑖𝑖 is even, we have that 𝛾_𝑖 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)𝛽₁𝑡^𝑚[𝐷]_𝑙,𝑡⁻¹𝐷𝛽=0,₁𝑡^−1−𝑙[𝐷]_𝑙,𝐷=0. (3.15) Then 𝛽₁=𝑎𝜓_𝒟 for some 𝑎∈ℂ. Now, using (3.4), we have that 𝛾_𝑖 satisfies (3.15); thus, we get 𝛾_𝑖=𝑎_𝑖𝜓_𝒟 for some 𝑎_𝑖∈ℂ, proving the first part of this lemma.
In order to prove the second part, consider 𝑖≠𝑗. Then 𝛽₁𝑡𝐸_𝑖𝑖,𝑡⁻¹𝐸_𝑖𝑖=𝛽₁𝑡𝐸_𝑖𝑖−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝐸_𝑗𝑗,𝑡⁻¹𝐸_𝑖𝑖byLemma3.1=𝛽₁𝐸_𝑖𝑗,𝑡𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑖=𝛽₁𝐸_𝑖𝑗,𝑡𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑖−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑡𝐸_𝑗𝑖,𝐸_𝑖𝑗,𝑡⁻¹𝐸_𝑖𝑖=𝛽₁𝐸_𝑖𝑗,𝐸_𝑗𝑖+(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑡𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑗. (3.16) Similarly, 𝛽₁𝑡𝐸_𝑗𝑗,𝑡⁻¹𝐸_𝑗𝑗=𝛽₁𝑡𝐸_𝑗𝑗,𝑡⁻¹𝐸_𝑗𝑗−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝐸_𝑖𝑖byLemma3.1=𝛽₁𝑡𝐸_𝑗𝑗,𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑗=𝛽₁𝑡𝐸_𝑗𝑗,𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑗+𝛽₁𝐸_𝑗𝑖,𝑡𝐸_𝑗𝑗,𝑡⁻¹𝐸_𝑖𝑗=𝛽₁𝑡𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑗−𝛽₁𝐸_𝑗𝑖,𝐸_𝑖𝑗=𝛽₁𝑡𝐸_𝑗𝑖,𝑡⁻¹𝐸_𝑖𝑗+(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑖𝑗,𝐸_𝑗𝑖. (3.17) Therefore, 𝛽₁(𝑡𝐸_𝑖𝑖,𝑡⁻¹𝐸_𝑖𝑖)=(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁(𝑡𝐸_𝑗𝑗,𝑡⁻¹𝐸_𝑗𝑗), which means that, 𝑎_𝑖=(−1)^{|𝐸𝑖𝑗|}𝑎_𝑗 for all 𝑖≠𝑗, finishing the proof.

Lemma 3.4. 𝛽₁(𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖)=𝛽₁(𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑖) for 𝑖≠𝑗 and 𝑔∈𝒟_as.

Proof. Since 𝑖≠𝑗, 𝛽₁𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖=𝛽₁𝐸_𝑖𝑗,𝐸_𝑗𝑖,𝑔𝐸_𝑖𝑖=𝛽₁𝐸_𝑖𝑗,𝐸_𝑗𝑖,𝑔𝐸_𝑖𝑖+(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑗𝑖,𝐸_𝑖𝑗,𝑔𝐸_𝑖𝑖=𝛽₁𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑗𝑗,𝑔𝐸_𝑖𝑖−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑗𝑖,𝑔𝐸_𝑖𝑗=𝑎_𝑖𝜓_𝒟1,𝑔+𝛽₁𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑖,byLemmas3.3and3.1=𝛽₁𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑖bydefinitionof𝜓_𝒟 (3.18)

Lemma 3.5. 𝛽₁(𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖)=𝛽₁(𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖) for 𝑖≠𝑗 and any 𝑓,𝑔∈𝒟_as.

Proof. Observe that 𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖=𝛽₁𝑓𝐸_𝑖𝑗,𝐸_𝑗𝑖,𝑔𝐸_𝑖𝑖byLemma3.1=𝛽₁𝑓𝐸_𝑖𝑗,𝐸_𝑗𝑖,𝑔𝐸_𝑖𝑖−(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑗𝑖,𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑖𝑖=𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖+(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝐸_𝑗𝑖,𝑔∘𝑓𝐸_𝑖𝑗=𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖−𝛽₁𝑔∘𝑓𝐸_𝑖𝑗,𝐸_𝑗𝑖. (3.19) Similarly, 𝛽₁𝑓𝐸_𝑖𝑖,𝑔𝐸_𝑖𝑖=(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑓𝐸_𝑗𝑗,𝑔𝐸_𝑗𝑗byLemma=()3.3−1^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑓𝐸_𝑗𝑗,𝑔𝐸_𝑗𝑖,𝐸_𝑖𝑗byLemma=()3.1−1^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑓𝐸_𝑗𝑗,𝑔𝐸_𝑗𝑖,𝐸_𝑖𝑗+(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑔𝐸_𝑗𝑖,𝑓𝐸_𝑗𝑗,𝐸_𝑖𝑗=(−1)^{|𝐸𝑖𝑗||𝐸𝑗𝑖|}𝛽₁𝑓∘𝑔𝐸_𝑗𝑖,𝐸_𝑖𝑗+𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖=−𝛽₁𝑓∘𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑖+𝛽₁𝑓𝐸_𝑖𝑗,𝑔𝐸_𝑗𝑖byLemma3.4. (3.20) Hence, from (3.19) and (3.20), we obtain 𝛽₁𝑓,𝑔𝐸_𝑖𝑗,𝐸_𝑗𝑖=0. (3.21) Since [𝒟,𝒟]=𝒟, we have 𝛽₁(𝒟𝐸_𝑖𝑗,𝐸_𝑗𝑖)=0. Therefore, (3.19) becomes the statement of this lemma.

Proof of Theorem 2.1. From the previous lemmas, one can easily see that 𝛽₁=𝑎₁Ψ, by observing that the relation between the 𝑎^′_𝑖𝑠 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).