Lie superalgebras graded by P(n) and Q(n)

1. Introduction

In ref. 1 Berman and Moody initiated the study of Lie algebras graded by a finite root system. Let 𝒢 be a finite dimensional split simple Lie algebra over a field F of characteristic zero and have a root space decomposition relative to a split Cartan subalgebra H.

Definition 1.1: A Lie algebra L over F is graded by the root system Δ if (i) L contains , (ii) L = ∑_α∈Δυ{0}L_α, where L_α = {x ∈ L|[h, x] = α(h)x for all h ∈ H} for α ∈ Δ υ {0}, and (iii) L₀ = ∑_α∈Δ[L_-α, L_α].

The motivation for the study of this class of algebras comes from the intersection matrix algebras of Slodowy (see ref. 2) and the extended affine Lie algebras (see ref. 3) that are graded by a finite root system.

Examples:

1. Let R be an associative unital F algebra, and let gl_n(R) be the Lie algebra of all n × n matrices over R with the commutator product. The subalgebra e_n(R) of gl_n(R) generated by the elements e_ij(a), 1 ≤ i ≠ j ≤ n, a ∈ R is graded by the root system A_n-1. For n = 2 one has to consider a Steinberg algebra (see ref. 1) st₂(R) where the algebra R may be assumed to be alternative (see refs. 4 and 5).

2. Let R be an associative commutative unital algebra, and let 𝒢 be the split simple Lie algebra with the root system Δ. Then is a Δ-graded algebra.

3. Generalized Tits constructions (see refs. 6 and 7) involving the split octonion algebra and the exceptional 27-dimensional split Jordan algebra (see ref. 4) are G₂- and F₄-graded, respectively.

It is appropriate to study root-graded Lie algebras up to central isogeny. Any perfect Lie algebra L has a universal central extension that is also perfect called a universal covering algebra of L (see ref. 8). We denote it as L̃.

Definition 1.2: Two perfect Lie algebras L₁ and L₂ are said to be centrally isogenous if L̃₁ ≃ L̃₂.

In ref. 1 Berman and Moody proved that an arbitrary A_n-graded algebra (n ≥ 2) is centrally isogenous with st_n+1(R) for some associative (for n = 2 alternative) algebra R. For other simply laced root systems Δ = D_n, n ≥ 4, E₆, E₇, E₈ a Δ-graded algebra is centrally isogenous with , where R is an associative commutative algebra. Root-graded Lie algebras for nonsimply laced root systems were classified in ref. 7 (for more Jordan approach to C_n, B_n see also ref. 9). Remark that the classification of root-graded algebras in refs. 1 and 7 led to a complete description of intersection matrix algebras. In refs. 10-12 Benkart and Elduque extended the theory discussed above to Lie superalgebras and completely determined root-graded Lie superalgebras for all finite dimensional split simple classical superalgebras except P(n), , and A(n, n) (for notation and results on Lie superalgebras see ref. 13).

Let 𝒢 be a finite dimensional split simple classical Lie superalgebra with even part . Let H be a Cartan subalgebra of , and let be the decomposition of 𝒢 into the sum of eigenspaces with respect to the action of H, .

Definition 1.3 (see ref. 10): A Lie superalgebra L over F is graded by the root system Δ if (i) L contains a subsuperalgebra ; (ii) see Definition 1.1; and (iii) see Definition 1.1.

In this article we discuss Lie superalgebras graded by the root systems of the superalgebras P(n), , n ≥ 2.

Recall that the Lie superalgebra is the superalgebra of 2(n + 1) × 2(n + 1) matrices of the type (), where a, b are (n + 1) × (n + 1) matrices, tr(b) = 0, and . The superalgebra is the universal central extension of .

The Lie superalgebra P(n - 1) is the superalgebra of 2n × 2n matrices of the type (), where a, k, h are n × n matrices over F, Tr(a) = 0, k^T = -k, h^T = h. The Cartan subalgebra H of consists of diagonal matrices H = {h = diag(a₁,..., a_n, -a₁,..., -a_n) | a_i ∈ F, a_i = 0}. The even and odd roots are Thus P(n) is graded by the free abelian group .

Let's fix the notation for the following weight elements: For n ≠ 3 the universal covering superalgebra of P(n) is P(n). However, P(3) has a unique nontrivial universal central extension with ¢.

Keeping in mind the intersection matrix superalgebras and our focus on subspaces of nonzero weight, we will modify part i of Definition 1.3, allowing the superalgebra to contain a central extension of 𝒢 instead of 𝒢 Let be a central extension of 𝒢; that is, , and there exists an epimorphism such that Kerπ Center(). It follows from the classification (see ref. 13) that if a classical simple Lie superalgebra 𝒢 has a nontrivial central extension , then is semisimple; hence is a direct summand of . We will consider the root decomposition of L with respect to the Cartan subalgebra of the ideal of .

Definition 1.4: A Lie superalgebra L over F is graded by the root system Δ if

1. L contains a central extension of 𝒢,

2. For a Cartan subalgebra H of we have, , where for all h ∈ H} for all , and

3. .

Theorem 1.1.A Lie superalgebra that is graded by P(n), n ≠ 3 is isomorphic to a tensor product , where R is an associative commutative superalgebra.

In ref. 14 Cheng and Kac introduced a new superconformal algebra that they denoted as CK(6). Simultaneously and independently this superalgebra was constructed by Grozman et al. (see ref. 15). In ref. 16 for an arbitrary associative commutative superalgebra R with an even derivation d : R → R we constructed a superalgebra CK (R, d) so that CK(6) ≃ CK(F[t^-1, t], d/dt).

Consider the associative Weyl algebra W = ∑_i≥0 Rdⁱ, where the variable d commutes with a coefficient a ∈ R via da = d(a) + ad. We will realize the CK(R, d) as a superalgebra of 8 × 8 matrices over W. Implicitly this realization was mentioned in ref. 17.

The Lie algebra of skew-symmetric 4 × 4 matrices K₄(F) is a direct sum of two ideals . For an arbitrary element k ∈ K₄(F) we consider its decomposition k = k′ + k″ and let .

The universal central extension of P(3) can be realized as a superalgebra of 8 × 8 matrices over F[d] of the type where a, k, h are 4 × 4 matrices over F, Tra = 0, k^T = -k, h^T = h, α ∈ F, and I is the identity matrix. The superalgebra CK(R, d) is a subsuperalgebra of 8 × 8 matrices over W generated by and by all matrices , where a ∈ R, 1 ≤ i ≠ j ≤ 4.

Theorem 1.2.A P(3)-graded Lie superalgebra is centrally isogenous with CK(R, d), where R is an associative commutative superalgebra, and d : R → R is an even derivation. The superalgebra CK(R, d) is centrally closed; that is, .

Remark: If a Lie superalgebra L is P(3)-graded in the sense of Definition 1.3, then , where R is an associative commutative superalgebra.

Classification of -graded Lie superalgebras immediately follows from the classification of A_n-1-graded Lie algebras by Berman and Moody (see ref. 1) (see section 3).

Theorem 1.3.A -graded Lie superalgebra is centrally isogenous to st_n+1(R), where is an associative or an alternative (if n = 2) unital superalgebra such that there exists.

2. Jordan Systems

Consider a system of vector spaces J = (J_α, α ∈ Δ) with transformations whenever α, β, α + β ∈ Δ, and for all α, β ∈ Δ. The trilinear operations ψ_α,-α,β give rise to the mappings , where the projection of to is the mapping .

We call this system a Jordan system (see refs. 7 and 18) if the direct sum where , becomes a Lie algebra with respect to the operation:

A homomorphism between two Jordan systems, J = (J_α, α ∈ Δ) and J′ = (J′_α, α ∈ Δ), is a family of linear mappings, f = (f_α, α ∈ Δ), f_α : J_α → J′_α, which preserves the bilinear and the trilinear operations.

Every Δ-graded Lie algebra gives rise to a Jordan system (L_α, α ∈ Δ) with the bilinear operations if α, β, α + β ∈ Δ and the trilinear operations .

In ref. 7 it was shown that two Δ-graded Lie algebras and are centrally isogenous if and only if the Jordan systems (L_α, α ∈ Δ) and (L′_α, α ∈ Δ) are isomorphic.

Everything that we said about Jordan systems and their connections with Δ-graded Lie algebras obviously extends to superalgebras.

Lemma 2.1.Let and be two P(n)-graded Lie superalgebras, n ≥ 2. Letbe a family of linear mappings such thatfor arbitrary roots α, β ∈ Δ, α + β ∈ Δ, and arbitrary elements x_α ∈ L_α, y_β ∈ L_β. Then f is a homomorphism of Jordan supersystems.

3. A_n-1-Graded Superalgebras

Definition 3.1 (see ref. 1): Let R be a unital (super)algebra, n ≥ 3. The Steinberg Lie (super)algebra st_n(R) is presented by generators X_ij(a), 1 ≤ i, j ≤ n, i ≠ j and relators X_ij(αa + βb) = αX_ij(a) = βX_ij(b); [X_ij(a), X_jk(b)] = X_ik(ab) if i, j, k are distinct; [X_ij(a), X_kt(b)] = 0 if i ≠ t and j ≠ k.

Let L be an A_n-1-graded Lie algebra, n ≥ 3. Thus . Let e_ij, 1 ≤ i ≠ j ≤ n denote the matrix units from sl_n(F). Berman and Moody (see ref. 1) proved that there exists a unital algebra R such that the Jordan system (L_α, α ∈ Δ) is isomorphic to the Jordan system (X_ij(R), 1 ≤ i ≠ j ≤ n) of the Steinberg Lie algebra st_n(R). If n ≥ 4, then the algebra R is associative. For n = 3, R has to be alternative. It is proved in ref. 19 that the maps R → X_ij(R) are injective.

Moreover, there exists an isomorphism Ψ = (ψ_ij, 1 ≤ i ≠ j ≤ n) between these Jordan systems such that ψ_ij(e_ij) = X_ij(1), 1 ≤ i ≠ j ≤ n.

If the algebra R is associative, in particular if n ≥ 4, then st_n(R) is the universal central extension of the Lie algebra sl_n(R) generated by the matrices e_ij(a), 1 ≤ i ≠ j ≤ n, a ∈ R.

Following the Berman-Moody proof (verbatim) we get the following.

Lemma 3.1.Letbe a Lie superalgebra, the even part of which contains sl_n(F) = H + Σ_α∈An-1 G_α. Suppose further thatL_α, where L_α = {x ∈ L | [h, x] = α(h)x}, and. Then there exists a superalgebraand an isomorphism of Jordan supersystems Ψ : (X_ij(R), 1 ≤ i ≠ j ≤ n) → (L_α, α ∈ A_n-1) such that ψ_ij(X_ij(1)) = e_ij.

This lemma implies Theorem 1.3. Indeed, the root system of the superalgebra is A_n. Hence the -graded Lie superalgebra L is A_n-graded. Therefore L is centrally isogenous to st_n+1(R), where is an associative or an alternative (if n = 2) superalgebra. An A_n-graded superalgebra -graded if and only if it contains a subsuperalgebra 𝒢 isomorphic to or to such that .

Let us show that if there is with ν² = 1, then L is -graded.

Consider an isomorphism of Jordan systems π: (X_ij(R), 1 ≤ i ≠ j ≤ n + 1) → (L_wi-wj, 1 ≤ i ≠ j ≤ n + 1). The Jordan system (X_ij(F1 + Fν), 1 ≤ i ≠ j ≤ n + 1) is isomorphic to the Jordan system . Hence the subsuperalgebra of L generated by the subspaces π(X_ij(F1 + Fν)), 1 ≤ i ≠ j ≤ n + 1, is isomorphic to .

Now suppose that . We will show that there exists with ν² = 1. Let be an isomorphism of Jordan systems.

By Lemma 3.1 we can assume that

Suppose that Then

Hence which implies ν² = 1. Theorem 1.3 is proved.

4. P(n - 1)-Graded Superalgebras: Associativity and Commutativity of R

Let be the root system be a Δ-graded Lie superalgebra. Then is the Lie superalgebra graded by A_n-1. By Lemma 3.1 there exists a superalgebra and an isomorphism of Jordan systems such that ψ_ij(X_ij(1)) = e_wi-wj. For arbitrary 1 ≤ i ≠ j ≤ n, arbitrary a ∈ R denote ψ_ij(e_ij(a)) as e_wi-wj(a). Clearly, .

The superalgebra L contains a central extension of the superalgebra P(n - 1). If n ≠ 4, then P(n - 1) does not have nontrivial central extensions, so . If n = 4, then L contains either P(3) or its universal central extension . Recall that ¢. The elements h_wi-wj, e_wi-wj, q_-wi-wj, q_wi+wj are multiplied as in P(3) except for . The element ¢ lies in the center.

For 1 ≤ i, j ≤ n, a ∈ R choose , and let .

Lemma 4.1.

1. For k, t ∈ {1, 2,...,n} - {i,j} we have Thus the definition of q_-wi-wj(a) does not depend on the particular choice of k.

2. q_-wi-wj(a) = q_-wj-wi(a).

Lemma 4.2. Let 1 ≤ i, j, k ≤ n be distinct. Then for arbitrary elements a, b ∈ R we have

Lemma 4.3.Let 1 ≤ i,j ≤ n be distinct. Then for arbitrary elements a, b ∈ R we have.

Lemma 4.4.For 1 ≤ i ≠ j ≤ n and arbitrary elements a, b ∈ R we have

Lemma 4.5..

Proof: Denote . By Lemmas 4.2 and 4.3 we have

Hence

An element has weight -2w_i - w_k - w_t, which is a root only when n = 3 or 4 and the integers i, k, t are distinct. In both cases and it remains to notice that -2w_i + w_k - w_t, -2w_i - 2w_k are not roots.

Hence . Lemma 4.5 is proved.

Let 1 ≤ i ≠ j ≤ n, a ∈ R. Assuming that n ≠ 4 denotes q_wi+wj(a) = , where 1 ≤ k ≤ n, k ≠ i,j.

Lemma 4.6.

1. The definition above does not depend on the choice of k. In other words, if 1 ≤ t ≤ n, t ≠ i, j, then.

2. q_wj+wi(a) = ^-q_wi+wj(a).

Remark: For n = 4 the expression may depend on the choice of k.

For n = 4 define q_w1+w2(a) = .

Lemma 4.7.

1. Let n ≠ 4. For arbitrary integers 1 ≤ i, j, k ≤ n, arbitrary a, b ∈ R we have

2. For n = 4, for arbitrary a, b ∈ R, we have

Lemma 4.8. The superalgebra R is associative and commutative.

Proof: Let a, b, c be arbitrary homogeneous elements from R. From and the Jacobi identity, it follows that

Let us determine the left- and right-hand sides separately.

The left-hand side is equal to

The right-hand side is equal to

We proved that

Let c = 1. Then ab + (-1)^|a||b|ba = (-1)^|a||b|(2ba). Hence ab = (-1)^|a||b|ba and the superalgebra R is commutative. Hence (-1)^|a||b|2(ba)c = 2(ab)c = 2(-1)^|a||b|b(ac), and Lemma 4.8 is proved.

We will consider separately the case n = 4 in the next section. From now on in this section we assume that n ≠ 4.

Lemma 4.9.For arbitrary distinct integers 1 ≤ i, j, k ≤ n and arbitrary elements a, b ∈ R we have

Lemma 4.10.For arbitrary 1 ≤ i ≠ j ≤ n, arbitrary elements a, b ∈ R we have

Proof: Choose k ≠ i, j. We have and Lemma 4.10 is proved.

Lemma 4.11.For arbitrary 1 ≤ i, j, k, t ≤ n, i ≠ j, k ≠ t we have

Proof of Theorem 1.1: We claim that e_wi-wj(R), 1 ≤ i ≠ j ≤ n, q_-wi-wj(R), 1 ≤ i, j ≤ n, and q_wi+wj(R), 1 ≤ i ≠ j ≤ n, exhaust all nonzero eigenspaces L_α, α ≠ 0. For α ∈ this is obvious. If α ∈ , then there exists 1 ≤ i ≠ j ≤ n such that (α | w_i - w_j) ≠ 0 and α ≠ ±(w_i + w_j). Hence which proves the assertion.

Lemmas 2.1, 4.2-4.5, 4.7, and 4.9-4.11 imply that the Jordan supersystems (L_α, α ∈ Δ) and (P(n)_αR, α ∈ Δ) are isomorphic. Hence L is centrally isogenous with P(n) R. From Proposition 6.1 (which will be proved later) it follows that L ≃ P(n) R, and Theorem 1.1 is proved.

5. Cheng-Kac Superalgebras

Let L be a P(3)-graded Lie superalgebra, , where 𝒢 = P(3) or , H is a Cartan subalgebra of , L = L_α is the decomposition into the sum of eigenspaces with respect to H, L₀ = ∑_α∈Δ . The elements e = e_w1-w3 + e_w2-w4, f = e_w3-w1 + e_w4-w2, h = [e, f] = h_w1-w3 + h_w2-w4 form an sl₂-triple with ad(h) : L → L having eigenvalues -2, 0, 2. Let L = L_(-2) + L₍₀₎ + L₍₂₎ be the decomposition of L into the sum of eigenspaces with respect to ad(h). It is known (see ref. 6) that L₍₂₎ with the new operation becomes a Jordan superalgebra (see refs. 13 and 16).

The Lie superalgebra L is centrally isogenous with the Tits-Kantor-Koecher construction of this Jordan superalgebra (see refs. 6, 20, and 21).

In ref. 13 it is shown that the subsuperalgebra (, ·) of J = (L₍₂₎, ·) is isomorphic to the Jordan superalgebra The Jordan superalgebra JP(2) has a basis 1, , 1 ≤ i ≤ 3 (see ref. 16) such that = = 1, = -1,x·ν_i = 0, 1 ≤ i ≤ 3, [x_i, x] = ν_i, 1 ≤ i ≤ 3, x_i·ν_j = x_i×j, where x_i×i = 0, x_1×2 = -x_2×1 = x₃, x_1×3 = -x_3×1 = x₂, -x_2×3 = x_3×2 = x₁.

The subspace ε = e_w1-w3(R) + e_w1-w4(R) + e_w2-w3(R) + e_w2-w4(R) is a subsuperalgebra of J, which is isomorphic to M₂(R)⁽⁺⁾ (recall that R is an associative commutative superalgebra).

Lemma 5.1.The superalgebra J is generated by JP(2) and ε.

Consider the following two commuting linear transformations on J: R(ν₃) the right multiplication by ν₃ and the inner derivation The superalgebra JP(2) decomposes into a sum of eigenspaces with respect to R(ν₃), D(ν₁, ν₂), the weights being (0, 0), .

We say that an element a belongs to the weight (α, β) if aR(ν₃) = αa, aD(ν₁, ν₂) = βa. The corresponding weight space is denoted as J_(α,β).

The superalgebra J also decomposes into a sum of eigenspaces with respect to R(ν₃), D(ν₁, ν₂) with the same weights. We have just changed a Cartan subalgebra in .

The elements are orthogonal idempotents, . If , then , hence y·e₁ = 0, y·e₂ = y. This implies that . Similarly, .

Denote .

Lemma 5.2.J′·J′ = (0).

Proof: We have The products belong to one of the eigenvalues with respect to D(ν₁, ν₂) and therefore are equal to zero, and Lumma 5.2 is proved.

We identify ε with M₂(R) and we identify R with its image in M₂(R).

Lemma 5.3.RD(J′, J′) = (0).

Proof: We have . Therefore and Lemma 5.3 is proved.

Lemma 5.4.If a ∈ J_(α1,β1), b ∈ J_(α2,β2), and c ∈ J_(α3,β3), then {a, b, c} ∈ J_{(α1-α2 + α3,β1 + β2 + β3)}.

Lemma 5.5.((Rν_i)x)x_i = (0), 1 ≤ i ≤ 3.

Lemma 5.6.Let ν be an even element of J such that ν² = 1. Then U(ν): a→{ν, a, ν} is an automorphism of order 2. The superalgebra J decomposes into the eigenspaces J = J(-1) + J(1), J(i) = {a ∈ J | aU(ν) = ia}. If a ∈ J(1), then (aν)ν = a, D(a, ν) = 0.

Proof: The Macdonald identity (see ref. 4) implies that U(ν) is an automorphism. Clearly, U(ν)² = U(ν²) = Id. If a ∈ J(1), then aU(ν) = a(2R(ν²) - R(ν²)) = 2(aν)ν - a = a, which yields (aν)ν = a. Furthermore, D(a, ν)=D((aν)ν, ν) = ½D(aν, ν²) = ½D(aν, 1) = 0, and Lemma 5.6 is proved.

Lemma 5.7.RD(J′, x) = (0).

Proof: The subspace R of M₂(R) is invariant under all even and odd derivations of M₂(R). Comparing weights it is easy to see that and therefore .. On the other hand, for a nonzero element a ∈ RD(J′, x) we always have aD(ν₁, ν₂) ≠ 0. Hence RD(J′, x) = (0), and Lemma 5.7 is proved.

Lemma 5.8.For an arbitrary element a ∈ R we have (ax_i)x = aν_i, 1 ≤ i ≤ 3.

Lemma 5.9.(Rν_i)(Rx_i) = (0), 1 ≤ i ≤ 3.

Proof: As stated above, it is sufficient to prove the assertion for i = 1. We have Hence we need only to check that (Rν₁)x₁ = (0). By Lemmas 5.7 and 5.8 for an arbitrary element a ∈ R we have aν₁ = (ax₁)x = -(ax)x₁. Hence (aν₁)x₁ = -(ax)R(). The element x₁ lies in J′, hence by Lemma 5.3 aR(x₁)² = 0 and xR(x₁)² = 0 in JP(2). Lemma 5.9 is proved.

Lemma 5.10.For arbitrary 1 ≤ i ≤ 3; a, b ∈ R we have a(bx_i) = (ab)x_i.

Proof: By Lemma 5.9 which implies the result and proves Lemma 5.10.

Denote ′: R → R, a′ = aR(x)².

Lemma 5.11.For arbitrary 1 ≤ i ≤ 3; a, b ∈ R we have (aν_i)(bx) = (a′b)x_i.

Proof: We have D(aν_i, b) = 0. Hence (aν_i)(bx) = (-1)^|a||b|b((aν_i)x). By Lemma 5.8 aν_i = (ax_i)x. Now ((ax_i)x)x = a′x_i and finally (aν_i)(bx) = (-1)^|a||b|b(a′x_i) = (a′b)x_i, by Lemma 5.10, and Lemma 5.11 is proved.

Lemma 5.12.For arbitrary 1 ≤ i ≤ 3; a, b ∈ R we have a(bx) = (ab)x.

Proof: Arguing as in Lemma 5.10 and using Lemma 5.9, we have Hence, and Lemma 5.12 is proved.

Lemma 5.13. For arbitrary 1 ≤ i ≠ j ≤ 3; a, b ∈ R we have

Proof: We have D(aν_i, b) = 0 and D(a, x_j) = D(±(aν_j)ν_j, x_j) = 0. This implies the assertion and proves Lemma 5.13.

Lemma 5.14. For arbitrary elements a, b ∈ R we have

Proof: We claim that (ax)(bx) ∈ R. Indeed, Comparing weights, we see that , and therefore . By Lemma 5.12 aR(bx)R(x) = ((ab)x)x = (ab)′ ∈ R. For an arbitrary element c ∈ R if cx₁ = 0, then by Lemma 5.8 (cx₁)x = cv₁ = 0, and by Lemma 5.6 c = (cv₁)v₁ = 0. Hence we need to verify that ((ax)(bx))x₁ = (a′b - ab′)x₁. We have Now, (ax)(bx₁) = a(x(bx₁)), because D(a, bx₁) = D((av₁)ν₁, bx₁) = 0. By Lemma 5.8 a(x(bx₁)) = -(-1)^|b|abv₁. Hence, By Lemma 5.11 Clearly, (ax)ν₁ = 0. Now, and Finally,

Lemma 5.14 is proved.

Proof of Theorem 1.2: From Lemmas 5.1 and 5.6 it follows that with multiplication defined by Lemmas 5.8-5.12 and 5.14. Comparing with the multiplication tables from ref. 16 we see that J is a Cheng-Kac Jordan superalgebra. Hence L is centrally isogenous with the Tits-Kantor-Koecher construction of the Cheng-Kac Jordan superalgebra, that is, with a Cheng-Kac Lie superalgebra. Central closeness will be considered in section 6. Theorem 1.2 is proved.

6. Universal Central Extensions

Let L be a Δ-graded Lie superalgebra, L ⊃ a central extension of a finite dimensional simple classical Lie superalgebra corresponding to the root system Δ, H a Cartan subalgebra in . Let L̃ be a central extension of L, ψ : L̃ → L an epimorphism, .. We claim that L̃ contains a finite dimensional subsuperalgebra that is a preimage of . Indeed, choose a basis e₁,...,e_s in and its preimage a₁,..., a_s ∈ L̃. Then . The subsuperalgebra is a finite dimensional preimage of . We have , and the kernel of each of these homomorphisms is central. Choose a Cartan subalgebra H′ in the Levi factor of . It is easy to see that L̃ is a sum of eigenspaces with respect to the action of H′. If H = ψ(H′), then ψ is a graded epimorphism, and the mapping of the Jordan systems (L̃_α, α ∈ Δ) → (L_α, α ∈ Δ) is an isomorphism. The finite dimensional perfect superalgebra ∑_α∈Δ is a central extension of 𝒢. We proved that L̃ is a Δ-graded Lie superalgebra. Throughout this section we will assume that Δ = P(n - 1), n ≥ 3, and keep the notation of section 2. In particular, the Jordan system is isomorphic to the Jordan system (e_ij(R), 1 ≤ i ≠ j ≤ n). Denote .

Lemma 6.1.Let 1 ≤ i < j ≤ n. Then.

Proof: We have In particular, if i = 1, then . Now for arbitrary 1 ≤ i ≠ j ≤ n we have e_wi-wj(R) = ,, and therefore

Lemma 6.1 is proved.

Lemma 6.2.If n ≠ 4, then L̃₀ = S.

Proof: By the previous lemma and the results of section 4 we need only to check that . We have by Lemma 6.1, and Lemma 6.2 is proved.

Proposition 6.1. For an associative commutative superalgebra R the superalgebra L = P(n - 1) R, n ≠ 4 is centrally closed.

Proof: Let x ∈ Kerψ. By Lemma 6.2 x = , a_i ∈ R. This implies that ψ(x) = h_wi-w1(a_i) = 0 in P(n - 1) R. Hence a₂ = ··· = a_n = 0, and Proposition 6.1 is proved.

Now let us assume that n = 4. Recall that . Denote E = ∑_{1≤i≠j≤n} e_wi-wj(R) + , 𝒬 = Fq_w1+w2 + Fq_w1+w3. Clearly, E, and by Lemma 6.1 . For an arbitrary odd root α ∈ , choose k = 2 or 3 such that (α | w₁ - w_k) ≠ 0. Then This implies that .. Hence Considering the components of weight zero we get

Proposition 6.2. For an arbitrary associative commutative superalgebra R with an even derivation d the superalgebra L = CK(R, d) is centrally closed.

Proof: Let x ∈ Kerψ. As we have seen above, a_i, a ∈ R. Computing the right-hand side in the superalgebra CK(R, d) we get Hence a₂ = a₃ = a₄ = a = 0 and Proposition 6.2 is proved.