JORDAN SUPERALGEBRAS DEFINED BY BRACKETS

Consuelo Martínez^{∗} Ivan Shestakov^{†} Efim Zelmanov^{‡}

Abstract

Jordan superalgebras defined by brackets on associative commutative superalgebras are studied. It is proved that any such a superalgebra is imbedded into a superalgebra defined by Poisson brackets. In particular, all Jordan superalgebras of brackets are i-special. The speciality of these superalgebras is also examined, and it is proved, in particular, that the Cheng—Kac superalgebra is special.

Introduction

Let Γ = Γ_{0} + Γ_{1} be an associative commutative superalgebra over a ground field F, chF ≠ 2, with a bracket {, } : Γ × Γ → Γ. Consider the direct sum of two copies of the vector space Γ

J = Γ + Γx

with the product

a · b = ab, a · bx = (ab)x,
(bx) · a = (−1)^{|a|}(ba)x, ax · bx = (−1)^{|b|}{a, b}

where a, b ∈ Γ_{0} ∪ Γ_{1}, juxtaposition stands for the product in Γ and (−1)^{|a|} = (−1)^{k} if a ∈ Γ_{k}. We will refer to J = J(Γ, {, }) as a Kantor double of (Γ, {, }).

A bracket {, } is called Jordan if the Kantor double J is a Jordan superalgebra. I. L. Kantor [K] proved that every Poisson bracket is Jordan.

∗Partially supported by DGES PB97-1291-C03-01, FICYT PGI-PB99-04 and FEDER IFD-97-0556

†Research at MSRI is supported in part by NSF grant DMS-9701755. Supported in part by KIAS (Seoul, Korea).

‡Partially supported by NSF grant DMS-9704132

In [KMZ] it was shown that all Jordan superalgebras that correspond to the so called ”superconformal algebras” are Kantor doubles or are embeddable in Kantor doubles (the Cheng-Kac series). Moreover, the same is true for all simple finitedimensional Jordan superalgebras over a field of prime characteristic with a nonsemisimple even part (see [MZ]).

A Jordan superalgebra J is called special if it can be embedded into a Jordan superalgebra of the type A^{+}, which is obtained from an associative superalgebra A via the new multiplication a·b = \frac{1}{2}(ab+(−1)^{|a||b|}ba). Furthermore, J is called i-special if it is a homomorphic image of a special one.

In [S2] it was proved that for every Poisson bracket the corresponding Jordan Kantor double J(Γ, {, }) is i-special. Moreover, in this case J(Γ, {, }) is special if and only if {{Γ, Γ}, Γ} = 0.

In this paper we consider the speciality and i-speciality problems for the general Jordan Kantor doubles. First, we prove that all Jordan brackets are embeddable into Poisson brackets. In view of [S2] this implies that all Jordan Kantor doubles are i-special. Then we examine speciality of Jordan superalgebras from [KMZ], [MZ]. In particular, we show that the Jordan superalgebras of the Cheng-Kac series are special.

1 Embedding of Jordan brackets into Poisson brackets

We start with some definitions and notations that will be used in the rest of the paper.

By a superalgebra we mean a ℤ/2ℤ-graded algebra A = A_{0} + A_{1}

Example. Let V be a vector space. The Grassmann (or exterior) algebra G(V) is the quotient of the tensor algebra T (V) modulo the ideal generated by symmetric tensors v ⊗ w + w ⊗ v; v, w ∈ V. Clearly G(V) = G¯0 + G¯1, where G¯0 (resp. G¯1 ) is spanned by products of elements of V of even (resp. odd) length.

Let V be a vector space of countable dimension. By the Grassmann envelope of a superalgebra A = A¯0 + A¯1 we mean the subalgebra G(A) = A¯0 ⊗ G¯0 + A¯1 ⊗ G¯1 of the tensor product A ⊗ G(V).

Let V be a homogeneous variety of algebras, that is, a class of F -algebras satisfying a certain set of homogeneous identities and all their partial linearizations (see [ZSSS]).

Definition 1.1 A superalgebra A = A¯0 + A¯1 is called a V-superalgebra if the Grassmann envelope G(A) lies in V.

In particular, if A = A¯0 + A¯1 is a V-superalgebra, then A¯0 ∈ V and A¯1 is a V-bimodule over A¯0 (see [J]).

In this way one can define Lie superalgebras, Jordan superalgebras, etc. Clearly, associative superalgebras are just ℤ/2ℤ-graded associative algebras.

If A is an associative superalgebra then the vector space A with a new operation a · b = \frac{1}{2}(ab + (−1)^{|a||b|}ba) is a Jordan superalgebra which is denoted A^{(+)}. A Jordan superalgebra is said to be special if it is embeddable into a superalgebra of type A^{(+)}. Otherwise J is said to be exceptional. If J ⊆ A^{(+)} and the associative superalgebra A is generated by the subspace J then A is said to be an associative enveloping superalgebra of J. A Jordan superalgebra is i-special if it is a homomorphic image of a special Jordan superalgebra.

Let Γ = Γ_{0}+Γ_{1} be a unital associative commutative superalgebra with a skewsymmetric bilinear mapping {, } : Γ × Γ → Γ, {Γ_{i}, Γ_{j}} ⊆ Γ_{i+j} which we call a bracket.

Consider the Kantor double J = Γ + Γx (see the Introduction) with the multiplication

a · b = ab, a · bx = (ab)x,
(bx) · a = (−1)^{|a|}(ba)x, ax · bx = (−1)^{|b|}{a, b},

where a, b ∈ Γ_{0} ∪ Γ_{1} and juxtaposition stands for the multiplication in Γ. The ℤ/2ℤ-gradation on Γ can be extended to a ℤ/2ℤ-gradation on J via J_{0} = Γ_{0} + Γ_{1}x, J_{1} = Γ_{1} + Γ_{0}x.

A bracket {, } is said to be Jordan if the Kantor double J is a Jordan superalgebra. For a Jordan bracket {, } the mapping D :↦ {a, 1} is a derivation of Γ (see [M]). Moreover, in [M], [KM] it was proved that a bracket {, } is Jordan if and only if for arbitrary elements a, b, c ∈ Γ_{0} ∪ Γ_{1}, x ∈ Γ_{1} hold the identities

{a, b} = −(−1)^{|a||b|}{b, a},  (1)
{a, bc} = {a, b}c + (−1)^{|a||b|}b{a, c} − D(a)bc,  (2)
J(a, b, c) := {{a, b}c} + (−1)^{|a||b|+|a||c|}{{b, c}, a} + (−1)^{|a||c|+|b||c|}{{c, a}, b}
= −{a, b}D(c) + (−1)^{|a||b|+|a||c|}{b, c}D(a) + (−1)^{|a||c|+|b||c|}{c, a}D(b),  (3)
{{x, x}, x} = −{x, x}D(x).  (4)

Identity (4) is needed only in characteristic 3 case, otherwise it follows from (3) (see [KM1]).

A Jordan bracket {, } is called a Poisson bracket if D(a) = 0 for any a ∈ Γ. Observe that in this case Γ is a Lie superalgebra with respect to the bracket.

We will now prove that every Jordan bracket can be embedded into a Poisson bracket.

Proposition 1.1 Let {, } be a Jordan bracket on Γ. On the superalgebra of Laurent polynomials Γ[t^{−1}, t] consider a bracket

< t^{i}a, t^{j}b >= t^{i+j−2}(−(i − 1)aD(b) + (−1)^{|a||b|}(j − 1)bD(a) + {a, b})

where a, b ∈ Γ_{0} ∪ Γ_{1}. Then <, > is a Poisson bracket. Moreover, the Jordan superalgebra J = (Γ, {, }) is isomorphic to a subsuperalgebra Γ + tΓx of the Kantor double J = (Γ[t^{−1}, t], <, >).

Proof: First of all observe that < ta, tb >= {a, b}. Hence Γ + tΓx is a subsuperalgebra of J(Γ[t^{−1}, t], <, >) which is isomorphic to J(Γ, {, }).

Passing to Grassmann envelopes we see that it is sufficient to prove that <, > is a Poisson bracket only for the case Γ = Γ_{0}.

Let a, b, c ∈ Γ = Γ_{0}, and let us check that <, > satisfies the Leibniz identity

< t^{i+j}ab, t^{k}c > = t^{i+j+k−2}(−(i + j − 1)abD(c) + (k − 1)cD(ab) + {ab, c})
= t^{i+j+k−2}(−(i + j − 1)abD(c) + (k − 1)caD(b) +
+ (k − 1)cbD(a) + {a, c}b + a{b, c} + abD(c))
= t^{i+j+k−2}a(−(j − 1)bD(c) + (k − 1)cD(b) + {b, c})
+ t^{i+j+k−2}b(−(i − 1)aD(c) + (k − 1)cD(a) + {a, c})
= t^{i}a < t^{j}b, t^{k}c > +t^{j}b < t^{i}a, t^{k}c > .

Now let us check the Jacobi identity. We have

<< t^{i}a, t^{j}b >, t^{k}c >=< t^{i+j−2}(−(i − 1)aD(b) + (j − 1)bD(a) + {a, b}), t^{k}c > = t^{i+j+k−4}(−(i + j − 3)(−(i − 1)aD(b) + (j − 1)bD(a) + {a, b})D(c)

+ (k − 1)(−(i − 1)D(a)D(b) − (i − 1)aD^{2}(b) + (j − 1)D(b)D(a)
+ (j − 1)bD^{2}(a) + D({a, b}))c
− (i − 1){aD(b), c} + (j − 1){bD(a), c} + {{a, b}, c})

= t^{i+j+k−4}((i + j − 4)(i − 1)aD(b)D(c) − (i + j − 4)(j − 1)bD(a)D(c)

+ (k − 1)(j − i)cD(a)D(b) − (k − 1)(i − 1)aD^{2}(b)c
+ (k − 1)(j − 1)D^{2}(a)bc − (i + j − 3){a, b}D(c)
+ (k − 1)D({a, b})c − (i − 1){a, c}D(b) − (i − 1)a{D(b), c}
+ (j − 1){b, c}D(a) + (j − 1)b{D(a), c} + {{a, b}, c}).

Let us compute the coefficients of all summands in \frac{1}{t^{i+j+k−4}} J(t^{i}a, t^{j}b, t^{k}c).

The summand aD(b)D(c) has the coefficient

(i + j − 4)(i − 1) − (k + i − 4)(i − 1) + (i − 1)(k − j) = 0.

The summand aD^{2}(b)c has the coefficient

−(k − 1)(i − 1) + (i − 1)(k − 1) = 0.

By symmetry, all the summands in J(t^{i}a, t^{j}b, t^{k}c) which do not contain the bracket {, } are zero.

Now the difference −(i − 1)a{D(b), c} + (j − 1)b{D(a), c} after cyclic summing gives zero.

The summand D(c){a, b} has the coefficient

−(i + j − 3) + (j − 1) + (i − 1) = 1,

and therefore

\frac{1}{t^{i+j+k−4}} J(t^{i}a, t^{j}b, t^{k}c) = D(c)\{a, b\} + D(b)\{c, a\} + D(a)\{b, c\}
+ {{a, b}c} + {{b, c}, a} + {{c, a}, b} = 0.

By the property (4) of Jordan brackets Proposition is proved.

In [S2] it was shown that a Kantor double of a Poisson bracket is always i-special. This and Proposition 1.1 immediately imply the following corollary.

Corollary 1.1 For an arbitrary Jordan bracket {, } on an associative commutative superalgebra Γ the superalgebra J(Γ, {, }) is i-special.

2 On speciality of bracket superalgebras

Let J = J_{0}+J_{1} be a Jordan superalgebra. For an arbitrary element x ∈ J let R(x) denote the operator of right multiplication by x. If x ∈ J_{1} then R(x)^{2} is a derivation of the superalgebra J. Let D(J_{1}, J_{1}) denote the linear span of all derivations R(x)^{2}, x ∈ J_{1}. Clearly D(J_{1}, J_{1}) is a Lie algebra.

Now our aim will be to prove the following proposition.

Proposition 2.1 Let J = J_{0} + J_{1} be a special Jordan superalgebra, I a nilpotent ideal of the even part J_{0} which is finitely generated as an ideal and is invariant under D(J_{1}, J_{1}). Then the ideal id_{J}(I) of the superalgebra J generated by I is nilpotent.

Corollary 2.1 Let J = J_{0} + J_{1} be a Jordan superalgebra which does not contain nonzero nilpotent ideals. If J_{0} contains a nonzero nilpotent ideal which is D(J_{1}, J_{1})- invariant and finitely generated as an ideal then J is exceptional.

Let J = J_{0} + J_{1} be a special Jordan superalgebra with an associative enveloping superalgebra R = R_{0} + R_{1}. Let I be an ideal of J_{0} which is invariant under D(J_{1}, J_{1}). Let I_{s} denote the linear span of all products a_{1} ⋯ a_{k}, k ≥ s, a_{i} ∈ I, in the algebra R.

Lemma 2.1 I_{s}RI_{s}RI_{s} ⊆ RI_{2s}R.

Proof: First let us notice that

I_{s}R_{0} ⊆ R_{0}I_{s}.

Indeed, the algebra R_{0} is generated by J_{0} and J_{1}J_{1}. For arbitrary elements u_{1}, …, u_{k} ∈ I, k ≥ s, a ∈ J_{0}; x, y ∈ J_{1} we have u_{1} ⋯ u_{k}a = u_{1} ⋯ u_{k−1}(u_{k}a + au_{k}) − u_{1} ⋯ u_{k−2}(u_{k−1}a + au_{k−1})u_{k} + ⋯ + (−1)^{k}au_{1} ⋯ u_{k} ∈ RI_{s}. Furthermore, xy = \frac{1}{2}([x, y] + (xy + yx)). The commutator [x, y], lies in J_{0} whereas the operator

J → J
u → [u, xy + yx]

lies in D(J_{1}, J_{1}).

Hence u_{1} ⋯ u_{k}(xy + yx) = (xy + yx)u_{1} ⋯ u_{k} + [u_{1} ⋯ u_{k}, xy + yx] ∈ RI_{s}. This implies that

I_{s}R_{0}I_{s} ⊆ R_{0}I_{2s}.

Let us show that I_{s}R_{i}I_{s}R_{j}I_{s} ⊆ RI_{2s}R. If i = 0 or j = 0 then the inclusion was proved above. If i = j = 1 then R_{i}I_{s}R_{j} ⊆ R_{0} and it remains to refer to the inclusion above.

Lemma is proved.

Corollary 2.2 In the assumptions of Lemma 2.1, for any k ≥ 0 holds the inclusion

(RI_{s}R)^{3^{k}} ⊆ RI_{2^{k}s}R.

Proof: The Lemma gives the base of induction for k = 1. Assume that the assertion is true for k − 1, then

(RI_{s}R)^{3^{k}} = ((RI_{s}R)^{3^{k−1}})^{3} ⊆ (RI_{2^{k−1}s}R)^{3} ⊆ RI_{2·2^{k−1}s}R = RI_{2^{k}s}R.

For elements a, b ∈ R denote a · b = \frac{1}{2}(ab + ba). Let I^{·n} denote the n-th power of the ideal I in J.

Lemma 2.2 For an arbitrary k ≥ 1 we have \underbrace{I^{·2} ⋯ I^{·2}}_{k} ⊆ RI^{·k+1}, where juxtapo-

sition stands for the multiplication in R.

Proof: The assertion is obvious for k = 1. By the induction assumption \underbrace{I^{·2} ⋯ I^{·2}}_{k} ⊆

RI^{·k}I^{·2}. If u ∈ I^{·k}, a, b ∈ I, then u(a·b) = u·(a·b)+\frac{1}{2}[u, a·b] and clearly u·(a·b) ∈ I^{·k+1}. Now, [u, a · b] = [u · a, b] + [u · b, a], u · a ∈ I^{·k+1}, u · b ∈ I^{·(k+1)} and it remains to notice that (u · a)b = 2(u · a) · b − b(u · a) ∈ RI^{·k+1}. Lemma is proved.

Lemma 2.3 Suppose that the ideal I of J_{0} is generated (as an ideal) by m elements a_{1}, …, a_{m}. Then I_{3(m+2)} ⊆ R_{0}I^{·3}.

Proof: Let A be the subalgebra of R_{0} generated by J_{0}. Let us show first that I_{m+2} lies in the ideal of A generated by I^{·2}. Let u_{1}, …, u_{m+2} ∈ I. We have u_{1} ⋯ u_{m+2} = \frac{1}{2}u_{1} ⋯ u_{m}[u_{m+1}, u_{m+2}] = ±u_{1} ⋯ û_{i} ⋯ u_{m+1}[u_{i}, u_{m+2}] mod id_{A}(I^{·2}).

If a, b ∈ I, x ∈ J_{0} then [a · x, b] = [a, b · x] + [x, a · b] = [a, b · x] mod id_{A}(I^{·2}). Hence, without loss of generality we can assume that u_{1}, …, u_{m+1} ∈ {a_{1}, …, a_{m}} and therefore u_{i} = u_{j} for some 1 ≤ i ≠ j ≤ m+1. This implies that u_{1} ⋯ u_{m+1} ∈ id_{A}(I^{·2}).

Now let us show that

(id_{A}(I^{·2}))^{3} ⊆ id_{A}(I^{·3})

Since for arbitrary elements a, b ∈ J_{0} we have ab = a·b+ \frac{1}{2}[a, b] and [I^{·2}, [a, b]] ⊆ I^{·2} it is sufficient to prove that

I^{·2}J_{0}I^{·2}J_{0}I^{·2} ⊆ id_{A}(I^{·3}).

Choose arbitrary elements u_{1}, u_{2}, u_{3} ∈ I^{·2} and a, b ∈ J_{0}. We have {I^{·2}, J_{0}, I^{·2}} ⊆ (I^{·2} · J_{0}) · I^{·2} + (I^{·2} · I^{·2}) · J_{0} ⊆ I^{·3}. This implies that the

expression u_{1}au_{2}bu_{3} + id_{A}(I^{·3})/id_{A}(I^{·3}) is skew symmetric in u_{1}, u_{2}, u_{3}.

We have also I^{·2}I^{·2} ⊆ I^{·2} · I^{·2} + [I^{·2} · I, I] ⊆ id_{A}(I^{·3}) and I^{·2}II^{·2} ⊆ (I^{·2} · I)I^{·2} + II^{·2}I^{·2} ⊆ id_{A}(I^{·3}).

Hence u_{1}(au_{2}b + bu_{2}a)u_{3} ⊆ I^{·2}II^{·2} ⊆ id_{A}(I^{·3}). Therefore the expression u_{1}au_{2}bu_{3} + id_{A}(I^{·3})/id_{A}(I^{·3}) is skewsymmetric in a and

b.

Now u_{1}au_{2}bu_{3} = 4(u_{1} · a)(u_{2} · b)u_{3} = 4(u_{2} · b)(u_{1} · a)u_{3} mod id_{A}(I^{·3}). Finally, u_{1}au_{2}bu_{3} = 4((u_{1} · a) · (u_{2} · b))u_{3} ∈ I^{·2}I^{·2} = (0) mod id_{A}(I^{·3}). The inclusions I_{m+2} ⊆ id_{A}(I^{·2}), (id_{A}(I^{·2}))^{3} ⊆ id_{A}(I^{·3}) imply I_{3(m+2)} ⊆ (I_{m+2})^{3} ⊆

(id_{A}(I^{·2}))^{3} ⊆ id_{A}(I^{·3}).

Lemma is proved.

Proof of Proposition 2.1. Suppose that the ideal I of J_{0} is D(J_{1}, J_{1})-invariant and generated (as an ideal) by m elements. Suppose further that I^{·n} = (0), n ≥ 2.

Find integers s_{1}, s_{2} such that 2^{s_{1}} ≥ n − 1, 2^{s_{2}} ≥ 3(m + 2). By Corollary 2.2 applied to the ideal I^{·3} we have

(RI^{·3}R)^{3^{s_{1}}} ⊆ R(I^{·3})^{2^{s_{1}}}R ⊆ R(I^{·3})^{n−1}R
⊆ R(I^{·2})^{n−1}R ⊆ RI^{·n}R = (0).

Similarly, by Lemma 2.3,

(RIR)^{3^{s_{2}}} ⊆ RI_{2^{s_{2}}}R ⊆ RI_{3(m+2)}R ⊆ RI^{·3}R.

Let s = s_{1} + s_{2}. Then

(RIR)^{3^{s}} = ((RIR)^{3^{s_{2}}})^{3^{s_{1}}} ⊆ (RI^{·3}R)^{3^{s_{1}}} = (0),

which proves the Proposition.

Corollary 2.3 Let Γ = Γ_{0} + Γ_{1} be an associative commutative finitely generated superalgebra such that Γ_{1}Γ_{1} ≠ (0). Let {, } be a Jordan bracket on Γ. Suppose that the Kantor double superalgebra J = J(Γ, {, }) does not contain nonzero nilpotent ideals. Then the superalgebra J is exceptional.

Proof: Let I the ideal of J_{0} generated by Γ_{1}Γ_{1} ≠ (0). Since Γ_{1}Γ_{1} is the product in the superalgebra J, to check that I is D(J_{1}, J_{1})-invariant it is sufficient to prove that Γ_{1}D(J_{1}, J_{1}) ⊆ Γ_{1}. We have

D(J_{1}, J_{1}) = D(Γ_{1}, Γ_{1}) + D(Γ_{1}, Γ_{0}x) + D(Γ_{0}x, Γ_{0}x).

It is easy to see that Γ_{1}D(Γ_{1}, Γ_{1}) = Γ_{1}D(Γ_{1}, Γ_{0}x) = (0) and that Γ_{1}D(Γ_{0}x, Γ_{0}x) ⊆ Γ ∩ J_{1} = Γ_{1}.

If Γ is generated by m even elements a_{1}, …, a_{m} and n odd elements b_{1}, …, b_{n}, then I is generated (as a J_{0}-ideal) by all products b_{i}b_{j}, 1 ≤ i < j ≤ n.

It is easy to see that I = Γ_{1}Γ_{1} + Γ_{1}Γ_{1}Γ_{1}x and that for any k

I^{·k} = \underbrace{Γ_{1}Γ_{1} ⋯ Γ_{1}}_{2k} + \underbrace{Γ_{1} ⋯ Γ_{1}}_{2k+1} x.

Hence, if 2k ≥ n+1 then I^{·k} = (0). By Corollary 1 the superalgebra J is exceptional.

Let now F be an algebraically closed field of characteristic p > 2, and let O(m) = F [a_{1}, …, a_{m} | a_{1}^{p} = ⋯ = a_{m}^{p} = 0] be the algebra of truncated polynomials. Furthermore, let G(n) =< 1, ξ_{1}, …, ξ_{n} | ξ_{i}ξ_{j} + ξ_{j}ξ_{i} = 0, 1 ≤ i, j ≤ n > be the Grassmann algebra on n variables. The tensor product

O(m, n) = O(m) ⊗_{F} G(n) = O(m) ⊗_{F} G_{0}(n) + O(m) ⊗_{F} G_{1}(n)

is an associative commutative superalgebra.

In [MZ] it was proved that an arbitrary simple finite dimensional Jordan superalgebra J over F with a nonsemisimple part J_{0} is isomorphic to one of the following superalgebras:

1) a Kantor double J = J(O(m, n), {, }) of an appropriate Jordan bracket on O(m, n),

2) a superalgebra of a Cheng-Kac type (see below for more details).

Corollary 2 implies that for n ≥ 2 the simple superalgebras J(O(m, n), {, }) are exceptional.

A classification along the same lines: Kantor doubles and Cheng-Kac superalgebras, is developed in [KMZ] for infinite dimensional Jordan superalgebras of zero characteristic that correspond to the so called ”superconformal algebras” ([KL], [CK]). Speciality problems for these superalgebras can be treated similarly.

Let Γ be again an arbitrary associative commutative superalgebra with a Jordan bracket {, }, D(a) = {a, 1}. The bracket {, } is said to be of vector type if {a, b} = D(a)b − aD(b). In [M],[KM] and in [S1] it was proved that the Kantor double of a bracket of vector type is a special superalgebra. Furthermore, in [M],[KM] two important examples of classical and Grassmann Poisson brackets were treated and it was showed that in both cases the Kantor double superalgebras are exceptional. Since both classical and Grassman Poisson brackets are not nilpotent, the last result follows from the following speciality criterium for the Kantor double of a Poisson bracket obtained in [S2]: a Poisson bracket {, } : Γ × Γ → Γ is special if and only if {{Γ, Γ}, Γ} = (0).

Proposition 2.2 Let Γ = Γ_{0} + Γ_{1} be a finitely generated associative commutative superalgebra with a Jordan bracket {, } such that the superalgebra J(Γ, {, }) does not contain nonzero nilpotent ideals. Suppose that either (i) Γ_{1} = (0) or (ii) Γ_{1} contains an element ξ such that Γ_{1} = Γ_{0}ξ and {Γ_{0}, ξ} = (0), {ξ, ξ} = −1. Then the superalgebra J(Γ, {, }) is special if and only if the restriction of {, } to Γ_{0} is of vector type.

Remark. Proposition 2.2 takes care of all superalgebras of the type J(O(m, n), {, }), n = 0 or 1, from the classification list of [MZ].

Proof: Let Γ = Γ_{0} + Γ_{1} be a finitely generated associative commutative superalgebra with a Jordan bracket {, }. Suppose that the Kantor double superalgebra J = Γ+Γx is special and does not contain nonzero nilpotent ideals. Let R = R_{0} +R_{1} be an associative enveloping superalgebra of J, J ⊆ R^{(+)}. Factoring out the Baer radical of R we can assume that R does not contain nonzero nilpotent ideals. For elements u, v ∈ R let uv denote their product in R and let [u, v] = uv−vu, u∘v = uv+vu. We will also denote by · the product in the superalgebra J.

We will use the following identities from [S2] which can be verified in an associative algebra by a straightforward computation:

[a ∘ x, b ∘ x] = [a, x^{2}] ∘ b − [b, x^{2}] ∘ a + ([a, b] ∘ x) ∘ x + [[a, x], [b, x]],  (5)
[u, [r, s]] = ((u ∘ r) ∘ s − (u ∘ s) ∘ r).  (6)

Let a, b ∈ J_{0}. Our aim is to prove that < a, b >= {a, b} − D(a) · b + a · D(b) = 0. It is easy to see that < a, b >= − < b, a > and <, > satisfies the Leibniz identity, that is, < a, b^{·2} >= 2 < a, b > ·b. Now, for an arbitrary element u ∈ J we have by (6)

[u, [a, b]] = 4((u · a) · b − (u · b) · a) = 0,

Hence [a, b] lies in the center of the algebra R. Moreover,

2[a, b]^{2} = [a, [a, b^{·2}]] − [a, [a, b]] ∘ b = 0.

Since R does not contain nonzero nilpotent ideals, it follows that [a, b] = 0.

Furthermore, observe that [a, x^{2}] = 2[a·x, 1·x] = 4{a, 1} = 4D(a) and [a·x, b·x] = 2{a, b}. Since [a, b] = 0, we get from (5)

2{a, b} = 2D(a) · b − 2D(b) · a + 1/4 [[a, x], [b, x]],

which yields

< a, b >= \frac{1}{8} [[a, x], [b, x]].

Consider 2[a, x]^{2} = [a, [a, x^{2}]] − [a, [a, x]] ∘ x. The second summond is zero by (6), and the first one is equal to 4[a, D(a)] = 0. Hence [a, x]^{2} = 0 and [a, x][b, x]+[b, x][a, x] = 0. Now by (6) again,

16 < a, b >^{2}= [[[a, x]^{2}, [b, x]], [b, x]] − [a, x] ∘ [[[a, x], [b, x]], [b, x]] = 0.

Let I be the ideal generated in J_{0} by all the elements < a, b >; a, b ∈ Γ_{0}. Since the algebra Γ is finitely generated it implies that the even part Γ_{0} is also finitely generated. Let Γ_{0} =< a_{1}, …, a_{m} >. Then the ideal I is generated (as an ideal) by elements < a_{i}, a_{j} >, 1 ≤ i < j ≤ m.

If Γ = Γ_{0} then J_{0} = Γ_{0}. If Γ = Γ_{0} + Γ_{0}ξ, {ξ, ξ} = −1, then J_{0} = Γ_{0} + Γ_{0}ξx, (ξx) · (ξx) = 1. In the latter case J_{0} ≃ Γ_{0} ⊕ Γ_{0}. In both cases J_{0} is an associative commutative algebra. This implies that the ideal I is nilpotent.

For elements a, b, c ∈ Γ_{0} consider

< a, b >_{c}= (a · cx) · (b · cx) − ((a · cx) · cx) · b + a · ((b · cx) · cx)
= {ac, bc} − {ac, c}b + a{bc, c} =< a, b > c^{2} ∈ I.

Since Γ_{0} and Γ_{0}x are invariant under D(Γ_{0}x, Γ_{0}x) this implies that the ideal I is invariant under D(Γ_{0}x, Γ_{0}x).

If Γ = Γ_{0} then ID(J_{1}, J_{1}) ⊆ I. In the case Γ = Γ_{0} + Γ_{0}ξ we have D(J_{1}, J_{1}) = D(Γ_{0}x, Γ_{0}x) + D(Γ_{0}x, Γ_{0}ξ) + D(Γ_{0}ξ, Γ_{0}ξ).

From Γ_{0}D(Γ_{0}ξ, Γ_{0}ξ) = Γ_{0}D(Γ_{0}x, Γ_{0}ξ) = (0) it follows that ID(J_{1}, J_{1}) ⊆ I. By Proposition 2.1 we have I = (0), that is,

{a, b} = D(a)b − aD(b)

for arbitrary elements a, b ∈ Γ_{0}.

Now suppose that the restriction {, }|_{Γ_{0}} is of vector type. If Γ = Γ_{0} then speciality of J(Γ, {, }) follows from the results of [M], [S1].

Let Γ = Γ_{0} +Γ_{0}ξ, {Γ_{0}, ξ} = (0), {ξ, ξ} = −1. Denote D = R(x)^{2}. In this case the superalgebra J(Γ, {, }) is embeddable into the Cheng-Kac superalgebra CK(Γ_{0}, D) (see below), which is special by Proposition 3.1 (also below). Proposition is proved.

3 Cheng-Kac Superalgebras

Recall the definition of Cheng-Kac Jordan superalgebras (see [MZ]). Let Z be an associative commutative algebra with a derivation D : Z → Z.

Let A = Z + ∑_{i=1}^{3} Zw_{i}, M = xZ + ∑_{i=1}^{3} x_{i}Z be two free Z-modules of rank 4. The multiplication on A is Z-linear, w_{i}w_{j} = 0 for i ≠ j, w_{1}^{2} = w_{2}^{2} = 1, w_{3}^{2} = −1.

Denote x_{i×i} = 0, x_{1×2} = −x_{2×1} = x_{3}, x_{1×3} = −x_{3×1} = x_{2}, −x_{2×3} = x_{3×2} = x_{1}. The bimodule structure A × M → M is defined via

g  w_{j}g
xf x(fg) x_{j}(fg^{d})
x_{i}f x_{i}(fg) x_{i×j}(fg)

The bracket on M is defined via

xg  x_{j}g
xf f^{d}g − fg^{d} −w_{j}(fg)
x_{i}f w_{i}(fg)  0

In [MZ] it was proved that the superalgebra CK(Z, D) = A + M is Jordan.

If Γ = Γ_{0} + Γ_{0}ξ, {Γ_{0}, ξ} = (0), {ξ, ξ} = −1, D(a) = {a, 1}, then the Kantor double superalgebra J = J(Γ, {, }) is generated by Γ_{0} and the elements w_{1} = ξx, x with w_{1}^{2} = 1. Clearly, J is embeddable into CK(Γ_{0}, D).

Proposition 3.1 A Cheng-Kac Jordan superalgebra CK(Z, D) is special.

Proof: Let W denote the algebra of differential operators on Z, that is, the algebra of linear transformations on Z generated by all right multiplications R(a), a ∈ Z, and by the derivation D. Consider the algebra R = M_{4}(W) of 4×4 matrices over W with the ℤ/2ℤ-gradation

R_{0} = ( \begin{matrix}W & 0 & W & 0 \\ 0 & W & 0 & W \\ W & 0 & W & 0 \\ 0 & W & 0 & W \end{matrix} ), R_{1} = ( \begin{matrix}0 & W & 0 & W \\ W & 0 & W & 0 \\ 0 & W & 0 & W \\ W & 0 & W & 0 \end{matrix} ).

Our embedding of the superalgebra CK(Z, D) will extend the King-McCrimmon embedding of the Kantor double of the vector type bracket {a, b} = D(a)b − aD(b) into M_{2}(W) (see [KM]).

For an element a ∈ Z let

φ(a) = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ), φ(w_{1}) = ( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ),

φ(w_{2}) = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ), φ(w_{3}) = ( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ),

φ(x) = ( \begin{matrix}0 & 0 & 0 & 2D \\ 0 & 0 & −1 & 0 \\ 0 & −2D & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ), φ(x_{1}) = ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ),

φ(x_{2}) = ( \begin{matrix}0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} ), φ(x_{3}) = ( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} ).

A straightforward computation shows that φ extends to a homomorphism. Proposition is proved.

Corollary 3.1 The Cheng-Kac Lie superalgebra (see [CK]) embedds into a superalgebra of 8 × 8 matrices over the Weyl algebra < t^{−1}, t, \frac{d}{dt} >.

References

[CK] S.J. Cheng and V.G. Kac, A new N=6 superconformal algebra, Commun. Math. Phys. 186, 219-231, 1997.

[J3] N. Jacobson, Structure and representation of Jordan Algebras, Amer. Math. Soc. Providence, R.I., 1969.

[KL] V.G. Kac and J.W. van de Leur, On classification of superconformal algebras, Strings 88, World Sci., 77-106, 1989.

[KMZ] V.G. Kac, C. Martinez and E. Zelmanov, Graded simple Jordan superalgebras of growth one, To appear in Memoirs of the AMS.

[K] I.L.Kantor, Jordan and Lie superalgebras defined by Poisson algebras, Algebra and Analysis (Tomsk, 1989), 55–79, Amer. Math. Soc. Transl. Ser. (2), 151, Amer. Math. Soc., Providence, RI, 1992.

[KM] D. King and K. McCrimmon, The Kantor Construction of Jordan Superalgebras, Comm. in Algebra 20(1), 109-126, 1992.

[KM1] D. King and K. McCrimmon, The Kantor doubling process revisited, Comm. in Algebra 23(1), 109-126, 1995.

[M] K. McCrimmon, Speciality and non-speciality of two Jordan superalgebras, J. of Algebra 149, no. 2, 326-351, 1992.

[MZ] C. Martinez and E. Zelmanov, Simple finite dimensional Jordan superalgebras of prime characteristic, Submitted.

[S1] I.P. Shestakov, Superalgebras and counter-examples, Sibirsk. Mat, Zh. 32 (1991), no.6, 187–196; English transl, in Siberian Math. J. 32 (1991), no. 6, 1052– 1060 (1992).

[S2] I.P. Shestakov, Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras, Algebra i Logika, 32 (1993), no.5, 571–584; English transl, in Algebra and Logic, 32 (1993), no. 5, 309–317.

[ZSSS] K.A. Zhevlakov, A.M.Slin’ko, I.P.Shestakov, A.I.Shirshov, Rings that are nearly associative, Academic Press, New York, 1982.

Consuelo Martínez, Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo, s/n, 33007, Oviedo SPAIN, e-mail: chelo@pinon.ccu.uniovi.es

Ivan Shestakov, Instituto de Mathemática e Estatística, Universidade de São Paulo, Caixa Postal 66281 - CEP 05315-970, São Paulo, BRASIL, and Sobolev Institute of Mathematics, Novosibirsk, 630090, RUSSIA, e-mail: shestak@ime.usp.br

Efim Zelmanov, Department of Mathematics, Yale University, New Haven, CT 06520, USA, and KIAS, Seoul 130-012, S. KOREA e-mail: zelmanov@pascal.MATH.YALE.EDU