Algebra i analiz St. Petersburg Math. J. Tom 24 (2012), 4 Vol. 24 (2013), No. 4, Pages 591–600 S 1061-0022(2013)01255-6 Article electronically published on May 24, 2013 NEW EXAMPLES OF SIMPLE JORDAN SUPERALGEBRAS OVER AN ARBITRARY FIELD OF CHARACTERISTIC 0 V. N. ZHELYABIN Abstract. In a joint paper with the author, I. P. Shestakov constructed a new example of a unital simple special Jordan superalgebra over the real number field. It turned out that this superalgebra is a subsuperalgebra of a Jordan superalgebra of the vector type J(Γ,D), but it is not isomorphic to a superalgebra of this type. Moreover, the superalgebra of quotients of the constructed superalgebra is isomorphic to a Jordan superalgebra of vector type. Later, a similar example was constructed for Jordan superalgebras over a field of characteristic 0 in which the equation t2 +1=0 is unsolvable. In the present paper, an example is given for a Jordan superalgebra with the same properties over an arbitrary field of characteristic 0. A similar example was discovered also for a Cheng–Kac superalgebra. §1. Introduction Jordan algebras and superalgebras form an important class of algebras in the theory of rings. Simple Jordan superalgebras were studied in [1, 2, 3, 4, 5, 6, 7, 8]. In [9, 10], a description was given for the unital simple special Jordan superalgebras with an associative even part A,theoddpartM of which being an associative A-module. The paper [11] in which the simple (−1, 1)-superalgebras of characteristic different from 2 and 3 were described influenced significantly the investigations carried out in [9]. In the Jordan case, if a superalgebra is not the superalgebra of a nondegenerate bilinear superform, then its even part A is a differentiably simple algebra relative to a certain set of derivations, and its odd part M is a finitely generated projective A-module of rank 1. Here, like in the case of (−1, 1)-superalgebras, multiplication in M is given with the help of fixed finite sets of derivations and elements of the algebra A. It turned out that each Jordan superalgebra is a subsuperalgebra of this sort in a superalgebra of vector type J(Γ,D). Under some restrictions on the algebra A,theoddpartM is a one-generated A-module, and thus, the initial Jordan superalgebra is isomorphic to a superalgebra J(Γ,D). So, for example, if A is a local algebra, then, by the well-known Kaplansky theorem, the odd part M is a free, and thus, one-generated A-module. If the ground field is of characteristic p>2, then, by [14], A is a local algebra; therefore, the odd part M is a one-generated A-module. If A is a ring of polynomials in a finite number of variables, then, by [15], the odd part M is a free and, thus, one-generated A-module. Naturally, the question arises as to whether the initial superalgebra is isomorphic to a superalgebra J(Γ,D). This is equivalent to the question as to whether the odd part M is a one-generated A-module. In [10, 12, 13], there are examples of a unital simple special 2010 Mathematics Subject Classification. Primary 16W10; Secondary 17A15. Key words and phrases. Jordan superalgebra, (−1, 1)-superalgebra, superalgebra of vector type, dif- ferentiably simple algebra, polynomial algebra, projective module. Supported by the RFBR grant 09-01-00157, by the analytic Departmental Special Program “Develop- ment of the scientific potential of Higher School” of the Federal Educational Agency (project 2.1.1.419), by the Special Federal Program “Scientific and Pedagogical Staff of Innovative Russia for 2009–2013” (state contracts nos. 02.740.11.0429, 02.740.11.5191, 14.740.11.0346). c 2013 American Mathematical Society 591 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 592 V. N. ZHELYABIN Jordan superalgebra with an associative even part and with the odd part M that is not a free module, i.e., a one-generated module. In those examples, the ground field is either the real number field or any field of characteristic 0 in which the equation t2 +1=0is not solvable. In the present paper, we construct a similar example of a Jordan superalgebra over an arbitrary field of characteristic 0. Also, a similar example of a Jordan superalgebra is given for the Jordan Cheng–Kac superalgebra. The examples of such superalgebras answer a question posed by N. Cantarini and V. Kac in [8]. §2. Generalized superalgebras of vector type Let F be a field of characteristic different from 2. The superalgebra J = J0 + J1 is a Z2-graded F algebra, i.e., 2 ⊆ 2 ⊆ ⊆ ⊆ J0 J0,J1 J0,J1J0 J1,J0J1 J1. We set A = J0 and M = J1.ThespaceA (M) is called the even (respectively, odd) part of the superalgebra J. The elements of the set A ∪ M are said to be homogeneous. The expression p(x), where x ∈ A ∪ M, is the parity index of a homogeneous element x: p(x)=0ifx ∈ A (x is even) and p(x)=1ifx ∈ M (x is odd). For an element x ∈ J, we denote by Rx the operator of right multiplication by x. A superalgebra J is said to be Jordan if the following operator identities are fulfilled for homogeneous elements: p(a)p(b) (1) aRb =(−1) bRa, (2) Ra2 Ra = RaRa2 , p(a)p(b)+p(a)p(c)+p(b)p(c) p(b)p(c) RaRbRc +(−1) RcRbRa +(−1) R(ac)b (3) p(a)p(b) p(a)p(c)+p(b)p(c) = RaRbc +(−1) RbRac +(−1) RcRab. In any Jordan superalgebra, the identity (4) (x, tz, y)=(−1)p(x)p(t)t(x, z, y)+(−1)p(y)p(z)(x, t, y)z, is valid for homogeneous elements, where (x, z, y)=(xz)y − x(zy) is the associator of the elements x, z, y. We give several examples of Jordan superalgebras. Let B = B0 + B1 be an associative Z2-graded algebra with multiplication ∗. Defining a supersymmetric product 1 a ◦ b = a ∗ b +(−1)p(a)p(b)b ∗ a ,a,b∈ B ∪ B , s 2 0 1 on the space B, we obtain the Jordan superalgebra B(+)s. A Jordan superalgebra J = A+M is said to be special if it is embeddable (as a Z2-graded algebra) in the superalgebra (+)s B for an appropriate associative Z2-graded algebra B. The superalgebra J(Γ,D) of vector type. Let Γ be an associative commutative F -algebra with unity 1 and nonzero derivation D.WedenotebyΓtheisomorphiccopy → of the space Γ with the isomorphism map a sa. Consider the direct sum of spaces J(Γ,D)=Γ+Γs and define multiplication (·)onJ(Γ,D) in accordance with the rules · · · · − a b = ab, a sb = ab,s sa b = ab,s sa sb = D(a)b aD(b), where a, b ∈ Γandab is the product in Γ. Then J(Γ,D) is a Jordan superalgebra with even part A = Γ and odd part M = Γ.s A superalgebra J(Γ,D)issimpleifandonly if the algebra Γ is D-simple [16] (i.e., Γ does not contain proper nonzero D-invariant ideals). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW EXAMPLES OF SIMPLE JORDAN SUPERALGEBRAS 593 Consider the associative superalgebra B = M 1,1(End Γ) with the even part 2 φ 0 B = , where φ, ψ ∈ End Γ 0 0 ψ and the odd part 0 φ B = , where φ, ψ ∈ End Γ . 1 ψ 0 In [17], it was proved that then the map R 4R D +2R a + sb → a b D(b) −Rb Ra is an embedding of the superalgebra J(Γ,D) in the superalgebra B(+)s.Consequently, the Jordan superalgebra J(Γ,D) is special. The Kantor double J(Γ, { , }). Let Γ = Γ0 +Γ1 be an associative supercommutative superalgebra with unity 1, and let { , } :Γ→ Γ be a superskewsymmetric bilinear map, which will be called a bracket. By the superalgebra Γ and the bracket { , },onecan construct a superalgebra J(Γ, { , }). Consider J(Γ, { , })=Γ⊕Γx, a direct sum of spaces, where Γx is an isomorphic copy of the space Γ. Let a and b be homogeneous elements of Γ. Then multiplication (·)onJ(Γ, { , }) is defined by the formulas a · b = ab, a · bx =(ab)x, ax · b =(−1)p(b)(ab)x, ax · bx =(−1)p(b){a, b}. We set A =Γ0 +Γ1x and M =Γ1 +Γ0x.ThenJ(Γ, { , })=A + M is a Z2-graded algebra. Abracket{ , } is said to be Jordan if the superalgebra J(Γ, { , }) is a Jordan superalge- bra. As is known (see [18]), { , } is a Jordan bracket if and only if the following relations are satisfied: (5) {a, bc} = {a, b}c +(−1)p(a)p(b)b{a, c}−{a, 1}bc, {a, {b, c}} = {{a, b},c} +(−1)p(a)p(b){b, {a, c}} + {a, 1}{b, c} (6) +(−1)p(a)(p(b)+p(c)){b, 1}{c, a} +(−1)p(c)(p(a)+p(b)){c, 1}{a, b}, (7) {d, {d, d}} = {d, d}{d, 1}, where a, b, c ∈ Γ0 ∪ Γ1, d ∈ Γ1. In particular, a superalgebra J(Γ,D)isanalgebraJ(Γ, { , })if {a, b} = D(a)b − aD(b). In [10], the following theorem was proved. Theorem. Let J = A + M be a simple special unital Jordan superalgebra; let its even part A be an associative algebra and its odd part M an associative A-module.