New Examples of Simple Jordan Superalgebras Over an Arbitrary Field of Characteristic 0

Home , Field (physics)

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, differentiably 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 superalgebra. 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. Assume that J is not a superalgebra of a nondegenerate bilinear superform. Then there exist elements x1,...,xn ∈ M such that

M = x1A + ···+ xnA and the product in M is given by the relation

(8) axi · bxj = γijab + Dij(a)b − aDji(b),i,j=1,...,n,

where γij ∈ A and Dij is a derivation of the algebra A. The algebra A is diﬀerentiably simple relative to the set of derivations Δ={Dij | i, j =1,...,n}. The module M is a projective A-module of rank 1. Moreover, the superalgebra J is a subalgebra of the superalgebra J(Γ,D).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 594 V. N. ZHELYABIN

In [10], there is an example of a Jordan superalgebra over the real number field that satisfies the assumptions of the theorem and is not isomorphic to the algebra J(Γ,D). A similar example of a Jordan superalgebra over a field of characteristic 0 in which the equation t2 + 1 = 0 is not solvable was constructed in [12, 13]. We give another example of such a superalgebra over an arbitrary field of characteristic 0. Let F be an arbitrary field of characteristic 0. Consider the polynomial algebra F [x, y] ∂ ∂ in two variables x and y.By∂x and ∂y we denote the operators of differentiation of 3 ∂ − ∂ the algebra F [x, y] with respect to the variables x and y.WesetD =2y ∂x x ∂y and f(x, y)=x2 + y4 − 1. Then D is a derivation of the algebra F [x, y]andD(f(x, y)) = 0. Let Γ = F [x, y]/f(x, y)F [x, y] be the quotient algebra of the algebra F [x, y] by the ideal f(x, y)F [x, y]. Clearly, the derivation D induces a derivation of the algebra Γ, which we also denote by D. We identify the images of the elements x and y under the canonical homomorphism F [x, y] → Γ with the elements x and y.ThenΓ=F [y]+xF [y], where F [y] is the polynomial ring of the variable y. Proposition 1. The algebra Γ is differentiably simple relative to the derivation D. Proof. Let I be a nonzero D-invariant ideal of the algebra Γ. If f(y) ∈ F [y]andf(y) ∈ I, then the polynomial D(f(y)) = −xf (y) belongs to I;heref (y)isthederivative of the polynomial f(y) with respect to the variable y.Then(1− y4)f (y) ∈ I and D (1 − y4)f (y) ∈ I. Therefore, −x −4y3f (y)+(1− y4)f (y) ∈ I. This implies that (1 − y4)2f (y) ∈ I. Continuing this process, we conclude that (1 − y4)kf (k)(y) ∈ I for any k,wheref (k)(y)isthekth derivative of f(y). Consequently, (1 − y4)k ∈ I for some k. 4 k Let k be the smallest number for which zk =(1− y ) ∈ I.Then 3 4 k−1 D(zk)=4kxy (1 − y ) ∈ I. Therefore, 1 x(1 − y4)k−1 = xz + yD(z ) ∈ I. k 4k k Consequently, D(x(1 − y4)k−1)=2y3(1 − y4)k−1 +(k − 1)4y3(1 − y4)k−1 =2(2k − 1)y3(1 − y4)k−1 ∈ I. Hence, we get y3(1 − y4)k−1 ∈ I and y4(1 − y4)k−1 ∈ I.Then 4 k 4 4 k−1 zk−1 =(1− y ) + y (1 − y ) ∈ I. Thus, we may suppose that F [y] ∩ I =0. Let f(y)+xg(y) ∈ I.Then (f(y)+xg(y))(f(y) − xg(y)) = f(y)2 − (1 − y4)g(y)2 ∈ I. By what we have proved above, f(y)2 =(1− y4)g(y)2. Then 1 − y4 = h(y)2 for some polynomial h(y) ∈ F [y], a contradiction. Consequently, the algebra Γ is differentiably simple relative to the derivation D. Consider the subalgebra A generated in Γ by the elements 1, y2, xy.Then D(y2)=−2xy ∈ A and D(xy)=3y4 − 1 ∈ A. Consequently, D(A) ⊆ A. Note that the elements 1, y2i, xy2i−1,wherei =1, 2,..., form a linear basis of the algebra A. Any element of A can be represented in the form f(y)+xyg(y), where f(y),g(y) ∈ F [y2]. Proposition 2. The algebra A is differentiably simple relative to the derivation D.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW EXAMPLES OF SIMPLE JORDAN SUPERALGEBRAS 595

Proof. Let I be a nonzero D-invariant ideal of the algebra A.Iff(y) ∈ F [y2]and f(y) ∈ I,thenxf (y)=−D(f(y)) ∈ I. Therefore, (1 − y4)yf(y)=(xy)(xf (y)) ∈ I. Since D(xf (y)) = 2y3f (y) − (1 − y4)f (y) ∈ I, we have (1 − y4)2f (y) ∈ I. By easy induction we derive the relations (1 − y4)2k−1yf(2k−1)(y) ∈ I and (1 − y4)2kf (2k)(y) ∈ I. This implies (1 − y4)2k ∈ I. Let k be the smallest number with (1 − y4)k ∈ I. Then D (1 − y4)k = −4kxy3(1 − y4)k−1 ∈ I. Consequently, xy(1 − y4)k−1 = xy(1 − y4)k + y2(xy3(1 − y4)k−1) ∈ I. Therefore, D xy(1 − y4)k−1 =(3y4 − 1)(1 − y4)k−1 +(k − 1)4y4(1 − y4)k−1 = (4k − 1)y4 − 1 (1 − y4)k−1 ∈ I. Then (4k − 2)(1 − y4)k−1 =(4k − 1)(1 − y4)k + (4k − 1)y4 − 1 (1 − y4)k−1 ∈ I. Thus, we may assume that F [y2] ∩ I =0. Let f(y)+xyg(y) ∈ I.Then f(y)2 − (1 − y4)y2g(y)2 =(f(y)+xyg(y))(f(y) − xyg(y)) ∈ I. By what we have proved above, f(y)2 − (1 − y4)y2g(y)2 = 0. We obtain a contradiction, because deg f(y)2 =4n, but deg(1 − y4)y2g(y)2 =4m +6. Thus, the algebra A is diﬀerentiably simple relative to the derivation D. In the algebra Γ, we consider the subspace M = xA + yA.ThenM is an associative A-module. Proposition 3. The module M is not a one-generated A-module. Proof. Let M be a one-generated A-module, and let z be its generator. Then z = xa+yb, where a, b ∈ A,andx = zc, y = zd,wherec, d ∈ A. Hence, (9) xd = yc, (10) x = x(ac + bd),y= y(ac + bd). Let a = f0 + xyf1,b= g0 + xyg1,c= e0 + xye1,d= h0 + xyh1, 2 where f0, f1, g0, g1, e0, e1, h0,andh1 are polynomials in F [y ]. From (9) we obtain 2 4 h0 = y e1 and e0 =(1− y )h1. By (10), 4 2 4 2 (11) f0e0 +(1− y )y f1e1 + g0h0 +(1− y )y g1h1 =1,

(12) f0e1 + f1e0 + g0h1 + g1h0 =0.

Let (e1,h1) be the greatest common divisor of the polynomials e1 and h1.Since 2 4 h0 = y e1 and e0 =(1− y )h1, by (11) we have 1=(1− y4)f h +(1− y4)y2f e + y2g e +(1− y4)y2g h 0 1 1 1 0 1 1 1 4 2 2 4 =(1− y )(f0 + y g1)h1 + y (1 − y )f1 + g0 e1.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 596 V. N. ZHELYABIN

Consequently, (e ,h ) = 1. By (12), 1 1 2 4 (f0 + y g1)e1 + (1 − y )f1 + g0 h1 =0. 2 This relation and the fact that (e1,h1)=1implythatf0 + y g1 = h1u,whereu ∈ F [y]. Then 4 h1ue1 + (1 − y )f1 + g0 h1 =0. Therefore, 4 ue1 + ((1 − y )f1 + g0)=0. By the foregoing, − 4 2 2 − 4 − 4 2 − 2 2 1=(1 y )(f0 + y g1)h1 + y (1 y )f1 + g0 e1 =(1 y )h1u y e1u. Then u ∈ F .Consequently, − 4 2 2 2 (1 y )h1u =1+y e1u. This is impossible, because there is a polynomial of degree 4k + 4 on the left-hand side and a polynomial of degree 4m + 2 on the right. Thus, the module M is not a one-generated A-module. We set 4 2 D11 =(1− y )D, D12 = xyD, D22 = y D.

Then D11, D12,andD22 are derivations of the algebra A. Proposition 4. The algebra A is diﬀerentiably simple relative to the set of derivations Δ={D11,D12,D22}. Proof. Let I be an ideal of the algebra A closed with respect to the set of derivations 2 2 Δ={D11,D12,D22}.Theny D22(I) ⊆ y I ⊆ I.Since 2 D = D11 + y D22, we have D(I) ⊆ I. Then, by Proposition 2, either I =0,orI = A.Consequently,A is diﬀerentiably simple relative to Δ = {D11,D12,D22}. Consider the superalgebra J(Γ,D). By Proposition 1, J(Γ,D) is a simple superalgebra. In this superalgebra, consider the subspace J(A, Δ) = A + M.Ď We recall that A is the subalgebra in Γ generated by the elements 1, y2, xy and M = xA + yA. Let a, b ∈ A. Then, in J(Γ,D) we have the relations · − 2 − − 2 xaĎ Ďxb = D(xa)xb D(xb)xa = D(x)axb + D(a)x b D(x)xab D(b)x a = D11(a)b − aD11(b) ∈ A. Similarly, · 2 − − 2 − ∈ yaĎ ybs = D(y)ayb + D(a)y b D(y)yab D(b)y a = D22(a)b aD22(b) A, · − − xaĎ ybs = D(x)ayb + D(a)xyb D(y)xab D(b)yxa 4 =(1+y )ab + D12(a)b − aD12(b) ∈ A. Consequently, J(A, Δ) is a subsuperalgebra in J(Γ,D). Therefore, J(A, Δ) is a Jordan superalgebra. Moreover, the multiplication of odd elements in the superalgebra J(Γ,D) 4 is done in accordance with relations (8), where Δ = {D11,D12,D22} and γ12 =1+y . By Proposition 3, the superalgebra J(A, Δ) is not isomorphic to a superalgebra of the type J(Γ0,D0).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW EXAMPLES OF SIMPLE JORDAN SUPERALGEBRAS 597

We show that J(A, Δ) is a simple superalgebra. Let I be a nonzero Z2-graded ideal of the superalgebra J(A, Δ). Then I = I0 + I1,whereI0 is an ideal of the algebra A.For ∈ · · · ∈ ∈ any r I0,wehaveD11(r)=(Ěxr) xs =(r xs) xs I0. Similarly, D12(r),D22(r) I0. Therefore, I0 is invariant relative to the set of derivations Δ. By Proposition 4, either I0 = A or I0 =0.IfI0 = A,then1∈ I0 ⊆ I and I = J(A, Δ). If I0 =0,thenI ⊆ MĎ and I · MĎ ⊆ I0 =0.Itisclearthat

A = AD11(A)+AD12(A)+AD22(A).

Therefore, 1= (a1i, x,s xs)b1i + (a2i, x,s ys)b2i + (a3i, y,s ys)b3i i i i

for some elements a1i, a2i, a3i, b1i, b2i,andb3i in A. With the help of (4), we deduce that 1 ∈ (A, M,Ď MĎ)and I · (A, M,Ď MĎ) ⊆ (A, I · M,Ď MĎ)+(A, I, MĎ) · MĎ =0. Then I =0.Consequently,J(A, Δ) is a simple superalgebra. The foregoing is summarized in the theorem below. Theorem 1. Let F be an arbitrary ﬁeld of characteristic 0. Consider the algebra F [x, y] 2 4 − 3 ∂ − of polynomials in two variables x and y.Wesetf(x, y)=x + y 1 and D =2y ∂x ∂ x ∂y .LetΓ=F [x, y]/f(x, y)F [x, y]. Then the derivation D induces a derivation of the algebra Γ, which we also denote by D. Weidentifytheimagesoftheelementsx and y under the canonical homomorphism F [x, y] → Γ with the elements x and y.LetA be the subalgebra in Γ generated by the elements 1, y2, xy,andletM = xA + yA.Weput 4 2 Δ={D11,D12,D22}, where D11 =(1− y )D, D12 = xyD, D22 = y D. Then the subspace J(A, Δ) = A + MĎ is a subsuperalgebra in J(Γ,D) and multiplication of odd elements in J(A, Δ) is given by the formulas · − · − xaĎ Ďxb = D11(a)b aD11(b), yaĎ ybs = D22(a)b aD22(b), · 4 − xaĎ ybs =(1+y )ab + D12(a)b aD12(b). Moreover, the superalgebra J(A, Δ) is simple, and MĎ is not a one-generated A-module, i.e., J(A, Δ) is not isomorphic to a superalgebra of the vector type J(Γ0,D0).

A superalgebra of the type JS(Γ,D). Let Γ = Γ0 +Γ1 be an associative supercommutative superalgebra with a nonzero odd derivation D, i.e., D(Γi) ⊆ Γ(i+1) mod 2 and D(ab)=D(a)b +(−1)p(a)aD(b)

for a, b ∈ Γ0 ∪ Γ1. We set A =Γ1, M =Γ0,andJS(Γ,D)=A + M.OnthespaceJS(Γ,D), we deﬁne multiplication by the formula a ◦ b = aD(b)+(−1)p(a)D(a)b. Then JS(Γ,D) is a Jordan superalgebra. If JS(Γ,D) is a simple superalgebra, then the superalgebra Γ is diﬀerentiably simple (see [8]). Proposition 5. The superalgebra JS(Γ,D) is not unital.

Proof. Let e be the unity of the superalgebra JS(Γ,D). Then e ∈ A ⊆ Γ1. For any a ∈ JS(Γ,D)weobtain a = e ◦ a = eD(a)+D(e)a.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 598 V. N. ZHELYABIN

2 Since the algebra Γ is supercommutative and e ∈ Γ1,wehavee =2eD(e)ande =0 1 in the algebra Γ. Consequently, ea = eD(e)a = 2 ea. This implies that eΓ=0.Then e =2eD(e)=0.

Corollary 1. The superalgebra J(A, Δ) is not isomorphic to the superalgebra JS(Γ,D).

§3. Generalized Cheng–Kac superalgebras The Cheng–Kac superalgebra. Let Γ be an associative commutative F -algebra with unity 1 and a nonzero derivation D. Consider two direct sums of vector spaces:

J0 =Γ+w1Γ+w2Γ+w3Γ and

J1 = Γ+s x1Γ+s x2Γ+s x3Γs, where Γs is an isomorphic copy of the space Γ. Let a, b ∈ Γ. On the space J0 we deﬁne multiplication by setting

a · b = ab, a · wib = wiab, w1a · w1b = w2a · w2b = ab,

w3a · w3b = −ab, wia · wjb =0 for i = j.

We put xi×i =0,x1×2 = −x2×1 = x3, x1×3 = −x3×1 = x2, x2×3 = −x3×2 = −x1 and deﬁne a bimodule action J0 × J1 → J1 by the formulas

a · sb = ab,s a · xisb = xiab,s wia · sb = xiDĞ(a)b, wia · xjsb = xi×jab.s

AbracketonJ1 is deﬁned in accordance with the rule · − · − · · sa sb = D(a)b aD(b), sa xisb = wi(ab),xisa sb = wi(ab),xisa xjsb =0.

Then the space J = J0 + J1 with the multiplication

(a0 + a1) · (b0 + b1)=(a0 · b0 + a1 · b1)+(a0 · b1 + a1 · b0),

where a0,b0 ∈ J0 and a1,b1 ∈ J1, is an algebra, denoted by CK(Γ,D). As is known (see [5, 8]), CK(Γ,D) is a Jordan superalgebra, which is simple if and only if Γ is D-simple. Now let Γ = F [x, y]/(f(x, y)F [x, y], where ∂ ∂ f(x, y)=x2 + y4 − 1andD =2y3 − x . ∂x ∂y Consider the Jordan superalgebra J(A, Δ) = A + MĎ constructed above. In the superalgebra CK(Γ,D), consider the subspace

GCK(A, Δ) = A + w1A + w2A + w3A + MĎ + x1MĎ + x2MĎ + x3M.Ď InthealgebraΓwehaveM 2 ⊆ A. Therefore, GCK(A, Δ) is a subsuperalgebra in CK(Γ,D). Thus, GCK is a Jordan superalgebra with the even part

GCK(A, Δ)0 = A + w1A + w2A + w3A and the odd part

GCK(A, Δ)1 = MĎ + x1MĎ + x2MĎ + x3M.Ď Theorem 2. Let F be an arbitrary ﬁeld of characteristic 0.ThenGCK(A, Δ) is a simple unital Jordan superalgebra.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW EXAMPLES OF SIMPLE JORDAN SUPERALGEBRAS 599

Proof. Let I = I0 + I1 be a nonzero ideal of the superalgebra GCK(A, Δ). Then K = A ∩ I0. Therefore, K + K · MĎ is an ideal of the superalgebra J(A, Δ). If K =0,then 1 ∈ K, because the superalgebra J(A, Δ) is simple. Consequently, I = GCK(A, Δ). Suppose A ∩ I0 =0andr = a + w1a1 + w2a2 + w3a3 ∈ I0.Then

w2(w2(w1r)) = a1 ∈ A ∩ I0.

Consequently, a1 = 0. Similarly, a2 = a3 = 0. Therefore, I0 = 0. This implies that I ⊆ GCK(A, Δ)1 and I · GCK(A, Δ)1 ⊆ I0 = 0. Since 1 ∈ (A, M,Ď MĎ), identity (4) shows that I ⊆ I · (A, M,Ď MĎ) ⊆ (A, I · M,Ď MĎ)+(A, I, MĎ) · MĎ =0. Consequently, GCK(A, Δ) is a simple superalgebra.

I take an opportunity to express my special gratitude to A. P. Pozhidaev whose re- marks helped to improve this paper.

References

[1]V.G.Kac,Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra 5 (1977), no. 13, 1375–1400. MR0498755 (58:16806) [2] I. L. Kantor, Jordan and Lie superalgebras, determined by a Poisson algebra, Algebra and Analysis (Tomsk, 1989), Amer. Math. Soc. Transl. Ser. 2, vol. 151, Amer. Math. Soc., Providence, RI, 1992, pp. 55–80. MR1191172 (93j:17004) [3] I. P. Shestakov, Prime alternative superalgebras of arbitrary characteristic, Algebra Logika 36 (1997), no. 6, 675–716; English transl., Algebra Logic 36 (1997), no. 6, 389–412. MR1657313 (99k:17006) [4] E. Zelmanov, Semisimple finite-dimensional Jordan superalgebras, Lie Algebras and Related Topics, Springer, New York, 2000, pp. 227–243. [5] C. Martinez and E. Zelmanov, Simple finite dimensional Jordan superalgebras of prime characteristic,J.Algebra236 (2001), no. 2, 575–629. MR1813492 (2002e:17042) [6] V. G. Kac, C. Martinez, and E. Zelmanov, Graded simple Jordan superalgebras of growth one,Mem. Amer. Math. Soc. 150 (2001), no. 711, 140 pp. MR1810856 (2001k:17052) [7] M. Racine and E. Zelmanov, Simple Jordan superalgebras with semisimple even part,J.Algebra 270 (2003), no. 2, 374–444. MR2019625 (2005b:17063) [8]N.CantariniandV.G.Kac,Classification of linearly compact simple Jordan and generalized Poisson superalgebras,J.Algebra313 (2007), no. 2, 100–124. MR2326139 (2008g:17033) [9] V. N. Zhelyabin, Simple special Jordan superalgebras with an associative nil-semisimple even part, Algebra Logika 41 (2002), no. 3, 276–310; English transl., Algebra Logic 41 (2002), no. 3, 152–172. MR1934537 (2003j:17048) [10] V. N. Zhelyabin and I. P. Shestakov, Simple special Jordan superalgebras with an associative even part, Sibirsk. Mat. Zh. 45 (2004), no. 5, 1046–1072; English transl., Siberian Math. J. 45 (2004), no. 5, 860–882. MR2108503 (2005i:17041) [11] I. P. Shestakov, Simple (−1, 1)-superalgebras, Algebra Logika 37 (1998), no. 6, 721–739; English transl., Algebra Logic 37 (1998), no. 6, 411–422. MR1680396 (2000e:17012) [12] V. N. Zhelyabin, Differential algebras and simple Jordan superalgebras,Mat.Tr.12 (2009), no. 2, 41–51. (Russian) MR2599424 (2011d:17054) [13] , Differential algebras and simple Jordan superalgebras, Siberian Adv. in Math. 20 (2010), no. 3, 223–230. [14] Shuen Yuan, Differentiably simple rings of prime characteristic, Duke Math. J. 31 (1964), no. 4, 623–630. MR0167499 (29:4772) [15] A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252; English transl. in Math. USSR-Izv. 11 (1977), no. 2. MR0472792 (57:12482) [16] D. King and K. McCrimmon, The Kantor construction of Jordan superalgebras, Comm. Algebra 20 (1992), no. 1, 109–126. MR1145328 (92j:17032)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 600 V. N. ZHELYABIN

[17] K. McCrimmon, Speciality and nonspeciality of two Jordan superalgebras,J.Algebra149 (1992), no. 2, 326–351. MR1172433 (93k:17060) [18] D. King and K. McCrimmon, The Kantor doubling process revisited, Comm. Algebra 23 (1995), no. 1, 357–372. MR1311793 (96b:17033)

S. L. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences 4, Academician Koptyug prospect, Novosibirsk 630090, Russia

Novosibirsk State University, Pirogov street 2, Novosibirsk, 630090, Russia E-mail address: [email protected] Received September 13, 2010

Translated by N. B. LEBEDINSKAYA

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use