Irreducible bimodules over alternative algebras and superalgebras Ivan Shestakov∗and Maria Trushina† Abstract The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field. 1 An Introduction Alternative algebras are the nearest generalization of associative algebras. An algebra A is called alternative if it satisfies the identities (x, x, y) = 0 (left alternativity), (x, y, y) = 0 (right alternativity), where (x, y, z) = (xy)z − x(yz) is the associator of the elements x, y, z. A classical example of nonassociative alternative algebra is the 8-dimensional algebra of octonions (Cayley numbers), or, more generally, a Cayley-Dickson algebra. The structure theory of finite-dimensional alternative algebras was developed by M. Zorn: if A is a finite-dimensional alternative algebra, then A contains a unique maximal nilpotent ideal R (the nil radical of the algebra A), and the quotient algebra A/R is semisimple [29]. Every semisimple algebra is a direct sum of ideals which are simple algebras [28]. Every simple alternative algebra is either an associative algebra or a Cayley-Dickson algebra over its ∗ † Supported by FAPESP, Proc. 2010/50347-9 and CNPq, Proc. 3305344/2009-9 Supported by FAPESP, Proc. 2008/50141-1 1 center (see [13, 28]). For principal results of the theory of infinite-dimensional alternative algebras we refer the reader to [27]. The theory of representations of alternative algebras was initiated in the papers by R.D.Schafer [14] and N. Jacobson [6], where birepresentations of finite-dimensional alternative algebras were studied. In particular, in [14] it has been shown that any birepresentation of a semisimple alternative algebra of characteristic 0 (without any restriction on dimension) is completely reducible, and in [6] the irreducible alternative bimodules over finite-dimensional algebras were classified. It should be noted that, contrary to the associative case, for nonassociative algebras the concept of birepresentation is more natural and easily defined than that of one-sided (right or left) representation. Right representations of alternative algebras were studied in K. A. Zhevlakov [26] and A. M. Slin’ko, I. P. Shestakov [20]. In particular, in [20] some description of right irreducible alternative modules was obtained and a characterization of the quasi-regular radical in terms of representation theory was given. However, even in the finite-dimensional case the classification of irreducible alternative right modules remains incomplete. For example, until now it is not clear whether the left regular representation of a Cayley-Dickson algebra (it is irreducible) is an alternative right representation. The simple alternative superalgebras were described by E. I. Zel’manov, I. P. Shestakov [25] (in the case of characteristic different from 2 and 3) and I. P. Shestakov [17] (for arbitrary characteristic). It turned out that nontrivial examples of such superalgebras appear in the case of characteristic 2 or 3 only. The study of representations of alternative superalgebras began in N. A. Pisarenko [9]. In that paper the finite-dimensional irreducible alternative superbimodules over finite-dimensional simple alternative superalgebras of characteristic different from 2 and 3 were described. It turned out that in this case nontrivial alternative non-associative superbimodules appear in the case of the minimal nontrivial simple superalgebra, i.e. doubled field F [u] = F ⊕ F u, u2 = 1 only. In [7, 21] irreducible bimodules over the simple alternative superalgebras B(1|2) and B(4|2) of characteristic 3 and of dimensions 3 and 6 were described. The aim of our paper is to describe irreducible birepresentations of alternative algebras of arbitrary dimension and characteristic, on one side, and on the other, to describe finite-dimensional irreducible alternative superbimodules. We also describe irreducible superbimodules of any dimension and characteristic over the simple alternative superalgebras. To complete the description of irreducible infinite-dimensional superbimodules, it remains to describe such superbimodules (or to prove their absence) over prime locally nilpotent alternative superalgebras of characteristic 3. The results of the paper were announced in [22]. 2 2.1 Definitions and known results Birepresentations of alternative algebras Let A be an alternative algebra over a field F and V be an A-bimodule, i. e. V is a vector space over F such that on V left and right bilinear multiplications by the elements of the algebra A are defined: A ⊗F V → V, (a ⊗ v) 7→ a · v, V ⊗F A → V, (v ⊗ a) 7→ v · a, where a ∈ A, v ∈ V . In the vector space direct sum E = A ⊕ V define a product ∗ by (a1 + v1 ) ∗ (a2 + v2 ) = a1 a2 + a1 · v2 + v1 · a2 , where a1 , a2 ∈ A, v1 , v2 ∈ V . So E becomes an algebra over F , in which A is a subalgebra and V is an ideal with zero multiplication. The algebra E is called the split null extension of the algebra A by its bimodule V . The bimodule V is called an alternative A-bimodule, if the split null extension E = A ⊕ V is an alternative algebra. It is easy to check that a bimodule V is alternative if and only if for any a, b ∈ A, v ∈ V the following equalities holds in the algebra E (a, b, v) = (b, v, a) = (v, a, b), (v, a, a) = 0. The description of irreducible (bi)modules is one of the main problem in the (bi)representation theory in any class of algebras. For an algebra A denote by Reg A the regular bimodule V = A with the action of A given by the product in A. It is clear that the bimodule Reg O over a Cayley-Dickson algebra O is an irreducible faithful alternative bimodule which is not associative. Another examples of alternative nonassociative bimodules provide the so-called Cayley bimodules over a generalized quaternion algebra H, which are defined as follows. Let a 7→ ā be the symplectic involution in H, then an alternative H-bimodule V is called a Cayley bimodule if it satisfies the identity av = vā for any a ∈ H, v ∈ V . The Cayley H-bimodules form a category which is isomorphic to the category of left associative H-modules via the following isomorphism (see [17, Lemmas 11,12]). Let L be a left associative H-module with the action (a, v) 7→ av. Then the vector space L with the bimodule operations v · a = av, a · v = āv, is an alternative Cayley bimodule over H which we will denote as Cay L. The results of the mentioned in the introduction papers of R. D. Schafer [14] and N. Jacobson [6] lead to the following description of finite-dimensional irreducible alternative bimodules. Theorem 2.1. Let A be a finite-dimensional alternative algebra over a field F of characteristic different from 2 and V be an irreducible alternative faithful A-bimodule. Then one of the following cases holds: • A is associative and V is an associative bimodule, • A = O and V = Reg O, • A = H and V = Cay L, where L is a minimal left ideal of H. Irreducible bimodules over generalized quaternion algebras were described in [6] under the assumption that the ground field has characteristic different from 2 and centralizes the bimodule. In [17] the second author extended this result, removing the restrictions on the characteristic, dimension, and the condition of centralization. 2.2 Alternative superalgebras An algebra A is called a Z2 -graded algebra or superalgebra, if A = A0̄ ⊕ A1̄ , where Ai Aj ⊆ Ai+j , i, j ∈ Z2 . The subspaces A0̄ and A1̄ are called respectively the even and odd parts of the superalgebra A. For example, the Grassmann algebra G = G0̄ ⊕ G1̄ becomes a superalgebra, if we denote by G0̄ (G1̄ ) the submodule generated by the words of even (odd) length from generators of the algebra G. A superalgebra A = A0̄ ⊕ A1̄ is called an alternative superalgebra, if its Grassmann envelope G(A) = G0̄ ⊗ A0̄ + G1̄ ⊗ A1̄ is an alternative algebra. From this definition it follows that the alternative superalgebras are defined by the identities (ai , aj , ak ) = (−1)jk+1 (ai , ak , aj ), (ai , aj , ak ) = (−1)ij+1 (aj , ai , ak ), (a0 , a0 , A) = 0, where as ∈ As , s = i, j, k ∈ {0̄, 1̄}. Let A be an algebra. Denote by A[u] the superalgebra obtained by doubling of the algebra A: A[u] = A ⊕ A · u, (A[u])0̄ = A, (A[u])1̄ = A · u, where u is an odd√central element, u2 = α ∈ F, α 6= 0. When u2 = 1, we will denote A[u] as A[ 1 ]. C. T. C. Wall [23] proved that every simple finite-dimensional associative superalgebra over an algebraically closed field F is isomorphic to one of the following superalgebras:       0 ⋆ m ⋆ 0 m , , A1̄ = • A = Mm|n (F ), A0̄ = ⋆ 0 n 0 ⋆ n √ • A = Mn (F )[ 1 ], the doubled matrix algebra. In [25] it was proved that every nontrivial simple alternative superalgebra of characteristic different from 2 and 3 is associative. In particular, the nontrivial simple finite-dimensional alternative superalgebras over an algebraically closed field F of characteristic different from 2, 3 are exhausted by the superalgebras √ Mm|n , Mn [ 1 ]. The simple alternative superalgebras of characteristic 2 and 3 were described in [17]. In these cases nonassociative nontrivial simple superalgebras appeared. Over a field of characteristic 2, any such a superalgebra is isomorphic to one of the following two superalgebras obtained from a Cayley-Dickson algebra: • the Cayley-Dickson superalgebra O(4|4) = O = H + vH with the Z2 grading induced by the Cayley-Dickson process applying to the generalized quaternion subalgebra H, • the doubled Cayley-Dickson algebra O[u]. Note that both these superalgebras are alternative algebras. A simple nonassociative nontrivial alternative superalgebra of characteristic 3 is isomorphic to one of the following superalgebras (see [16, 17]): • superalgebra B(1|2). Let V be a two-dimensional vector space over a field F with a nonzero skew-symmetric bilinear form h·, ·i. The unital superalgebra B(1|2) = F ·1⊕V with an identity 1 is given by the grading B(1|2)0̄ = F · 1, B(1|2)1̄ = V and the supercommutative multiplication xy = hx, yi · 1 for x, y ∈ V ; • superalgebra B(4|2). Let V be the same space as above; set B(4|2)0̄ = End V ∼ = M2 (F ), B(4|2)1̄ = V with the multiplication v · a = a(v) = ā · v, u · v = h·, uiv ∈ End V, where a ∈ End V , u, v ∈ V and a 7→ ā denote the symplectic involution in End V (i. e. ha(u), vi = hu, ā(v)i for any u, v ∈ V ). It is easy to check that V = B(4|2)1̄ is a Cayley bimodule over M2 (F ) = B(4|2)0̄ ; • twisted superalgebra of vector type B(Γ, D, γ). Let Γ be an associative commutative algebra over a field F , D be a nonzero derivation of Γ, γ ∈ Γ. Denote by Γ̄ an isomorphic copy of the vector space Γ with respect to the isomorphism a 7→ ā and set B(Γ, D, γ) = Γ ⊕ Γ̄ with the multiplication a · b = ab, a · b̄ = ā · b = ij, ā · b̄ = γab + 2D(a)b + aD(b), where a, b ∈ Γ, ab is the product of the elements a and b in Γ, and with the grading B(Γ, D, γ)0̄ = Γ, B(Γ, D, γ)1̄ = Γ̄. The superalgebra B(Γ, D, γ) is simple if and only if the algebra Γ does not contain proper D-invariant ideals (i.e. D-simple). 2.3 Alternative superbimodules The alternative superbimodules are defined similarly to the non-graded case. A bimodule V = V0̄ ⊕ V1̄ over an alternative superalgebra A = A0̄ ⊕ A1̄ is called an alternative A-superbimodule, if the split null extension A ⊕ V = (A0̄ ⊕ V0̄ ) ⊕ (A1̄ ⊕ V1̄ ) is an alternative superalgebra. We will call an A-superbimodule V op = V0̄op + V1̄op opposite to an Asuperbimodule V = V0̄ + V1̄ , if V0̄op = V1̄ , V1̄op = V0̄ and A acts on it by the following rule: a · v = (−1)|a| av, v · a = va, where v ∈ V op , a ∈ A0̄ ∪ A1̄ , and |a| means the parity of a, that is, |a| = i if a ∈ Ai , i = 0̄, 1̄. It is easy to check that for any superbimodule V the identical application V → V op , v 7→ v, is an odd isomorphism between V and V op . In particular, if V is alternative, the opposite superbimodule V op is alternative as well. We sometimes will say that the bimodule V op is obtained from V by changing of parity. As in the case of algebras, the main problem of the theory of representations of superalgebras is the description of irreducible superbimodules. For alternative algebras, such bimodules are always defined either over simple algebras or over a direct sum of two simple algebras, and in the latter case the bimodule is associative. It is an important difference of the supercase that there are irreducible superbimodules over nilpotent superalgebras. This first was noted in [16], where this fact was used to construct finite-dimensional solvable non-nilpotent alternative and Jordan superalgebras. Let A = A1̄ = F x, x2 = 0. Assume that the field F contains a primitive 3-th root of 1, i. e. such element ε, that ε2 + ε + 1 = 0. Consider the Abimodule V ε (1|1) = F v0 ⊕ F v1 , where V0̄ = F v0 , V1̄ = F v1 and the action of the element u is given by the equalities v0 · x = v1 , x · v0 = εv1 , v1 · x = v 0 , x · v1 = (ε + 1)v0 . It is easy to check that V ε (1|1) is an irreducible alternative superbimodule; 2 furthermore, V ε (1|1)op ∼ = V ε (1|1). Note that the commutator bimodule V ε (1|1)(−) over the Malcev superalgebra A(−) appeared in the paper by A. Elduque and the second author [5]. It is clear that the bimodule V ε (1|1) can be considered also as a unital superbimodule over the superalgebra A♯ = F ⊕ F x obtained from A by adjoining an identity 1; obviously, it remains irreducible. If the field F does not contain a primitive 3-th root of 1 and char F 6= 3, then a four-dimensional irreducible alternative superbimodule over A can be constructed (see [16]). It is clear that for any simple alternative superalgebra A the regular superbimodule Reg A and its opposite companion (Reg A)op are irreducible alternative superbimodules. N. A. Pisarenko [9] investigated the structure of alternative superbimodules over finite-dimensional semisimple superalgebras of characteristic different from 2, 3. We call V nontrivial, if V0̄ 6= 0 and V1̄ 6= 0. Theorem 2.2 ( [9]). Let A be a finite-dimensional semisimple alternative superalgebra over an algebraically closed field F of characteristic different from 2, 3, thus A = A1 ⊕ A2 ⊕ . . . ⊕ Ak , where each summand Ai is isomorphic to either a √ Cayley-Dickson algebra O or a matrix superalgebra of types Mm|n and Mk [ 1 ] over F . Then • any alternative superbimodule over A is a direct sum of associative (Ai , Aj )bimodules and unital alternative Ai -bimodules; • any nontrivial faithful irreducible alternative superbimodule V over the simple superalgebra Ai is either associative and of the √ isomorphic to√one op op bimodules Reg M , (Reg M ) , Reg (M [ 1 ]), Reg (M [ 1 ]) , or k k m|n m|n √ Ai = F [ 1 ] = F [u], dim V = 2, V0̄ = F v0 , V1̄ = F v1 and the (unital ) action of Ai on V has one of the following forms: 1. v0 u = v1 , v1 u = 0, uv0 = −2v1 , uv1 = −v0 , 2. v0 u = v1 , v1 u = 2v0 , uv0 = 0, uv1 = v0 , 3. v0 u = 1+α v1 , δ v1 u = δv0 , uv0 = v1 , uv1 = (1 − α)v0 , where α, δ ∈ F are non-zero roots of the equation α2 + αδ + δ 2 = 1. Remark 2.1. Note that in the theorem above, the bimodules of types 1) - 3) can be defined in a unified manner as the following series of unital bimodules √ over F [ 1 ] = F ⊕ F u: V = Vλ,µ (1|1) = F v0 ⊕ F v1 , V0̄ = F v0 , V1̄ = F v1 , with the action v0 u = (3µ − λ)v1 , uv0 = 2λv1 , v1 u = 2µv0 , uv1 = (λ + µ)v0 , where λ, µ ∈ F , λ2 + 3µ2 = 1. In fact, the bimodule of type 1) is the bimodule V−1,0 (1|1), the bimodule of type 2) is isomorphic to the bimodule V0,1/√3 (1|1), √ the bimodules of type 3) are isomorphic to Vλ,µ (1|1) for λ = 2 − 2α − δ/2, µ = δ/(4λ). The irreducible unital superbimodules over the superalgebras B(1|2) and B(4|2) were described by M. C. López-Dı́az and the second author in [7]. (The classification of irreducible finite-dimensional alternative superbimodules over B(1|2) was also obtained independently by the first author in [21].) Theorem 2.3 ( [7, 21]). Any irreducible alternative bimodule over the superalgebra B = B(1|2), which is not associative, is isomorphic either to Reg B, or to (Reg B)op , or belongs to the series of irreducible bimodules V (λ, µ)(3|3) = V0̄ ⊕ V1̄ , V0̄ = VectF hv0 , v1 Ry , v0 Ry2 i, V1̄ = VectF hv1 , v0 Ry , v1 Ry2 i, with the action vRyj · y = ( vRyj+1 , µv s , j < 2, j = 2, vRyj · x = λv s Ryj + jvRyj−1 , j = 0, 1, 2, where v ∈ {v0 , v1 }, vis = v1−i and λ, µ are nonzero scalars. In addition, the superbimodules V (λ, µ) and V (λ′ , µ′ ) are
opisomorphic if and only if (λ, µ) = ′ ′ ∼ ±(λ , µ ). Moreover, V (λ, µ) = V (λ, µ) . Theorem 2.4 ( [7]). Every alternative bimodule over the superalgebra B(4|2) is completely reducible, and every irreducible bimodule up to the change of parity is isomorphic to the regular bimodule. 2.4 Useful identities We recall some useful identities that hold in any alternative algebra (see [3], [17], [27]): [ab, c] (ab, c, d) [(a, b, c), d] ((c, a, b), a, b) ((a, b, c), x, y) = = = = = a[b, c] + [a, c]b + 3(a, b, c), a(b, c, d) + (a, c, d)b − (a, b, [c, d]), (ab, c, d) + (bc, a, d) + (ca, b, d), [a, b](c, a, b), ((a, x, y), b, c) + (a, (b, x, y), c) + (a, b, (c, x, y)) −[b, (a, c, [x, y])] + ([a, c], b, [x, y]). (z, x, ty) = −(z, t, xy) + (z, x, y)t + (z, t, y)x, (z, x, yt) = −(z, t, yx) + x(z, t, y) + t(z, x, y).. (1) (2) (3) (4) (5) (6) (7) (8) We will also need the identities z(yxy) = ((zy)x)y, (zy)(xz) = (z(yx))z, (9) (10) known as right and central Moufang identities (see [27]). 3 Reduction to prime superalgebras and superalgebras with A21̄ = 0. Let A be an alternative superalgebra and V be an irreducible A-superbimodule. Denote by Ann V the annihilator of V : Ann V = {a ∈ A | a · V = V · a = 0}. It is easy to see that Ann V is an ideal of A and V is an irreducible faithful bimodule over A/Ann V. A bimodule V is called associaive if (V, A, A) = 0. One can easily check that if V is associative than (A, A, A) ⊆ Ann V , hence faithful associative bimodules exist only over associative algebras. Below V always will denote an irreducible faithful A-superbimodule which is not associative. We will use the following notations: • E = E(A, V ) = A ⊕ V , the split null extension; • (λ, ρ), the birepresentation of A associated with V , that is, λ, ρ are linear mappings from A to End V such that for any a ∈ A, v ∈ V holds λ(a) : v 7→ (−1)|a||v| a · v, ρ(a) : v 7→ v · a; • Z(V ) = ZA (V ) := {ϕ ∈ End V | [ϕ, ρ(a)]s = [ϕ, λ(a)]s = 0 ∀a ∈ A}, the centralizer of the A-bimodule V , where [a, b]s = ab − (−1)|a||b| ba; • E := Z(Reg E), the supercentroid of E; • Γ = {α ∈ E | V α ⊆ V , Aα ⊆ A}. Proposition 3.1 ( [5]). The centralizer Z = Z(V ) is a graded division algebra and Γ is a commutative superalgebra which can be embedded in Z via the application π : α 7→ α|V . Proof. By the graded version of the Schur Lemma, for an irreducible bimodule V the centralizer Z(V ) is a graded division algebra. Consider the homomorphism π : Γ → End V , α 7→ α|V , and let α ∈ ker π. Then V (Aα) = (V α)A = 0, hence Aα = 0 and α = 0. It is clear also that π(Γ ) ⊆ Z. Furthermore, let α, β ∈ Γ , v ∈ V , x ∈ A, then we have ((vα)β)x = (−1)|β||x| (vα)(xβ) = (−1)|β||x|+|α|(|x|+|β|) (v(xβ))α = (−1)|α||x|+|α||β| ((vβ)x)α = (−1)|α||β| ((vβ)α)x. Therefore, (V [α, β]s )A = 0, which implies that [α, β]s = 0 and Γ is a supercommutative subsuperalgebra of Z. ✷ Theorem 3.1. Let A = A0̄ ⊕ A1̄ be an alternative superalgebra. If there exists an irreducible faithful A-superbimodule V which is not associative then A is prime or A1̄ 6= 0, A21̄ = 0. We first prove two lemmas. Lemma 3.1. Let nonzero elements a, b ∈ A0̄ ∪A1̄ satisfy [a, b]s = (a, A, b) = 0. Then the subset (V, a, b) is an A-subbimodule of V . Moreover, if ab = ba = 0 and at least one of the elements a, b is even than (V, a, b) = 0. Proof. In fact, for any v ∈ V0̄ ∪ V1̄ , r ∈ A0̄ ∪ A1̄ we have by (2) and its superization (v, a, b)r = ±(vr, a, b) ± v(r, a, b) ± (v, r, [a, b]s ) = ±(vr, a, b) ∈ (V, a, b), r(v, a, b) = (rv, a, b) ± (r, a, b)v + (r, v, [a, b]s ) = (rv, a, b) ∈ (V, a, b), which proves that (V, a, b) is a subbimodule of V . Now, let ab = ba = 0 and a ∈ A0̄ . If (V, a, b) 6= 0 than by irreducibility (V, a, b) = V . By the Moufang identities, we have a(v, a, b) = (v, a, ba) = 0, (v, a, b)a = (v, a, ab) = 0. Therefore, 0 6= a ∈ Ann V , a contradiction. ✷ Lemma 3.2. Let nonzero ideals I, J of A satisfy IJ = JI = 0. Then I + J ⊆ A1̄ . Moreover, if A = A0̄ then A is prime. Proof. Assume that I0̄ 6= 0. By the previous lemma, the set N = (V, I, J) is an A-subbimodule of V and (V, I0̄ , J) = (V, I, J0̄ ) = 0. If i ∈ I1̄ , j ∈ J1̄ , a ∈ I0̄ , then, by the Moufang identities, for any homogeneous v ∈ V (v, i, j)a = (v, i, aj) + (v, a, ij) + (v, a, j)i = 0, a(v, i, j) = (v, i, ja) − (v, a, ji) − (−1)|v| i(v, a, j) = 0. Thus 0 6= I0̄ ⊆ Ann N , and N = 0. Consider now the sets I · V, V · I. It is easy to see that they are subbimodules of V and at least one of them is nonzero. Assume that I · V 6= 0, then I · V = V and we have J · V = J · (I · V ) ⊆ (JI) · V + (J, I, V ) = 0. Similarly, if V · I 6= 0 then we would have V · J = 0 and 0 6= J ⊆ Ann V = 0, a contradiction. Therefore, V · I = 0, and consequently (V, A, I) = 0. Furthermore, by the Moufang identity we have I · (V, A, A) ⊆ (V, A, AI) + (V, I, AA) + A · (V, I, A) = 0, which proves that (V, A, A) ⊆ Ann V (I) = {v ∈ V | v · I = I · v = 0}. It is clear that Ann V (I) is a subbimodule of V which should be zero since I 6= 0. Therefore, (V, A, A) = 0 which contradicts to non-associativity of V . The contradiction proves that I0̄ = 0. Similarly, J0̄ = 0. In particular, if I is an ideal of A with I 2 = 0 then I ⊆ A1̄ . We continue by considering the case A = A0̄ . Assume that I, J are ideals of A with IJ = 0. Then (I ∩J)2 = 0, I ∩J ⊆ A1̄ = 0, and we have JI ⊆ I ∩J = 0. Therefore, I = 0 or J = 0, and A is prime. ✷ 2 The proof of the theorem. Assume that A1̄ 6= 0 and prove that in this case A is prime. As above, it suffices to prove that if I, J are ideals in A with IJ = JI = 0 then I = 0 or J = 0. Assume that I 6= 0, J 6= 0, then by Lemma 3.2, we have I + J ⊆ A1̄ . Therefore, IA1̄ + A1̄ I ⊆ I0̄ = 0 and 0 6= A1̄ ⊆ Ann I. Since the annihilator of an ideal in A is an ideal, we may apply Lemma 3.2 to the ideals I, Ann I and to get Ann I ⊆ A1̄ . Then ✷ A21̄ ⊆ Ann I ⊆ A1̄ and A21̄ = 0, a contradiction. 4 The case A1̄ = 0. The objective of this section is to generalize the results by N.Jacobson and R.D.Schafer on irreducible alternative bimodules given in Theorem 2.1 to arbitrary dimension and characteristic. Theorem 4.1. Let A be an alternative algebra and V be an irreducible alternative faithful A-bimodule which is not associative, both A and V to be of arbitrary dimension. Then one of the following cases holds: • A = O and V = Reg O, • A = H and V = Cay L, where L is a minimal left ideal of H. Proof. By Theorem 3.1, A is prime. Hence, by [27], A is either an associative prime algebra or a Cayley-Dickson ring or is degenerate, that is, contains nonzero absolute zero divisors. (Recall that an element a ∈ A is called an absolute zero divisor (a.z.d.) if aAa = 0). In the last case, let 0 6= a ∈ A be an a.z.d. We next show that a ∈ Ann V. Indeed, let u = v · a 6= 0 for some v ∈ V . Then V is generated by u. Let (λ, ρ) : A → End V be the birepresentation of A associated with the A-bimodule V and M (A) be the subalgebra of the associative algebra End V generated by the set {λ(a), ρ(a) | a ∈ A}. Then V = uM (A), and there exists W ∈ M (A) such that v = uW = vρ(a)W = · · · = v(ρ(a)W )n . From [15, Corollary of Lemma 1, Corollary 1 of Theorem 1], it follows that the operator ρ(a)W is nilpotent. Hence v = 0, a contradiction. Therefore, A is a prime associative algebra or a Cayley-Dickson ring. Let us prove that in the first case A is a central order in a generalized quaternion algebra. Show first that A is not commutative. In fact, if it were true then by Lemma 3.1 for any a, b ∈ A we would have an A-subbimodule (V, a, b). Since (V, A, A) 6= 0, there exist a, b ∈ A such that V = (V, a, b). But then V = (V, a, b) = ((V, a, b), a, b) = [a, b](V, a, b) = 0 by (4), a contradiction. Let Vas = {v ∈ V | (v, A, A) = 0}. Since A is associative, it follows from (2) that Vas is a subbimodule of V which should be zero since V is irreducible and not associative. Since A is prime and noncommutative, A does not satisfy the identity [x, y]4 = 0. Let a, b ∈ A such that n = [a, b]4 6= 0. Assume that there exists c ∈ A such that [n, c] 6= 0. By [27, Lemma 7.5], we have ((V [n, c], A, A), A, A) = 0. Therefore, ((V [n, c], A, A) ⊆ Vas = 0 and V [n, c] ⊆ Vas = 0. Similarly, [n, c]V = 0 and [n, c] ⊆ Ann V = 0. The contradiction shows that A satisfies the identity [[x, y]4 , z] = 0. Since A is prime, the Posner theorem [12, x] implies that A is a central order in a generalized quaternion algebra. Therefore, Z(A) 6= 0 and the algebra of quotients (Z ∗ )−1 A is a generalized quaternion algebra H or a Cayley-Dickson algebra O over (Z ∗ )−1 Z. Let us show that V is a (Z ∗ )−1 A−bimodule. Lemma 4.1. Let A be a prime superalgebra, Z = Z(A)0̄ 6= 0. Then λ(z) = ρ(z) ∈ Z(V ) for any z ∈ Z and V is an irreducible faithful (Z ∗ )−1 A-bimodule. Proof. By Lemma 3.1, for any z ∈ Z, a ∈ A the set (V, a, z) is an Asubbimodule of V . If (V, a, z) = V, then, by (4) we have V = (V, a, z) = ((V, a, z), a, z) = 0, a contradiction. Hence (V, a, z) = 0, and we have (V z)a = V (za) = V (az) = (V a)z ⊆ V z, a(V z) = (aV )z ⊆ V z, which proves that V z is an A-subbimodule of V. Since V is faithful, we have V z = V for any nonzero z ∈ Z. Moreover, Ann z is an A-subbimodule of V too. Hence Ann z = 0. In other words, the aplication ρ(z) ∈ End V is inversible. We proceed to show that [V, z] = 0. By (1), the set [V, z] is an A-subbimodule of V. If [V, z] = V, then, by (2), for any a, b ∈ A (V, a, b) = ([V, z], a, b) ⊆ (ab, V, z) + a(b, V, z) + (a, V, z)b = 0, a contradiction. Thus [V, z] = 0, ρ(z) = λ(z) ∈ Z(V ) and V has a structure of a vector space over the field (Z ∗ )−1 Z. Clearly, V has also a natural structure of a (Z ∗ )−1 A−bimodule, which is evidently faithful and irreducible. ✷ If (Z ∗ )−1 A = O, a Cayley-Dickson algebra, then V ∼ = Reg O, the regular bimodule [6]. Consider an element m ∈ V corresponding to the identity 1 of O; then V = m · O. On the other hand, m · A is a nonzero A-subbimodule of V. Hence m · A = V = m · O. Consequently, for every z ∈ Z there exists a ∈ A such that m · z −1 = m · a. We have m · (1 − az) = 0. Therefore 1 = az and z −1 = a. Thus A = O is a Cayley-Dickson algebra. If (Z ∗ )−1 A = H, a generalized quaternion algebra, then V is the Cayley bimodule Cay L, that is, V is isomorphic to the left irreducible associative H-module L on which H acts as follows: v · a = av and a · v = āv, where av is the associative action of H on L and a 7→ ā is the symplectic involution in H. In this case, consider I = Ann l L = {a ∈ A | aL = 0}. Obviously, I is an ideal of A and V · I = 0. Hence (A, V, I) = 0. Using the Moufang identities, we obtain that (V, A, A)I = I(V, A, A) = 0. Note that, by (2), (V, A, A) is an A−subbimodule of V. It follows that I = 0, because V is not associative. Thus L is a faithful irreducible left A-module. Therefore, A is a primitive P I-algebra and, since A can be embedded into H, A is a P I-algebra. By Kaplansky’s Theorem, it follows that the center of A is a field. Hence A = (Z ∗ )−1 A = H. ✷ 5 5.1 The case A21̄ = 0, A1̄ 6= 0. A = F 1 ⊕ F x, x2 = λ. We will first consider the minimal possible case when dimF A1̄ = 1, A1̄ = F x, x2 = 0. The category of alternative bimodules over the superalgebra A = A1̄ = F x is isomorphic to the category of unital alternative bimodules over the superalgebra A♯ = F 1 ⊕ F x, therefore we will study the last √ one. In fact, in order to include also Pisarenko’s result on bimodules over F [ 1], we will consider the more general case when A = F 1 ⊕ F x, x2 = λ ∈ F. Recall the definition of the universal multiplicative enveloping superalgebra U (A) of an alternative superalgebra A. Let Ā be an isomorphic copy of the vector space A under the isomorphism a 7→ ā. Consider the vector space direct sum A ⊕ Ā with the Z2 -grading induced by that on A. Then the tensor algebra T (A ⊕ Ā) has a natural structure of an associative superalgebra. Denote by Ialt the ideal of T (A ⊕ Ā) generated by the set of elements a ⊗ b − ab + (−1)|a||b| (b ⊗ a − ba), ā ⊗ b̄ − ab + (−1)|a||b| (b̄ ⊗ ā − ba), ab − a ⊗ b − ā ⊗ b + (−1)|a||b| b ⊗ ā, ab + ā ⊗ b − (−1)|a||b| (b̄ ⊗ ā + b ⊗ ā), c ⊗ c − c2 , (11) where a, b ∈ A0̄ ∪ A1̄ , c ∈ A0̄ ; the elements of the last type are needed only in case of char F = 2. Since the generators of the ideal Ialt are homogeneous, the quotient algebra U (A) = T (A ⊕ Ā)/Ialt inherits the superalgebra structure from T (A). Consider a pair of linear mapping (L, R) : A → U(A) defined as follows: L : a 7→ La := ā + Ialt , R : a 7→ Ra := a + Ialt ; then for any alternative birepresentation (λ, ρ) : A → End V there exists a unique (super)algebra homomorphism ϕ : U (A) → End V such that λ = ϕ ◦ L, ρ = ϕ ◦ R. Conversely, every homomorphism ϕ : U (A) → End V defines a structure of an alternative A-bimodule on V . In other words, the categories BimodAlt –A and ModAs –U (A) are isomorphic. The (super)algebra U (A) is called the universal multiplicative enveloping (super)algebra of A. If A has a unit 1 then the quotient algebra U1 (A) = U (A)/(1 − R1 , 1 − L1 ) is called the universal unital multiplicative enveloping (super)algebra of A. Proposition 5.1. Let A = F 1 + F x, x2 = λ, A0̄ = F 1, A1̄ = F x. Then the universal unital multiplicative enveloping superalgebra U1 (A) is a free 4dimensional module over its center which is isomorphic to the polynomial ring F [t]. More exactly, it has a basis {1, a, b, ab} over F [t], where a2 = t, b2 = t − 2λ, a ◦ b = λ − t. If char F = 3 and λ = 0 then U1 (A) has a graded ideal I = F [t](ab √ −t− λ) + F [t](a − b) such that I 2 = 0, and in this case U1 (A)/I ∼ = (F [t])[ t]. If char F 6= 3 or λ 6= 0 then U1 (A) is a prime superalgebra whose central closure is a generalized quaternion algebra over the field F (t). Proof. Denote a = Rx , b = Lx , then U1 (A) = algh1, a, bi. It follows from (11) that the elements a, b satisfy the relations a2 − λ = b2 + λ = −a ◦ b. Denote t = a2 , then t is an even central element of U1 (A), and one can easily check that U1 (A) is spanned as an F [t]-module by the elements 1, a, b, ab. Moreover, a standart use of the Groebner-Shirshov basis method [2] (known also as The Diamond Lemma) shows that U1 (A) is a free F [t]-module and F [t] is a polynomial ring. If char F = 3 and λ = 0 then one can easily check that I is an ideal in U1 (A) and I 2 = 0. The quotient superalgebra Ū = U1 (A)/I has the form Ū = F [t] ⊕ F [t] ā, ā2 = t, where Ū0̄ = F [t], Ū1̄ = F [t] ā. Now let char F 6= 3 or λ 6= 0. Since U1 (A) is a free module over the center F [t], we may consider the central closure Ũ = (F [t])−1 U1 (A) which has the 1 same basis {1, a, b, ab} over the field F (t). Denote u = − t−λ ab, then u2 = and the subalgebra F (t)[u] has a structure of a composition algebra u − t(t−2λ) (t−λ)2 over F (t) with the involution u 7→ ū = 1 − u. One can easily check that Ũ ∼ = (F (t)[u], t), that is, Ũ is obtained from F (t)[u] by the Cayley-Dickson process with the parameter t [27, section 2.2]. Hence Ũ is a generalized quaternion algebra. ✷ We can now classify irreducible bimodules over the superalgebra A = F 1 + F x, x2 = λ. Theorem 5.1. Let V be a unital irreducible faithful alternative non-associative bimodule over the superalgebra A = F 1 + F x, x2 = λ. Then there exists a simple algebraic field extension K = F (α), α 6= λ, such that V is a K-vector space of dimK V ≤ 4, and up to the changing of parity we have the following possibilities. If the polynomial f (u) = u2 + (α − λ)u + α(α − 2λ) has a root ε 6= 0 in K, then V = Kv ⊕ Kvx, V0̄ = Kv, V1̄ = Kvx, and the action of A is given as follows: ε Vα,λ (1|1) : if α 6= 0 then v · x = vx, x · v = αε vx, vx · x = αv, x · vx = (α + ε − λ) v; Vλ (1|1) : if α = 0 then v · x = vx, x · v = −2vx, vx · x = 0, x · vx = −λ v. If the polynomial f (u) above is irreducible over K, then U (α) := U1 (A)/(t− α) is a graded division algebra over K, and V is isomorphic to the left regular super-module over U (α). More exactly, Vα,λ (2|2) : dimK V = 4, V has a basis {v, xv, vx, x(vx)}, V0̄ = Kv ⊕ Kx(vx), V1̄ = Kvx ⊕ Kxv, and the action of A is given as follows: v · x = vx, vx · x = αv, xv · x = x(vx) + (λ − α)v, x(vx) · x = (α − λ)vx + αxv, x · v = xv, x · vx = x(vx), x · xv = (2λ − α)v, x · x(vx) = (2λ − α)vx. Conversely, for any simple algebraic field extension K = F (α), α 6= λ, the ε above defined superbimodules Vα,λ (1|1) and Vα,λ (2|2) are irreducible and alternative. Proof. Notice first that every irreducible alternative bimodule V over a unital (super)algebra is necessary unital. In fact, by Lemma 4.1, λ(1) = ρ(1) ∈ Z(V ), which implies that W = {v − v · 1 | v ∈ V } is a subbimodule of V . Moreover, W 6= V since 1 ∈ Ann W and Ann V = 0. Thus W = 0, and V is unital. Therefore, the bimodule V in the theorem is a unital associative irreducible U -module for U = U1 (A). Let ϕ : U → End V be the representation of U corresponding to the U -supermodule V , then ϕ(t) = α lies in the even part of the centralizer of V which is a graded division algebra. Thus we may consider ϕ(U ) ⊆ End V as a homomorphic image of the superalgebra U (α) = U /(t − α) over the field K = F (α) with the same basis {1, a, b, ab}, where a2 = α, b2 = α − 2λ, a ◦ b = λ − α. It is easy to see that if α = λ then V is associative. Thus α − λ 6= 0. Furthermore, since ϕ(U ) is finitely generated over F and finite dimensional over K = F (α), it is easy to see that α is algebraic over F (see, for example, [10, Lemma 1.1]). If char F = 3 and λ = 0, then by Proposition 5.1 the superalgebra U (α) has the ideal I(α) with I(α)2 = 0. Since V is irreducible, ϕ(I) = 0, thus ϕ(U ) ∼ = U (α)/I(α) = K ⊕ K ā, ā2 = α. It is clear that in this case V as a ϕ(U )-module is isomorphic to the regular supermodule, which coincides with ε the bimodule Vα,λ (1|1) for ε = α and λ = 0. Let char F 6= 3 or λ 6= 0; then the superalgebra U (α) is simple and isomorphic to a generalized quaternion superalgebra over K, with U (α)0̄ = K1 + Kab, U (α)1̄ = Ka + Kb. Therefore, in this case ϕ(U ) = U (α), and we have the two possibilities, according to the structure of U (α)0̄ . If U (α)0̄ is a field, that is, the polynomial f (u) = u2 + (α − λ)u + α(α − 2λ) is irreducible over K, then V ∼ = Reg U (α) as a right U (α)-supermodule, and ∼ V = Vα,λ (2|2) as an A-bimodule. If the polynomial f (u) has a root ε in K then the K-subspace L = vecthab− ε, (λ − α − ε) a − α bi is a minimal graded right ideal of U (α). It produces the ε irreducible superbimodule Vα,λ (1|1). Since Ũ is simple and finite-dimensional, any other irreducible graded right U (α)-module up to the changing of parity is isomorphic to L. ✷ Remark 5.1. Observe that we have the following isomorphisms: ε2 ε1 Vα,λ (1|1)op ∼ (1|1), = Vα,λ Vα (2|2)op ∼ = Vα (2|2), where f (u) = (u − ε1 )(u − ε2 ). Remark 5.2. If the field F is algebraically closed then we have K = F ; λ ∈ {0, 1}, f (u) reducible and dimF V = 2. Moreover, if λ = 0, we may take α = 1, and for α = 0 we may take λ = 1. Thus up to the changing of ε ε parity, V = V1 (1|1), V = Vα,1 (1|1), or V = V1,0 (1|1), where in the last case 2 ε + ε + 1 = 0. 0 For λ = 1 observe also that the bimodules V1 (1|1), V2,1 (1|1) are isomorphic ε to the bimodules of type 1,2 from Theorem 2.2, and Vα,1 (1|1) is isomorphic to the bimodule of type 3 for the pair (α − 1, ε). Remark 5.3. In case when A = A1 = F x, x2 = 0, we have the isomorphism U (A) ∼ = U1 (A♯ ), hence A has the same irreducible superbimodules as A♯ does. 5.2 A21̄ = 0, the general case. We will consider now the general case of superalgebras with A21 = 0. Theorem 5.2. Let V be an irreducible faithful alternative bimodule over the superalgebra A with A21̄ = 0. Then there exist a field extension K of F , a nonzero algebraic element α ∈ K, an F -subspace T and an F -subalgebra S of K such that: i) V is a unital irredicuble faithful alternative bimodule over the superalgebra B = K ⊕ Ku, u2 = 0, and the structure of V as a B-bimodule is given by Theorem 5.1, with ρ(u)2 = α. ii) S + T u is an F -subsuperalgebra of B which is isomorphic to A. iii) K = algF hαT 2 i. iv) The action of A on V is inherited by that of B. Conversely, let K be a field extension of F , and let T, S and α be an F subspace, F -subalgebra and F -algebraic element of K such that ST + T S ⊆ T and algF hαT 2 i = K. Consider the superalgebra B = K ⊕ Ku, u2 = 0, then A = S + T u is a subsuperalgebra of B and every irreducible faithful alternative birepresentation (λ, ρ) of B with ρ(u)2 = α is so over A. Proof. Consider the subsuperalgebra Γ of the centroid E = E(E) of the split null extension E = V ⊕ A: Γ = {α ∈ E | V α ⊆ V , Aα ⊆ A}. By Proposition 5.1, Γ is supercommutative and isomorphic to a subsuperalgebra of the centralizer Z = Z(V ) which is a division superalgebra. We will identify Γ with its image in Z. Let (L, R) : A → End E be the regular birepresentation of A in E, then for any x, y ∈ A1̄ the operator D = Rx Ry is a derivation of the superalgebra E due to (2). Since D|A = 0, it follows that D ∈ Γ ⊆ Z. If Rx Ry = 0 for all x, y ∈ A1̄ then (V A1̄ )A1̄ = 0. But V A1̄ is a subbimodule of V , and if V = V A1̄ then V = (V A1̄ )A1̄ = 0, a contradiction. Thus V A1̄ = 0. It follows from (11) that Lx Ly = Rx Ry = 0, hence we have A1̄ (A1̄ V ) = 0 and A1̄ V = 0 as well. Thus A1̄ ⊆ Ann V = 0, a contradiction. Let Rx Ry 6= 0, show that then Rx2 6= 0. In fact, we have seen that 0 6= D = Rx Ry ∈ Z. Therefore, V = V D = V D2 = V Rx Ry Rx Ry . But Rx Ry = Ry Rx by (11), hence V = V Rx2 Ry2 and Rx2 6= 0. Since Rx2 is invertible on V , so is Rx . Assume now that Rz Rt = 0 for some z, t ∈ A1̄ , then Rt is not invertible and hence Ru Rt = Lu Lt = 0 for any u ∈ A1̄ . Thus V A1̄ Rt = 0 = (A1̄ V )Lt and t = 0. We proved that for any 0 6= x ∈ A1̄ the operator Rx is invertible on V . In particular, we have V1̄ = V0̄ x, V0̄ = V1̄ x. Let below v ∈ V, x, y ∈ A1̄ , a, b, c ∈ A0̄ . From super-linearized identity (10) we have (vx)(ay) ± (yx)(av) = (v(xa)y ± (y(xa))v, which gives (vx)(ay) = (v(xa))y or Rx Ray = Rxa Ry . (12) For y = x we have Rx Rax = Rx Rxa , which by invertibility of Rx implies Rxa = Rax or R[a,x] = 0, and eventually [a, x] = 0, that is, [A0̄ , A1̄ ] = 0. Furthermore, applying (12) and (13), we get R(ax)b Rx = Rax Rbx = Rax Rxb = Rxb Rax = R(xb)a Rx = Ra(xb) Rx , (13) which implies R(a,x,b) = 0 and therefore (A0̄ , A1̄ , A0̄ ) = 0. (14) Let R be the F -subalgebra of Z0 generated the elements {Rx Ry | x, y ∈ A1̄ }, then R ⊆ Γ0̄ is a commutative domain. Denote by K the field of fractions of R. Fix 0 6= x ∈ A1̄ and let α = (Rx )2 . Then for any y ∈ A1̄ , Ry = (Rx Ry )α−1 Rx . Consider the F -linear mapping τ : A1̄ → K given by τ (y) = (Rx Ry )α−1 , then Ry = τ (y)Rx . Clearly, ker τ = 0, hence T = τ (A1̄ ) is an F -vector subspace of K isomorphic to A1̄ . Furthermore, define an F -linear mapping σ : A0̄ → K by σ(a) = τ (ax) and denote S = σ(A0̄ ). Lemma 5.1. The mappings τ and σ satisfy the conditions: i) τ (ay) = σ(a)τ (y), ii) σ(ab) = σ(a)σ(b), iii) ker σ = 0, iv) Ly = τ (y)Lx , for all a, b ∈ A0̄ , y ∈ A1̄ . Proof. We have σ(a)τ (y) = (Rx Rax α−1 )(Rx Ry α−1 ) = Rax Ry Rx Rx α−2 (12) = Rax Ry α−1 = Rx Ray α−1 = τ (ay), proving i). Furthermore, σ(a)σ(b) = (12) = τ (ax)τ (bx) = (Rx Rax α−1 )(Rx Rbx α−1 ) = Rax Rbx α−1 (14) Rx Ra(bx) α−1 = Rx R(ab)x) α−1 = σ(ab), which proves ii). Let now a ∈ ker σ, then by i), τ (ay) = 0 for any y ∈ A1̄ and aA1̄ = 0. By Lemma 3.1, (V, a, x) is a subbimodule of V . If (V, a, x) = 0 then we have Ra Rx = 0 and Ra = 0. Similarly, La = 0 and a ∈ Ann V = 0. If (V, a, x) = V then V a = (V, a, ax) = 0 and aV = 0, and again a = 0. To prove iv), consider Ly = Ly αα−1 = Ly Rx Rx α−1 = Ly Lx Lx α−1 = Ry Rx Lx α−1 = τ (y)Lx ; ✷ Corollary 5.1. Let B = K ⊕ Ku, u2 = 0, with B0̄ = K, B1̄ = Ku, then the application a0 + y1 7→ σ(a0 ) + τ (y1 )u is an isomorphism of A and S + T u ⊆ B. In particular, A0̄ is an associative and commutative domain. Lemma 5.2. For any a ∈ A0̄ , s ∈ A0̄ ∪ A1̄ hold i) Ra = La ∈ Z, ii) Ra Rs = Ras . Proof. By Lemma 3.1, (V, a, b) is a subbimodule of V . If it is not zero then by (4) we have V = (V, a, b) = ((V, a, b), a, b) = [a, b](V, a, b) = 0, a contradiction. Therefore Ra Rb = Rij = Rba = Rb Ra . Furthermore, in view of (2), (13) and (14), (V, a, x) is a subbimodule of V for any x ∈ A1̄ . If (V, a, x) = 0 then Ra Rx = Rax = Rxa = Rx Ra . Moreover, in this case we have σ(a)Rx = Rax = Ra Rx , implying Ra = σ(a). Similarly La = σ(a) and La = Ra ∈ Z. Assume that (V, a, x) = V . Then for any v ∈ V we have (v, a, x)a = (v, a, ax) = (v, a, xa) = a(v, a, x), that is, Ra = La . Furthermore, we have Ra ◦ Rx = 2Rax , [Ra , Rx ] = [La , Rx ] = −Ra Rx + Rax . Summing the two equations we get 3Ra Rx = 3Rax . By (2), Ra Rx − Rax ∈ Γ1̄ ⊆ Z1̄ . Since Γ is a supercommutative domain, Γ1̄ may be non-zero only if char F = 2. Hence we have Ra Rx = Rax . ✷ Lemma 5.3. If chaF 6= 3 then Lx Rx 6∈ Z. If Lx Rx ∈ Z then char F = 3 and R x = Lx . Proof. Assume that Lx Rx ∈ Z, then we have Lx Rx Rx = Rx Lx Rx which implies Lx Rx = Rx Lx . By (11), 2Lx Rx = [Lx , Rx ]s = −(Rx )2 = −(Lx )2 . Therefore, 2Lx = −Rx , 2Rx = −Lx , implying Lx = Rx . Since x ∈ A1̄ is arbitrary, V is supercommutative, and we have (1) 3vRx Ry = 3(v, x, y) = 3(x, y, v) = [xy, v] − x[y, v] − (−1)|y||v| [x, v]y = 0, which is impossible if char F 6= 3. If char F = 3 then we have (Lx Rx − α)2 = −α2 − αLx Rx − 2αLx Rx + α2 = 0 (see Section 5.1). Therefore, if Lx Rx ∈ Z then Lx Rx = α = Rx Rx , and Rx = Lx . ✷ Return to the proof of the Theorem. Assume that Lx Rx 6∈ Z then there exists v ∈ V0̄ such that v and w = vLx Rx are linearly independent over the division algebra Z0̄ . For y, z ∈ A1̄ we have Ly Rz = (Ry Rz α−1 )(Lx Rx ) ∈ R[α−1 ]Lx Rx . Since V is irreducuble, it follows from Lemma 5.2 and (11) that vR + wR[α−1 ] = V0̄ . Thus we have V0̄ = vR + wR[α−1 ] ⊆ vK + wK ⊆ vZ0̄ + wZ0̄ ⊆ V0̄ , which implies that R = K = Z0̄ . If Lx Rx ∈ Z then char F = 3, Lx = Rx , and for any 0 6= v ∈ V0̄ we have V0̄ = vR, which implies as above R = K = Z0̄ . Notice that Ry Rz = τ (y)τ (z)α ∈ T 2 α, hence K = alghT 2 αi. Define a unital K-linear action of B on V by setting v · u = vRx , u · v = vLx , then evidently V is an alternative irreducible faithful B-bimodule. The restriction of this action on S + T u ∼ = A coincides with the A-bimodule structure of V . Conversely, let K be a field extension of F , and let T, S and α be an F subspace, F -subalgebra and F -algebraic element of K such that ST + T S ⊆ T and algF hαT 2 i = K. Consider the superalgebra B = K ⊕ Ku, u2 = 0, then A = S + T u is a subsuperalgebra of B. Let V be an irreducible unital faithful alternative B-superbimodule such that ρ(u)2 = α, and assume that W ⊆ V is an A-subsuperbimodule of V . We have for any w ∈ W, t1 , t2 ∈ T : (w · t1 u) · t2 u = (t1 t2 )(w · u) · u = (αt1 t2 )w ∈ W , hence W is a K subspace of V . Let 0 6= t ∈ T , then W ⊇ t−1 W · (tu) = W · u, and W = V . ✷ 6 6.1 Bimodules over the simple superalgebras. Reduction of prime superalgebras to simple ones We start this section with reduction of the case of prime associative superalgebras to the simple finite dimensional ones. We will say that a superalgebra A with the center Z(A) = Z0̄ ⊕ Z1̄ is an even central order in a superalgebra B if B = (Z0̄∗ )−1 A. In this case we will also call B to be the even central closure of A. Lemma 6.1. Let A be a prime associative superalgebra and let V be a faithful irreducible alternative A-superbimodule which is not associative. Then Z0̄ = Z(A)0̄ 6= 0, and the even central closure B = (Z0̄∗ )−1 A is a simple superalgebra which is finite dimensional over the even part of its supercenter (Z0̄∗ )−1 Z. Moreover, (Z0̄∗ )−1 Z ⊆ Z(V ), and V has a natural structure of a B-bimodule. Proof. Since A is associative, we can repeat the arguments from Section 4 and to get Vas := {v ∈ V | (v, A, A) = 0} = 0 and V · [N, A0̄ ] = 0, where N is the associative center of the split null extension E = A ⊕ V . Therefore A0̄ satisfies the identity [[x, y]4 , z] = 0. By [1], A is also a P I-algebra. Consider the quotient algebra A/B(A), where B(A) is the Baer radical of the algebra A. Since A/B(A) is semiprime (as algebra) and P I, it follows that there exists a nontrivial central polynomial g(x1 , . . . , xn ) of A/B(A) [12]. Hence [g(a1 , . . . , an ), A] ⊆ B(A) for arbitrary homogeneous elements ak . But the Baer radical of a prime superalgebra does not contain nonzero homogeneous elements (see [4]), therefore [g(a1 , . . . , an ), A] = 0 and Z0̄ 6= 0. Consider the prime superalgebra B = (Z0̄∗ )−1 A. Let ITbe a nonzero graded ideal of B. Then I is a prime superalgebra as well, and I Z0̄ (B) = Z0̄ (I) 6= 0. The proof of these statements repeats verbatim the non-graded case. Hence I = B and B is a simple superalgebra. It is also clear that Z0̄ (B) = (Z0̄∗ )−1 Z is a field. Due to [23], B is simple as an algebra or there exists a proper (non-graded) ideal I in B such that B = π0 (I) ⊕ π1 (I), where πi : I −→ Bi are the natural projections. It is easy to see that in this case B ∼ = B0̄ [u] where B0̄ is simple. In the first case, since B is a simple PI-algebra it follows that B is finitedimensional over Z(B). But Z(B) is finite-dimensional over Z0̄ (B). Indeed, if Z1̄ (B) 6= 0 then, fixing 0 6= z1 ∈ Z1̄ (B), we have z12 = α 6= 0 ∈ Z0̄ (B) and then for any x ∈ Z1̄ (B) we get x = x(α−1 z12 ) = (α−1 xz1 )z1 ∈ Z0̄ (B) · z1 . In the second case B0̄ is a simple P I-algebra, hence B0̄ is finite-dimensional over Z(B0̄ ). It is easily seen that B is finite-dimensional over Z(B0 ) too and, in this case, Z(B) = Z(B0̄ )[u]. By Lemma 4.1, Z(A) is contained in the centralizer Z(V ) of the bimodule V , which is a graded division superalgebra. Therefore, (Z0̄∗ )−1 Z ⊆ Z and V has a natural structure of a B-bimodule. Clearly, V is irreducible and faithful over B. ✷ In non-associative case we have Lemma 6.2. Let A be a prime alternative non-associative non-trivial superalgebra of char 6= 3 and let V be a faithful irreducible alternative A-superbimodule. Then char F = 2, Z0̄ = Z(A)0̄ 6= 0, and the even central closure B = (Z0̄∗ )−1 A is a simple superalgebra of type O(4|4) or O[u]. As above, (Z0̄∗ )−1 Z ⊆ Z(V ), and V has a natural structure of a B-bimodule. Proof. The first part of the Lemma follows from the classification of prime alternative superalgebras in [17]. The second part follows from Lemma 4.1 as above. ✷ Therefore, except for the case of char 3, the classification of irreducible alternative non-associative superbimodules over prime superalgebras is reduced to bimodules over simple central finite-dimensional superalgebras, where we call a superalgebra A over a field F central if Z(A)0̄ = F . We will show next that, modulo the superalgebras of type B = K[u], for the case of char 6= 2 it suffices to consider only non-associative superalgebras. Lemma 6.3. In the conditions of Lemma 6.1, if char F 6= 2 then the even central closure B of A is of the type B = K + Ks, s2 = λ ∈ K, λ 6= 0, and the structure of V and A is given by Theorem 5.1. Proof. By Lemma 6.1, Z = Z0̄ (A) 6= 0, A is an even central order in a simple central superalgebra B over the field K = (Z0̄∗ )−1 Z0̄ , and V is a faithful irreducible alternative bimodule over B. Let K̃ be the algebraic closure of K; then B̃ = K̃ ⊗ B is a simple central superalgebra over K̃ and Ṽ = K̃ ⊗ V is an alternative unital bimodule over B̃. If dimK̃ B̃ > 2, then by [9] Ṽ is completely reducible and all its irreducible components are associative. Hence in this case V is associative, contrary to the assumption. Therefore, dimK̃ B̃ = dimK B = 2 and B = K + Ks, s2 = λ ∈ K ∗ . ✷ Remark 6.1. Observe that, contrary to the case of algebras in Section 4, A may be not simple and √ hence in general B 6= A. As an example, consider the superalgebra B = Q[ 1] = Q ⊕ Qu, u2 = 1 and let V = Qv0 ⊕ Qv1 be the irreducible supermodule of type 3 from Theorem 2.2 for α = − 37 , δ = 78 . Let √ A = Q7 [ 1] = Q7 ⊕ Q7 u, u2 = 1, where Q7 stands for the subring of rational numbers whose denominators are not divisible by 7. Then V is an irreducible faithful alternative A-bimodule, and B = (Z0̄∗ )−1 A 6= A. If one wanted A to be an algebra over then one √just a ring, √ a field and not −x2 +1 may take in this example A = F [[x]][ 1], B = F ((x))[ 1], α = x2 +x+1 , δ = x(1 + α) where F is a field, F [[x]] and F ((x)) are the algebra of formal series and the field of formal Laurent series over F . 6.2 Superbimodules in characteristic 2. In this section, we will classify irreducible alternative non-associative superbimodules of characteristic 2. Recall that every Z2 -graded alternative algebra of characteristic 2 is an alternative superalgebra. The simple superalgebras O(4|4) and O[u] provide us the irreducible superbimodules Reg O(4|4), Reg (O[u]). Furthermore, consider a generalized quaternion algebra H with the Z2 grading coming from the Cayley-Dickson process; then it is an alternative superalgebra which we denote as H(2|2). As in the non-graded case, the category of Z2 -graded Cayley bimodules over H(2|2) is isomorphic to the category of (graded) left H(2|2)-modules. For a left H(2|2)-module L we again denote by Cay L the corresponding Cayley H(2|2)-bimodule. Then Cay L is an alternative superbimodule over H(2|2) in the characteristic 2 case. Finally, let A be an alternative algebra and V be an A-bimodule; consider the split null extension E = E(A, V ). The double E[u] = E + Eu, u2 = α 6= 0 is an alternative superalgebra with the even part E and the odd part Eu (recall that char F = 2). Moreover, the ideal V [u] = V + V u may be considered as a bimodule over the alternative superalgebra A[u] = A + Au, and if V√ is irreducible then so is V [u]. We will denote this bimodule as V [u] or as V [ 1] when u2 = 1. Theorem 6.1. Let V be an irreducible faithful alternative non-associative bimodule over a nontrivial prime alternative superalgebra A of characteristic 2 with dim(Z ∗ )−1 Z (Z ∗ )−1 A > 2. Then, up to the graded isomorphism, we have the following possibilities: • A = H(2|2), V = Cay L, L is a minimal Z2 -graded left ideal of H(2|2); • A = H[u], V = (Cay L)[u], L is a minimal left ideal of H; • A = O(4|4), V = Reg (O(4|4)); • A = O[u], V = Reg (O[u]). Proof. Consider first the associative case. By Lemma 6.1, B = (Z ∗ )−1 A is a simple central superalgebra over the field K = (Z ∗ )−1 Z. Let K̃ be the algebraic closure of K, then B̃ = K̃ ⊗K B is a simple central superalgebra √ over K̃ and by [23] B̃ is isomorphic to Mm|k (K̃), 1 ≤ k ≤ m, or to Mn (K̃)[ 1], n > 1. Let us show that it should necessary hold m+k = n = 2. Consider Ṽ = K̃ ⊗K V ; it is a bimodule over B̃ and by [27, Lemma 7.5] we have [N, B̃]s ⊆ Ann (Ṽ , B̃, B̃), where N is the associative center of the split null extension E(B̃, Ṽ ). Notice that since B is associative and V is irreducuble, we have V = (V, B, B). Therefore V [N, B̃]s = 0, which implies easily that [N, B̃]s = 0. In particular, the following inclusion holds in B̃: [[a, x] ◦s [a, y], a] ∈ Z0̄ (B̃) = K̃ (15) for any a ∈ B̃0̄ , x, y ∈ B̃1̄ . In fact, by [27, (7.25) and Lemma 7.1], we have the inclusion [[a, b]2 , a] ∈ N (A) in any alternative algebra A. Superlinearizing this inclusion on b, we get the inclusion [[a, x] ◦s [a, y], a] ∈ N (E), (16) for any a ∈ E0̄ , x, y ∈ E0̄ ∪ E1̄ , which implies (15). Assume now that B̃ = Mm|k (K̃) with m > 1 and substitute in (15) a = e11 +e22 +e12 , x = e1,m+1 , y = em+1,1 . We obtain [[a, x] ◦s [a, y], a] = [[e1,m+1 , em+1,1 + em+1,2 ], a] = [e11 + e12 + em+1,m+1 , e11 + e22 + e12 ] = e12 ∈ 6 K̃, √ a contradiction. Similarly, assume that B̃ = Mn (K̃)[ 1] with n > 2 and substitute in (15) a = e11 + e22 + e12 , x = e13 u, y = e31 u: [[a, x] ◦s [a, y], a] = [[ue13 , u(e31 + e32 )], a] = [e11 + e12 + e33 , e11 + e22 + e12 ] = e12 6∈ K̃, a contradiction again. Returning to the superalgebra B we conclude that generalized quaternion superalgebra H(2|2) and in the Let B = H(2|2), B0̄ = K + Ki, B1̄ = Kj + Kk, j(i + 1), j 2 = β, where α, β ∈ K, (4α + 1)β 6= 0 Substituting in (16) a = i, x = j, y = v ∈ V , we get in the first case B is a second case B0̄ ∼ = H. 2 i = i + α, k = ij = (see [27, Chapter 2]). ([[i, j] ◦s [i, v], i], B, B) = ([[[v, i], j], i], B, B) = 0, that is, [[[V, i], j], i] ⊆ Vas = {v ∈ V | (v, B, B) = 0} which is a subbimodule of V . Since V is nonassociative, Vas = 0, and we have [[[V, i], j], i] = 0. Let W = {w ∈ V | [w, i] = 0}, show that W = 0. For any w ∈ W, b ∈ B we have (w, b, i) = (w, b, i2 − α) = (w, b, i2 ) = (w ◦ i, b, i) = ([w, i], b, i) = 0, that is, (W, B, i) = 0. Further, (W, B, B)◦i ⊆ (W ◦i, B, B)+W ◦(i, B, B) = 0, thus (W, B, B ◦i) ⊆ (W, B, B)◦i+(W, B, i)◦B = 0. Observe that j = j ◦i, k = k ◦ i, which implies that (W, B, B) = 0 and W ⊆ Vas = 0. Therefore, we have [[V, i], j] = 0 and similarly [[V, i], k] = 0. Let us prove that U = [V, i] = V ◦ i is a subbimodule of V . Clearly, U i + iU ⊆ U . Note that (V, B, i) = (V, B, i2 ) = (V, B, i) ◦ i ⊆ U . We have also [U, j] = [U, k] = 0. Consider [v, i]j = [v, i2 ]j = [v ◦ i, i]j = [v ◦ i, ij] − i[v ◦ i, j] + 3(v ◦ i, i, j) = [v ◦ i, k] + (v ◦ i, i, j) = (v ◦ i, i, j) ∈ (V, B, i) ⊆ U. Thus U j ⊆ U and similarly U k ⊆ U , hence U is a subbimodule of V . We have already showed that the mapping v 7→ v ◦ i is injective. Therefore U = V , and we have [V, j] = [V, k] = 0. Consider 3(v, j, j) = [vj, j] − v[j, j] + [v, j]j = 0. Similarly, we have (V, k, k) = 0, which proves that V is an alternative Z2 graded B-bimodule. Denote T = {v − [v, i] | v ∈ V }, then clearly T i + iT ⊆ T . Furthermore, [v, i]j = [v, ij] − i[v, j] + 3(v, i, j) = 3(v, i, j) = [vi, j] − v[i, j] − [v, j]i = v[i, j] = vj, and similarly [v, i]k = vk. Thus T j = T k = 0 and T is a subbimodule of V . Since V is faithful, T = 0 and we have v ◦ i = v for any v ∈ V . Therefore, bv = v b̄ for any b ∈ B, and V is a Cayley bimodule over B. Since V is irreducible, V = Cay L, where L is an irreducible left H(2|2)-module. It is isomorphic to a minimal graded left ideal of B = H(2|2). As at the end of the proof of Theorem 4.1, we get that L is an irreducible graded left A-module and A is a graded primitive P I-algebra. By the graded version of Kaplansky’s theorem, A is graded simple. Hence Z(A) is a graded division algebra, K = Z0̄ and A = B = H(2|2). √ Consider now the case when B̃ = M2 (K̃)[ 1] = M2 (K̃) + M2 (K̃)u. We have Z1̄ (B̃) = K̃u 6= 0, therefore Z1̄ (B) 6= 0 and B is not simple as an algebra. Consequently B = B0̄ [u] = H[u] = H + Hu, u2 = α 6= 0. Take a ∈ H and consider W = (V, a, u). It follows from (2) that W is a subbimodule of V . Assume that W = V , then V = ((V, a, u), a, u). But we have for any v ∈ V by super-linearized (4) ((v, a, u), a, u) = ((u, a, u), a, v) + [a, u](v, a, u) + [a, v](u, a, u) = 0, a contradiction. Therefore (V, H, u) = 0, and by (3) [u, (V, H, H)] ⊆ (u, V, H) = 0. Let us show that if [u, v] = 0 then (u, u, v) = 0. In fact, by (1), 3(u, u, v) = [u2 , v] − u[u, v] − [u, v]u = 0. Thus ((V, H, H), u, u) = 0. Consider now the space (V, u, u); it is also a subbimodule of V . Assume that (V, u, u) = V , then we have (V, H, H) = ((V, u, u), H, H) = ((V, H, H), u, u) = 0. We have already seen that (V, H, u) = 0 as well. Take a, b ∈ H and consider (8) (v, a, bu) = (v, u, ba) + u(v, a, b) + a(v, u, b) = 0. Therefore (V, H, B) = 0, which imlies by (2) that [V, H] ⊆ Vas = 0. Consequently, V is an associative and commutative H-bimodule. But it is impossible since for any a, b ∈ H and v ∈ V we would have (ab)v = a(bv) = a(vb) = (vb)a = v(ba) = (ba)v, [a, b]v = v[a, b] = 0, and [a, b] = 0, a contradiction. Therefore, (V, u, u) = 0. It easy now to see that (V, u, B) = 0 and then [V, u] ⊆ Vas = 0. Therefore, u ∈ Z(E(V, B)) and V = V0̄ + V0̄ u = V0̄ [u]. Clearly, V0̄ is an irreducible alternative H-bimodule. If V0̄ were associative then V would be associative as well, which is not the case. Therefore, V ∼ = Cay L where L is a minimal left ideal of H. As in the previous case, A0̄ is a primitive P I-algebra and Z(A0̄ ) = Z(A)0̄ is a field. Therefore A = B. Assume now that B = O[u] = O + Ou, u2 = λ ∈ K, λ 6= 0, where O is a central Cayley-Dickson algebra over K. As above, we have (V, u, O) = [u, (V, O, O)] = ((V, O, O), u, u) = 0. Assume that (V, u, u) = V , then (V, O, O) = 0 and V is a unital associative bimodule over O, which is impossible. Hence (V, u, u) = 0. As above, we have (V, u, O) = [V, u] = 0, hence u ∈ Z1̄ (E(V, B)) and V = V0̄ [u]. It is clear now that V0̄ ∼ = Reg O. As in the proof of Theorem 4.1, we have also A = B. Let finally B = O(4|4). Then B0̄ = H and B1̄ is a Cayley bimodule over H. By [17, Lemma 12], V = Va ⊕ Vc , where Va is an associative Hbimodule and Vc is a Cayley H-bimodule. Moreover, by [7, Lemma 3.2], we have Va B1̄ + B1̄ Va ⊆ Vc , Vc B1̄ + B1̄ Vc ⊆ Va . Since V is faithful, this implies that Va 6= 0, Vc 6= 0. Let Za = {v ∈ Va | [v, H] = 0}. As in the proof of Lemma 3.3 in [7] we have that Za 6= 0 and [Za , B] = 0. Furthermore, 3(Za , B, B) ⊆ [BB, Za ]+B[B, Za ]+[B, Za ]B = 0. Hence 0 6= Za ⊆ Z(E(V, B)). Choose some homogeneous element 0 6= u ∈ Za , then the subspace u · B is a B-subbimodule of V and the mapping ϕ : a 7→ u · a is a B-bimodule homomorphism of Reg B onto uB, in case when u is even, or of (Reg B)op onto uB, in case when u is odd. Since both Reg B and (Reg B)op are irreducible, and ϕ(1) = u 6= 0, we have that uB = V is isomorphic to Reg B or to (Reg B)op . Clearly, in both cases we have A = B. ✷ 6.3 Superbimodules over the superalgebra B(Γ, D, γ). In this section we consider the last remained case of superbimodules in characteristic 3. Unlike the other cases, we can not classify irreducible alternative superbimodules of characteristic 3 in any dimension, since the prime alternative superalgebras of characteristic 3 are classified only under certain restrictions. Nevertheless, due to [8], every finite dimensional non-associative prime alternative superalgebra A is simple. Thus, if char A = 3 then A is one of the superalgebras B(1|2), B(4|2), B(Γ , D, γ). The supermodules over the su- peralgebras B(1|2), B(4|2) were classified in [7, 21] (see Theorems 2.3, 2.4). Therefore, to finish the classification of irreducible finite dimensional superbimodules, it suffices to classify them over the superalgebra B(Γ , D, γ). We will do this for any dimension. Below V denote an irreducible faithful bimodule over the superalgebra B = B(Γ , D, γ) which is not associative. We start with the studying the structure of V as a bimodule over B0̄ = Γ . Lemma 6.4. (V, Γ , Γ ) = [V , Γ ] = 0 . Proof. Fix a, b ∈ Γ . Since Γ is commutative and (B, Γ , Γ ) = 0 , by (2) it follows that (V, a, b) is a subbimodule of V. If (V, a, b) = V, then V = (V, a, b) = ((V, a, b), a, b) = 0 by (4). Thus, (V, a, b) = 0. Consider [V, a]. Since [Γ , B ] = 0 and char B = 3, it follows from (1) that [V, a] is a subbimodule of V. Hence, [V, a] = 0 or [V, a] = V. Note that D(a3 ) = 3a2 D(a) = 0. We will show next that if D(b) = 0 for some b ∈ Γ then [V, b] = 0. Let c, d ∈ Γ . We have ¯ = bc ¯ · d¯ − b · (γcd + 2D(c)d + cD(d)) (b, c̄, d) = γbcd + 2D(bc)d + bcD(d) − γbcd − 2bD(c)d − bcD(d) = 2D(bc)d − 2bD(c)d = 0. It follows now from (2) that (V, b, x) is a subbimodule of V for any x ∈ Γ . If (V, b, x) = V then V = (V, b, x) = ((V, b, x), b, x) = 0, by superized (6). Thus (V, b, Γ̄ ) = 0. If [V, b] = V then again from (2) we get (V, B, B) = ([V, b], B, B) ⊆ (BB, V, b) + B(B, V, b) + (B, V, b)B = 0, which contradicts to the nonassociativity of V. Hence, [V, b] = 0. Now, if [V, a] = V, then V = [V, a] = [[V, a], a] = [[[V, a], a], a] = [V, a3 ] = 0, by the above. Hence, [V, a] = 0. ✷ The next proposition is an adaptation of the results from [17] to the bimodule situation. Proposition 6.1. Let V = V0̄ ⊕ V1̄ be an irreducible bimodule over an alternative superalgebra A = A0̄ ⊕ A1̄ . If N is a proper A0̄ -subbimodule of Vi , and (N, A1̄ , A1̄ ) ⊆ N, then V = (N ⊕ A1̄ N A1̄ ) ⊕ (N A1̄ ⊕ A1̄ N ). Moreover, if [N, A1̄ ]s = 0 then N = Vi . Proof. It is clear that (N +A1̄ N A1̄ )⊕(N A1̄ +A1̄ N ) is a proper subbimodule of V and hence it is equal to V . Let us prove that the sums in the summands are direct. Suppose that N ∩ A1̄ N A1̄ = L 6= 0. Then L + L ◦ A1̄ is a nonzero subbimodule of V. Hence, L + L ◦ A1̄ = V, but this contradicts to the fact that L is a proper A0̄ -subbimodule of Vi . Therefore, N ∩ A1̄ N A1̄ = 0. Now let L = N A1̄ ∩A1̄ N. Then LA0̄ ⊆ N A1̄ ·A0̄ ⊆ N A1̄ . On the other hand, LA0̄ ⊆ A1̄ N · A0̄ ⊆ (A1̄ , N, A0̄ ) + A1̄ N ⊆ A1̄ N. Hence, LA0̄ ⊆ L. Similarly, A0̄ L ⊆ L. Thus L + L ◦ A1̄ = V . But LA1̄ ⊆ N A1̄ · A1̄ ⊆ N and, similarly, A1̄ L ⊆ N. Therefore V = L + N, a contradiction. So, N A1̄ ∩ A1̄ N = 0. Finally, assume that [N, A1̄ ] = 0. Then we have (A1̄ N )A1̄ ⊆ (N A1̄ )A1̄ ⊆ (N, A1̄ , A1̄ ) + N A0̄ ⊆ N, therefore V = N ⊕ N ◦ A1̄ and N = Vi . ✷ Lemma 6.5. (V0̄ , Γ , Γ̄ ) = 0 or (V1̄ , Γ , Γ̄ ) = 0 .  Proof. Let W = v ∈ V | (v, Γ , Γ̄ ) = 0 . We show first that W 6= 0. Recall that (Γ , Γ , Γ̄ ) = 0 . By superization of (4), in view of Lemma 6.4 we have ((V, Γ , Γ̄ ), Γ , Γ̄ ) ⊆ ((Γ , Γ , Γ̄ ), V , Γ̄ ) + ((V , Γ̄ , Γ̄ ), Γ , Γ ) + ((Γ , Γ̄ , Γ̄ ), V , Γ ) + [V , Γ̄ ](Γ̄ , Γ , Γ ) ⊆ (V , Γ , Γ ) = 0 . Therefore, W 6= 0. It follows from (2) that W Γ ⊆ W . Moreover, by (6) we have ((W, Γ̄ , Γ̄ ), Γ , Γ̄ ) ⊆ (W , Γ , Γ̄ ) + (W , Γ̄ , Γ ) = 0 . Consequently, (W, Γ̄ , Γ̄ ) ⊆ W . It is clear that W is a graded subspace of V , and we see that the components Wi , i = 0, 1, satisfy the condition of Proposition 6.1. Consider [W, Γ̄ ]. Since char B = 3, it follows from (1) that [W, Γ̄ ] is a Γ -submodule of V. By superization of (7) and (8), ([W, Γ̄ ], Γ̄ , Γ̄ ) ⊆ ([Γ̄ , Γ̄ ], Γ̄ , W ) + [(Γ̄ , Γ̄ , Γ̄ ), W ] + [(Γ̄ , Γ̄ , W ), Γ̄ ] ⊆ (Γ , Γ̄ , W ) + [W , Γ̄ ] = [W , Γ̄ ]. Consequently, the subspaces [Wi , Γ̄ ], i = 0, 1, satisfy the condition of Proposition 6.1 as well. Using (1) and Lemma 6.4, we obtain (Γ̄ [W, Γ̄ ])Γ̄ ⊆ ([W, Γ̄ ]Γ̄ )Γ̄ ⊆ [W, Γ̄ ] + [W, Γ̄ ]Γ ⊆ [W , Γ̄ ]. Therefore, for every i = 0, 1, by Proposition 6.1 we have [Wi , Γ̄ ] = 0 or V = [Wi , Γ̄ ]Γ̄ + [Wi , Γ̄ ]. If [Wi , Γ̄ ] = 0, then by Proposition 6.1 Wi = Vi and (Vi , Γ , Γ̄ ) = 0 . In the second case we have V1−i = [Wi , Γ̄ ]. Then by (9), (10) we have (V1−i , Γ̄ , Γ ) = ([Wi , Γ̄ ], Γ̄ , Γ ) ⊆ [(Γ̄ , Γ̄ , Γ ), W ] + [(Γ̄ , Γ̄ , W ), Γ ] + ([Γ , Γ̄ ], Γ̄ , W ) ⊆ [V , Γ ] = 0 , hence V1−i ⊆ W . ✷ Lemma 6.6. If (V0̄ , Γ , Γ̄ ) = 0 , then [V0̄ , Γ̄ ] = 0. Proof. Similarly as it was done for W in the proof of Lemma 6.5, we can show that [V0̄ , Γ̄ ] = 0 or V = [V0̄ , Γ̄ ]Γ̄ + [V0̄ , Γ̄ ]. Consider the last case. As in the proof of Lemma 6.5 we obtain that (V1̄ , Γ , Γ̄ ) = ([V0̄ , Γ̄ ], Γ , Γ̄ ) = 0 , so (V, Γ , Γ̄ ) = 0 . Note that for any nonzero a ∈ Γ we have V a = V, hence V = V a = (V a)a = V a3 . Thus a3 6= 0. Since Γ is D-simple [17], it follows from [19] that Γ is a field. By (2) we have 0 = (V0̄ , Γ , Γ̄ ) = ([V0̄ , Γ̄ ]Γ̄ , Γ , Γ̄ ) = [V0̄ , Γ̄ ](Γ̄ , Γ , Γ̄ ). In particular, for any a ∈ Γ we have 0 = [V0 , Γ̄ ](a, 1̄, 1̄) = 2[V0 , Γ̄ ]D(a). (It was proved in [11] that any D-simple algebra contains an identity element 1.) By choosing a such that D(a) 6= 0 we obtain that [V0 , Γ̄ ] = 0. ✷ Let (A, D) be an algebra with a derivation D. An A-bimodule (V, d) with a linear mapping d : V → V is called a bimodule with derivation or a Dbimodule over the algebra (A, D), if the linear mapping D + d : a + v 7→ D(a) + d(v) is a derivation of the split null extension E = A ⊕ V . If, in addition, V has no proper d-invariant A-subbimodules, then (V, d) is called a D-simple A-bimodule. Let (V, d) be an associative and commutative D-bimodule over an algebra with a derivation (Γ , D). Consider the split null extension E = Γ ⊕ V with the derivation D +d and construct the alternative superalgebra B(E, D +d, γ). We have B(E, D + d, γ) = (Γ ⊕ V ) ⊕ Γ ⊕ V = (Γ ⊕ Γ̄ ) ⊕ (V ⊕ V̄ ) = B ⊕ W , where W = V ⊕ V̄ is an alternative bimodule over the superalgebra B = B(Γ , D, γ). We denote this bimodule by B(V, d, γ). From the definition, it is easy to recover the explicit action of B(Γ , D, γ) on B(V, d, γ): a · v = v · a = av, ā · v = v · ā = a · v̄ = v̄ · a = av, ā · v̄ = 2D(a)v + ad(v) + γav, v̄ · ā = 2d(v)a + vD(a) + γva. We are now ready to prove Theorem 6.2. Let V be an irreducible faithful nonassociative alternative bimodule over the simple superalgebra B(Γ , D, γ). Then, up to the changing of parity, (V0̄ , d) is a D-simple faithful associative and commutative bimodule over the algebra with derivation (Γ , D), where d(v) = −(v, 1̄, 1̄) is a derivation of V0̄ ; V1̄ = V0̄ · 1̄ is an isomorphic copy of V0̄ and V ∼ = B(V0̄ , d, γ). Proof. By Lemmas 6.4, 6.5, 6.6, we may assume, up to the changing of the parity, that (V0̄ , Γ , B ) = [V0̄ , B ] = 0 . Let d : V0̄ → V0̄ , d : v 7→ 2(v, 1̄, 1̄), then it follows from (2) that (V0̄ , d) is an associative and commutative D-bimodule over (Γ , D). For v ∈ V0̄ , denote v̄ = v · 1̄, then we have av̄ = a(1̄v) = (a1̄)v = āv = (1̄a)v = 1̄(av) = av, āv̄ = ā(1̄v) = (ā1̄)v − (ā, 1̄, v) = (2D(a) + γa)v − (a1̄, 1̄, v) (7) = (2D(a) + γa)v − (1̄ · 1̄, a, v) − (1̄, 1̄, v)a − (1̄, a, v)1̄ = 2D(a)v + d(v)a + γav, and similarly v̄ā = 2d(v)a + vD(a) + γva. Therefore, V ∼ = B(V0̄ , d, γ). The converse is also true: ✷ Proposition 6.2. For any D-simple faithful associative and commutative bimodule (V, d) over (Γ , D) and any γ ∈ Γ the bimodule B(V, d, γ) is an irreducible faithful alternative superbimodule over the simple superalgebra B = B(Γ , D, γ). Moreover, the B-superbimodules B(V, d, γ) and B(V ′ , d′ , γ) are isomorphic if and only if the differential bimodules (V, d) and (V ′ , d′ ) over (Γ , D) are isomorphic. Proof. Let W be a nonzero subbimodule of V , then W = W0̄ + Ū for some Γ -subbimodule U of V0̄ . We show that W0̄ 6= 0. Assume on the contrary that W0̄ = 0. Then v̄ · 1̄ = 1̄ · v̄ = 0 for any v̄ ∈ W1̄ = U . Hence d(v) = 0 for any v ∈ U. It follows that U is a nonzero D-submodule of V0̄ . Thus U = V0̄ and 0 = d(va) = vD(a) for any v ∈ V0̄ , a ∈ Γ . Consequently, D(Γ ) = 0 , a contradiction. Therefore W0̄ is a nonzero Γ -subbimodule of V0̄ and W0̄ ⊆ W1̄ . For any v ∈ W0̄ , 1̄ · v̄ = γv + d(v) ∈ W0̄ . Hence d(W0̄ ) ⊆ W0̄ and W0̄ = V0̄ . Finally, if the superbimodules B(V, d, γ) and B(V ′ , d′ , γ) are isomorphic then their even parts V0̄ and V0̄′ are isomorphic associative and commutative bimodules over Γ . Moreover, the derivations d and d′ are defined via the same mapping v 7→ 2(v, 1̄, 1̄), hence (V, d) ∼ ✷ = (V ′ , d′ ). Clearly, for any D-simple algebra (Γ , D) the regular bimodule with derivation Reg (Γ , D) is a D-simple bimodule over (Γ , D). Moreover, for any a ∈ Γ the map D + Ra : Γ → Γ is a derivation of the bimodule Reg Γ . In fact, for any v ∈ Reg Γ , b ∈ Γ we have (D + Ra )(vb) = D(vb) + vba = D(v)b + vD(b) + vba = (D + Ra )(v)b + vD(b). It is clear that the differential bimodule (Reg Γ , D + Ra ) is D-simple as well. Proposition 6.3. The D-bimodules (Reg Γ , D + Ra ) and (Reg Γ , D + Rb ) over (Γ , D) are isomorphic if and only if there exists an invertible c ∈ Γ such that D(c) = (a − b)c. In particular, if a, b ∈ F then (Reg Γ , D + Ra ) ∼ = (Reg Γ , D + Rb ) if and only if a − b ∈ Spec D. Proof. Let ϕ : (Reg Γ , D + Ra ) → (Reg Γ , D + Rb ) be an isomorphism of D-bimodules over (Γ , D). Let c = ϕ(1), then for any x ∈ Γ we have ϕ(x) = ϕ(1 · x) = ϕ(1)x = cx. Since ϕ is an isomorphism, cΓ = Γ and c is invertible. Futhermore, ϕ((D + Ra )(x)) = cD(x) + cax = (D + Rb )(ϕ(x)) = D(cx) + bcx = D(c)x + cD(x) + bcx, which gives (D(c) − (a − b)c)Γ = 0 . Since Γ is D-simple, this proves that D(c) = (a − b)c. Conversly, if c is an invertible element that satisfies this relation then the application x 7→ cx is an isomorphism of D-bimodules (Reg Γ , D + Ra ) and (Reg Γ , D + Rb ). It remains to notice that if λ ∈ Spec D, that is, D(c) = λc for some c ∈ Γ then cΓ is a D-ideal in Γ , hence cΓ = Γ and c is invertible. ✷ We now describe the faithful D-simple bimodules over a finite dimensional D-simple associative commutative algebra (Γ , D) in the case when the ground field F is algebraically closed. Observe that due to [19] in this case Γ ∼ = 3 3 F [t1 , . . . , tn ]/(t1 , . . . , tn ). Theorem 6.3. Let (Γ , D) be a D-simple associative commutative algebra over an algebraically closed field F of characteristic 3, and let (V, d) be a D-simple finite dimensional associative and commutative bimodule over (Γ , D). Then there exist λ ∈ F and v ∈ V such that V = vΓ and the action of the derivation d is given by d(va) = v(λa + D(a)). In other words, the bimodule (vΓ , d ) is isomorphic to the D-bimodule (Reg Γ , D + λ). The parameter λ is defined uniquely modulo Spec D. Proof. Since the field F is algebraically closed and V is finite dimensional, the linear operator d : V → V has an eigen vector v with an eigen value λ ∈ F . For any a ∈ Γ we have d(va) = d(v)a + vD(a) = v(λa + D(a)), which proves that vΓ is a d-subbimodule of V . Clearly, vΓ 6= 0 and therefore V = vΓ . Consider the mapping ϕ : Reg Γ → V , a 7→ va. Clearly, ϕ is an isomorphism of Γ -modules. Moreovere, we have ϕ((D + λ)(a)) = v(D(a) + λa) = d(va) = d(ϕ(a)), which shows that (V, d) ∼ = (Reg Γ , D + λ). The last statement follows from Proposition 6.3. ✷ As a result, for the finite dimensional alternative superalgebras over an algebraically closed field we have a complete description of irreducible nontrivial bimodules, which are not associative. Theorem 6.4. Let A be a finite dimensional alternative superalgebra over an algebraically closed field F , V be an irreducible faithful nontrivial alternative superbimodule over A, which is not associative. Then either dim A ≤ 2, A1̄ = F x, x2 = 0 and V ∼ = V ε (1|1), ε2 + ε + 1 = 0, or the superalgebra A is simple and V is either isomorphic to one of the bimodules Reg A, (Reg A)op , or one of the following cases holds: √ • A = F [ 1 ] and V is one of the two-dimensional bimodules of types 1) – 3) described in Theorem 2.2; • char F = 3, A = B(1|2), V = Vλ,µ (3|3); • char F = 3, A = B(Γ , D, γ), V = B(Reg Γ , D + λ, γ), λ 6∈ Spec D; • char F = 2, A = M1|1 (F ), V = Cay L, L is a minimal Z2 -graded left ideal of M1|1 (F ); √ √ • char F = 2, A = M2 (F )[ 1], V = (Cay L)[ 1], L is a minimal left ideal of M2 (F ). 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