On a problem by Nathan Jacobson

arXiv:2007.09313v1 [math.RA] 18 Jul 2020 On a problem by Nathan Jacobson Victor H. López Solís Ivan P. Shestakov Departamento Académico de Matemática Instituto de Matemática e Estatística Facultad de Ciencias Universidade de São Paulo, Universidad Nacional Santiago Antúnez de Mayolo São Paulo. Brasil Huaraz, Perú and vlopezs@unasam.edu.pe Sobolev Institute of Mathematics, Novosibirsk, Russia ivan.shestakov@gmail.com Abstract We prove a coordinatization theorem for unital alternative algebras containing 2 × 2 matrix algebra with the same identity element 1. This solves an old problem announced by Nathan Jacobson [2] on the description of alternative algebras containing a generalized quaternion algebra H with the same 1, for the case when the algebra H is split. In particular, this is the case when the basic field is finite or algebraically closed. Keywords: Alternative algebras, Kronecker factorization theorem, quaternion algebra, Cayley bimodule, Plücker relations 1 Introduction. The classical Wedderburn Coordinatization Theorem says that if a unital associative algebra A contains a matrix algebra Mn (F ) with the same identity element then it is itself a matrix algebra, A ∼ = Mn (D), “coordinated” by D. Generalizations and analogues of this theorem were proved for various classes of algebras and superalgebras [2, 4, 6, 7, 8, 9, 10, 12, 13]. The common content of all these results is that if an algebra (or superalgebra) contains a certain subalgebra (matrix algebra, octonions, Albert algebra) with the same unit then the algebra itself has the same structure, but not over the basic field rather over a certain algebra that “coordinatizes” it. The Coordinatization Theorems play important role in structure theories, especially in classification theorems, and also in the representation theory, since quite often an algebra A coordinated by D is Morita equivalent to D, though they could belong to different classes (for instance, Jordan algebras are coordinated by associative and alternative algebras). 1 In this paper we consider alternative algebras. Recall that an algebra A is called alternative if it satisfies the following identities: x2 y = x(xy), (xy)y = xy 2 . (1) for all x, y ∈ A. All associative algebras are clearly alternative. A classical example of a non-associative alternative algebra is the Cayley (or generalized octonion) algebra O (see [3, 13, 5, 16]). Kaplansky [4] proved an analogue of Wedderburn’s theorem for alternative algebras containing the Cayley algebra. He showed that if A is an alternative algebra with identity element 1 which contains a subalgebra B isomorphic to a Cayley algebra and if 1 is contained in B, then A is isomorphic to the Kronecker product B ⊗ T , where T is the center of A. Jacobson gave a new proof of Kaplansky’s result, using his classification of irreducible alternative bimodules, and in addition proved an analogue of this theorem for Jordan algebras [2], where the role of Cayley algebra is played by the Albert algebra, the exceptional simple Jordan algebra of dimension 27. These results have important applications in the theory of representations of alternative and Jordan algebras [1, 3]. The Wedderburn coordinatization theorem in the case n ≥ 3 admits a generalization for alternative algebras, since every alternative algebra A which contains a subalgebra Mn (F ) (n ≥ 3) with the same identity element is associative (see [13, Corollary 11, Chapter 2]). The result is not true for n = 2, the split Cayley algebra and its 6-dimensional subalgebra are counterexamples. The problem of description of alternative algebras containing M2 (F ) or, more generally, a generalized quaternion algebra H with the same identity element was posed by Jacobson [2]. In this paper, we solve this problem for the split case H ∼ = M2 (F ), without any restriction on the dimension and characteristic of the base field F . Our M2 (F )-coordinatization involves two ingredients: an alternative M2 (F )-algebra A is “coordinated” by an associative algebra B and by a commutative B-bimodule V (that is, V is annihilated by any commutator of elements of B), on which a skew-symmetric mapping is defined with values in the center of B, satisfying Plücker relations. More exactly, A = M2 (B) ⊕ V 2 , with a properly defined multiplication. The details are given in the Main Theorem 5.1. The paper is organized as follows. In Sections 1-2 we give definitions and some known results on alternative algebras ans bimodules. In Section 3 we prove that a unital alternative algebra A containing the generalized quaternion algebra H with the same unit admits a Z2 grading A = Aa ⊕ Ac with associative 0-component Aa . In the next section we determine multiplication in the 1-component Ac . In Section 5 we prove The Main Theorem on M2 (F )coordinatization of alternative algebras. Section 6-7 are devoted to examples and open questions. Throughout this paper the ground field F is of arbitrary characteristic. 2 2 Definitions and known results Let A be an arbitrary algebra. Denote by (x, y, z) = (xy)z − x(yz) the associator of the elements x, y, z ∈ A and by [x, y] = xy − yx the commutator of the elements x, y ∈ A. For subsets B, C, D of A, we denote by (B, C, D) the associator space generated by all the associators (b, c, d), b ∈ B, c ∈ C, d ∈ D. The associative center N (A), the commutative center K(A) and the center Z(A) are respectively defined as follows: N (A) = {a ∈ A|(a, A, A) = (A, a, A) = (A, A, a) = 0}, K(A) = {a ∈ A|[a, A] = 0}, Z(A) = N (A) ∩ K(A). In term of associators, identities (1) defining alternative algebras can be written as (x, x, y) = 0, (x, y, y) = 0. The first of them is called the left alternative identity and the second one, the right alternative identity. Linearizing the left and right alternative identities, we obtain (x, z, y) + (z, x, y) = 0, (x, y, z) + (x, z, y) = 0, which show that in an alternative algebra the associator is an antisymmetric function of its arguments. Also, these identities can be written as (x ◦ z)y − x(zy) − z(xy) = 0, (xy)z + (xz)y − x(y ◦ z) = 0, (2) where a ◦ b = ab + ba. Throughout the article we will make use of some identities that are valid in any alternative algebra and will be mentioned at the time be required. 2.1 Alternative bimodules Let be A an alternative algebra over F and V a bimodule over A, this is, V is a vector space over F equipped with the applications A ⊗ V −→ V , a ⊗ v 7−→ av, V ⊗ A −→ V , v ⊗ a 7−→ va, a ∈ A, v ∈ V . Define on the vector space E = A ⊕ V a binary operation · : E × E −→ E by (a + v) · (b + w) = ab + (av + wb), where a, b ∈ A, v, w ∈ V . Then E with the operation (product) · becomes an algebra, the split null extension of A by bimodule V , where A is a subalgebra and V is an ideal such that V 2 = 0. Now, V is called an alternative bimodule over A if E is an alternative algebra with respect to ·. 3 Due to identities (1), a bimodule V over A is an alternative bimodule if and only if the following relationships are satisfied: (a, a, v) = 0, (a, v, b) + (v, a, b) = 0, (v, b, b) = 0, (a, v, b) + (a, b, v) = 0, for all a, b ∈ A, v ∈ V . Let A be a composition algebral (see [3, 5, 13, 16]). Recall that A is a unital alternative algebra, it has an involution a 7→ a∗ such that the trace t(a) = a+a∗ and norm n(a) = aa∗ lie in F . An alternative bimodule V over a composition algebra A is called a Cayley bimodule if it satisfies the relation av = va∗ , (3) where a ∈ A, v ∈ V , e a → a∗ is the canonical involution in A. Typical examples of composition algebras are the algebras of (generalized) quaternions H and octonions O with symplectic involutions. Recall that O = H ⊕ vH, with the product defined by a · b = ab, a · vb = v(a∗ b), vb · a = v(ab), va · vb = (ba∗ )v 2 , (4) where a, b ∈ H, 0 6= v 2 ∈ F, a 7→ a∗ is the symplectic involution in H. The subspace vH ⊂ O is invariant under multiplication by elements of H and it gives an example of a Cayley bimodule over H. If H is a division algebra then vH is irreducible, otherwise H ∼ = M2 (F ) and vH = hve22 , −ve12 i ⊕ h−ve21 , ve11 i, where M2 (F )-bimodules hve22 , −ve12 i and h−ve21 , ve11 i are both isomorphic to the 2dimensional Cayley bimodule Cay = F · m1 + F · m2 , with the action of M2 (F ) given by eij · mk = δik mj , m · a = a∗ · m, (5) where a ∈ M2 (F ), m ∈ Cay, i, j, k ∈ {1, 2} and a 7→ a∗ is the symplectic involution in M2 (F ). We will denote the Cayley bimodule vH for division H as Cay H, and the regular (associative) H-bimodule by Reg . 3 Z2-grading A = Aa + Ac. The statement of the following result follows from [15, Lemma 11] and its proof. 4 Proposition 3.1. [15, Lemma 11]. Let A be a unitary alternative algebra over the field F which contains a composition subalgebra C with the same identity element. Suppose that a subspace V of A is C−invariant and satisfies (3). Then, the following identities are valid for any a, b ∈ C, r ∈ A, u, v ∈ V ; (ab)v = b(av), v(ab) = (vb)a, (6) a(ur) = u(a∗ r), (7) a(uv) = u(va), (uv)a = (au)v, (8) (u, v, a) = [uv, a] (9) It is important to know the structure of unitary alternative H−bimodules. Their structure is given by the following result: Proposition 3.2. [15, Lemma 12]. Every unitary alternative H−bimodule V is completely reducible and admits decomposition V = Va ⊕ Vc , where Va is an associative H−bimodule and Vc is a Cayley bimodule over H; furthermore, the subbimodule Vc coincides with the associator subspace (V, H, H). Every irreducible component of the subbimodule Va is isomorphic to the regular H-bimodule Reg , and every irreducible component of the subbimodule Vc is isomorphic to Cay H if H is a division algebra and to Cay if H ∼ = M2 (F ). Let A be an alternative algebra such that A contains H with the same identity element, so we can consider A as a unitary alternative H−bimodule. Then, by Proposition 3.2, A is completely reducible and admits the decomposition A = Aa ⊕ A c , where Aa is a unitary associative H−bimodule and Ac is a unitary Cayley H−bimodule. Denote Za = {u ∈PAa |[u, H] = 0}. Since Aa is isomorphic to a direct sum of bimodules Reg , we have Aa = i ⊕Reg i , Reg i ∼ = Reg for all i. This implies that Aa contains a set of elements {ui } (the images of 1 under the isomorphisms with Reg) such that RegiP = ui H with ui ∈ Za , and each element of Aa can be written in only one way in the form u i ai , ai ∈ H. Now, of course, Za 6= 0 and Aa = Za H. Also, by Proposition 3.2, the bimodule Ac coincides with (A, H, H) and is completely P g g reducible; this is, Ac = j ⊕Cay j , where Cay is equal to Cay H or to Cay. Therefore, A=( X ⊕Regi ) ⊕ ( X g j ). ⊕Cay The statements and demonstrations of Lemmas 3.3 and 3.4 are similar to Lemmas 3.1 and 3.2 of [7] given there for superbimodules over the superalgebra B(4, 2) = H + Cay. 5 Lemma 3.3. Let A = Aa ⊕ Ac be the decomposition of A from above. Then for any m, n ∈ Ac , a ∈ H, (mn)a = (am)n, a(mn) = m(na) (10) and for any u ∈ Aa , m ∈ Ac , a, b ∈ H, (um)a a(mu) ((um)a)b b(a(mu)) (um, a, b) (b, a, mu) = = = = = = (ua∗ )m m(a∗ u), (um)(ba), (ab)(mu), (um)[b, a], [b, a](mu). (11) (12) (13) (14) (15) (16) Proof: First, let us consider m, n ∈ Ac , a ∈ H. By (3), (mn)a − (am)n = (mn)a − (ma∗ )n = (m, n, a) − (m, a∗ , n) + m(na − a∗ n) = (m, n, a) + (m, a, n) = 0. Analogously a(mn) − m(na) = 0. This proves (10). Now let u ∈ Aa , m ∈ Ac , a, b ∈ H. Then (um)a − (ua∗ )m = (u, m, a) − (u, a∗ , m) + u(ma − a∗ m) = 0, and similarly a(mu) − m(a∗ u) = 0, which proves (11) and (12). In addition by (11) (um)a.b = (ua∗ .m)b = (ua∗ .b∗ )m = (u.(ba)∗ )m = (um)(ba), which proves (13). Similarly, by (12), we get (14). Finally, by (13) and (14) we have (um, a, b) = ((um)a)b − (um)(ab) = (um)(ba) − (um)(ab) = (um)[b, a] (b, a, mu) = (ba)(mu) − b(a(mu)) = (ba)(mu) − (ab)(mu) = [b, a](mu), which proves (15) and (16).  Lemma 3.4. The products Aa Aa , Aa Ac , Ac Aa and Ac Ac are H−invariants subspaces. Moreover Aa Ac + Ac Aa ⊆ Ac and Ac Ac ⊆ Aa . Proof: Since Aa and Ac are H−invariante, in order to prove the first part of the Lemma it suffices to show that the product of any H−invariants subspaces U and W is again H−invariant. We have by the linearized identity of the right alternativity (2) (U W )H ⊆ U (W ◦ H) + (U H)W ⊆ U W, and similarly H(U W ) ⊆ U W. 6 Now, let us demonstrate that Aa Ac + Ac Aa ⊆ Ac . Recall that by Proposition 3.2, Ac = (A, H, H). Choose a, b ∈ H such that 0 6= [a, b]2 ∈ F , then by (16) Ac Aa = [a, b]2 (Ac Aa ) ⊆ [a, b](Ac Aa ) = (a, b, Ac Aa ) ⊆ (H, H, A) = Ac , and similarly Ac Aa ⊆ Ac . Finally, for any m, n ∈ Ac and a ∈ H, we have by (10) and (6) ((mn)a)b = ((am)n)b = (b(am))n = ((ab)m)n = (mn)(ab), which proves Ac Ac ⊆ Aa .  Lemma 3.5. Aa is an associative subalgebra of A. Proof: Recall the following identities valid in every alternative algebra (see [16, 15]). (xy)(zx) [x, yz] (xy, z, t) 2[(x, y, z), t] [x, y](x, y, z) ((z, w, t), x, y) = = = = = = + x(yz)x, [x, y]z + y[x, z] − 3(x, y, z), x(y, z, t) + (x, z, t)y − (x, y, [z, t]), ([x, y], z, t) + ([y, z], x, t) + ([z, x], y, t), (x, y, (x, y, z)) = −(x, y, z)[x, y], ((z, x, y), w, t) + (z, (w, x, y), t) (z, w, (t, x, y)) − [w, (z, t, [x, y])] + ([z, t], w, [x, y]). (17) (18) (19) (20) (21) (22) (23) Let us fix arbitrary elements u, v, w ∈ Za and a, b, c ∈ H. Then by (20) ([a, b], u, v) = 2[(a, b, u), v] − ([b, u], a, v) − ([u, a], b, v) = 0. So by (19) (uv, a, b) = u(v, a, b) + (u, a, b)v − (u, v, [a, b]) = −([a, b], u, v) = 0, which implies (Za Za , H, H) = 0. By linearization of (21), choosing a, b ∈ H such that [a, b]2 = α ∈ F , α 6= 0, we have for any x ∈ Aa [a, b](u, x, c) = −[u, b](a, x, c) − [a, x](u, b, c) − [u, x](a, b, c) + (a, b, (u, x, c)) + (u, b, (a, x, c)) + (a, x, (u, b, c)) + (u, x, (a, b, c)) = (a, b, (u, x, c)) (22) = ((u, a, b), x, c) + (u, (x, a, b), c) + (u, x, (c, a, b)) − [x, (u, c, [a, b])] + ([u, c], x, [a, b]) = 0. So α(u, x, c) = [a, b]2 (u, x, c) = [a, b]([a, b](u, x, c)) = 0, thus (u, x, c) = 0, which implies (Za , Aa , H) = 0. In particular, (Za , Za , H) = 0. Then, by (18) [a, uv] = [a, u]v + u[a, v] − 3(a, u, v) = 0, 7 and so [H, Za Za ] = 0. Therefore Za Za ⊆ Za . By linearization of (21), we have [a, b](u, v, w) = −[a, v](u, b, w) − [u, b](a, v, w) − [u, v](a, b, w) + (a, b, (u, v, w)) + (a, v, (u, b, w)) + (u, b, (a, v, w)) + (u, v, (a, b, w)) = 0 Choose again a, b ∈ H such that [a, b]2 = α ∈ F , α 6= 0. Then α(u, v, w) = [a, b]2 (u, v, w) = [a, b]([a, b](u, v, w)) = 0, and so (u, v, w) = 0. Thus Za is an associative algebra. Consequently, by linearization of the central Moufang identity (17) and using the fact that Aa is an associative H−bimodule, we have (ua)(vb) = −(ba)(vu) + (u(av))b + (b(av))u = −(ba)(vu) + (u(va))b + ((ba)v)u = −(ba)(vu) + ((uv)a)b + (ba)(vu) = (uv)(ab). Therefore Aa Aa ⊆ Aa , that is, Aa is a subalgebra of A. Remembering that (Za , Aa , H) = 0, then for all x, y ∈ Aa we have (19) (ua, x, y) = u(a, x, y) + (u, x, y)a − (u, a, [x, y]) = u(a, x, y) + (u, x, y)a. Thus, using the last equality several times and the fact that Za is associative, we have (ua, vb, wc) = u(a, vb, wc) + (u, vb, wc)a = −u(v(b, a, wc) + (v, a, wc)b) − (v(b, u, wc) + (v, u, wc)b)a = −((w(c, v, u) + (w, v, u)c)b)a = 0. Therefore, Aa is an associative subalgebra of A.  It follows immediately from Lemmas 3.4 and 3.5, the following result. Corollary 3.6. A = Aa ⊕ Ac is a Z2 −graded algebra, where Aa is the even part and Ac is the odd part of the Z2 −grading of A. In what follows we will use in a permanent way the fact that A is an Z2 −graded alternative algebra. Thus, we have (Ac , H, Ac ) ⊆ (Ac , Aa , Ac ) ⊆ Aa , and [Za , Ac ] ⊆ Ac . Lemma 3.7. [Za , Ac ] = (Za , A, A) = 0. 8 Proof: Let us fix arbitrary elements u, v, w ∈ Za , m, n ∈ Ac and a, b, c ∈ H. In the proof of the previous Lemma we have shown that (Za , Aa , H) = 0. So, let us generalize the previous equality and show first (Za , A, H) = 0. (24) By the fact that Ac is a Cayley H−bimodule, we have (a, u, m) = (au)m − a(um) = (au)m − (um)a∗ = (au)m − (ua)m = [a, u]m = 0, which proves (H, Za , Ac ) = 0. Thus, (Za , A, H) ⊆ (Za , Aa , H) + (Za , Ac , H) = 0, which proves (24). In addition, consider the identity ([x, y], y, z) = [y, (x, y, z)] (25) which is valid in every alternative algebra. Using its linearization, we obtain ([u, m], a, b) = −([u, a], m, b) + [m, (u, a, b)] + [a, (u, m, b)] = 0. Thus ([Za , Ac ], H, H) = 0, that is, [Za , Ac ] ⊆ Aa . Therefore [Za , Ac ] ⊆ Aa ∩ Ac = 0, which implies [Za , Ac ] = 0. By linearization of (21), by (22), choosing a, b ∈ H such that 0 6= [a, b]2 = α ∈ F , we have [a, b](u, m, n) = −[u, b](a, m, n) − [a, m](u, b, n) − [u, m](a, b, n) + (a, b, (u, m, n)) + (u, b, (a, m, n)) + (a, m, (u, b, n)) + (u, m, (a, b, n)) = (u, m, (a, b, n)) = ((u, m, a), b, n) − ((u, b, n), m, a) − (u, (m, b, n), a) + [m, (u, a, [b, n])] − ([u, a], m, [b, n]) = 0, hence α(u, m, n) = [a, b]2 (u, m, n) = [a, b]([a, b](u, m, n)) = 0 and (u, m, n) = 0; therefore, (Za , Ac , Ac ) = 0. Also [a, b](u, n, v) = −[u, b](a, n, v) − [a, n](u, b, v) − [u, n](a, b, v) + (a, b, (u, n, v)) + (u, b, (a, n, v)) + (a, n, (u, b, v)) + (u, n, (a, b, v)) = (a, b, (u, n, v)) = −(u, (a, n, v), b) − (u, a, (b, n, v)) + ((u, a, b), n, v) + [a, (u, b, [n, v])] − ([u, b], a, [n, v]) = 0, 9 so α(u, n, v) = [a, b]2 (u, n, v) = [a, b]([a, b](u, n, v)) = 0 and (u, n, v) = 0; thus, (Za , Ac , Za ) = 0. Then by (19) and (24) (ua, v, m) = u(a, v, m) + (u, v, m)a − (u, a, [v, m]) = 0; so (Za H, Za , Ac ) = 0. Therefore (Aa , Za , Ac ) = 0, and we have (Za , A, A) ⊆ (Za , Aa , Aa ) + (Za , Ac , Aa ) + (Za , Ac , Ac ) = 0, so Za ⊆ N (A).  Corollary 3.8. Aa = Za ⊗F H. P P Proof: As Aa = ⊕ui H, every element of Aa can be written uniquely in the form ui ai P with ai ∈ H. We know that Aa is associative. On the other hand, let x = ui ai ∈ Za be then ax = xa for all a ∈ H. Therefore, by [Za , H] = 0 we have X X ui aai = ui ai a; P so, aai = ai a. But as H is central, we have ai = αi 1, αi ∈ F . Then Za = F ui and Aa = Za ⊗F H.  Lemma 3.9. [Za , Za ]Ac = Ac [Za , Za ] = 0. Proof: In the proof of Lemma 3.7 we have obtained [Za , Ac ] = 0. Thus, by (18) and again by Lemma 3.7 [Za , Za ]Ac ⊆ [Za , Za Ac ] − Za [Za , Ac ] + 3(Za , Za , Ac ) = 0 and similarly Ac [Za , Za ] = 0.  Remark 3.10. Note that in general Za is not commutative. For example, if A = Mn (H) then Za ∼ = Mn (F ). If A is prime and nonassociative then by [16, Corollary to Theorem 8.11] N (A) = Z(A), hence Za ⊆ Z(A) is commutative. In fact, in this case A is a Cayley-Dickson ring (see [16]). 4 Multiplication in Ac. In the previous section we described, in particular, the structure of the associative part Aa . This section is devoted to description of the multiplication in the Cayley part Ac . Here and below we will assume that the quaternion algebra H is split, that is, H ∼ = M2 (F ). We have already mentioned that the Cayley H-bimodule Ac is completely reducible and is a direct sum of bimodules isomorphic to the Cayley bimodule Cay = F · m1 + F · m2 10 from (5). Denote by V (1) and V (2) the subspaces of Ac spanned by the elements of type m1 and m2 , respectively; then the mappings π12 : V (1) → V (2), v 7→ e12 · v, π21 : V (2) → V (1), v → 7 e21 · v are mutually inverse and establish isomorphisms between V (1) and V (2). Clearly, Ac = V (1) ⊕ V (2). Let V = V (1), for any v ∈ V we denote v(1) = v, v(2) = π12 (v), then Cay(v) = F · v(1) + F · v(2) ∼ = Cay. Proposition 4.1. For any u, v ∈ V we have Cay(u) · Cay(v) = hu, vi H where h, i : V ×V → Z(A) is a skew-symmetric bilinear mapping. In particular, Cay(v)2 = 0 for any v ∈ V . Proof: We have by identities of right and left alternativity (u(1)v(1))e11 = −(u(1)e11 )v(1) + u(1)(v(1) ◦ e11 ) = u(1)v(1), e11 (u(1)v(1)) = −u(1)(e11 v(1)) + (e11 ◦ u(1))v(1) = −u(1)v(1) + u(1)v(1) = 0, which shows that u(1)v(1) ∈ e22 Aa e11 = Za e21 , hence u(1)v(1) = ze21 for some z ∈ Za . Furthermore, we have ze22 = (ze21 )e12 = (u(1)v(1))e12 = −(u(1)e12 )v(1) + u(1)(v(1) ◦ e12 ) = u(2)v(1), ze11 = e12 (ze21 ) = e12 (u(1)v(1)) = −u(1)(e12 v(1)) + (e12 ◦ u(1))v(1) = −u(1)v(2), ze12 = e12 (ze22 ) = e12 (u(2)v(1)) = −u(2)(e12 v(1)) + (e12 ◦ u(2))v(1) = −u(2)v(2), which proves that Cay(u) · Cay(v) = z H. Since z ∈ Za , we have [z, H] = [z, Ac ] = 0. Hence in order to prove that z ∈ Z(A), it remains to show that [z, Za ] = 0. Observe that z = z(e11 + e22 ) = u(2)v(1) − u(1)v(2) ∈ A2c . Therefore, [z, Za ] ⊂ [A2c , Za ] ⊆ Ac [Ac , Za ] + [Ac , Za ]Ac + 3(Ac , Ac , Za ) = 0. Finally, denote z = z(u, v) and consider e11 (u(1) ◦ v(1)) = (e11 u(1))v(1) + (e11 v(1))u(1) = u(1) ◦ v(1). On the other hand, e11 (u(1) ◦ v(1)) = e11 ((z(u, v) + z(v, u))e21 ) = 0. Hence u(1) ◦ v(1) = 0 and z(u, v) = −z(v, u). Denote hu, vi = z(u, v), then we have as above hu, vi = z(u, v) = u(2)v(1) − u(1)v(2), (26) wich proves that hu, vi is a bilinear function of u, v.  11 Lemma 4.2. For any u, v, w, t ∈ V the following identities hold hu, viw + hv, wiu + hw, uiv = 0, hu, vihw, ti + hv, wihu, ti + hw, uihv, ti = 0. (27) (28) Proof: Recall that in the proof of Proposition 4.1 we obtained the equalities u(1)v(1) u(1)v(2) u(2)v(1) u(2)v(2) = = = = hu, vi e21 , −hu, vi e11 , hu, vi e22 , −hu, vi e12 . (29) (30) (31) (32) Therefore, using the fact that hu, vi ∈ Z(A), by the linearized right alternative identity we have 0 = (u(1), v(1), w(2)) + (u(1), w(2), v(1)) = hu, vie21 w(2) + u(1)hv, wie11 − hu, wie11 v(1) − u(1)hw, vie22 = hu, viw + 0 − hu, wiv − hw, viu = hu, viw + hw, uiv + hv, wiu. which proves (27). Multiplying (27) by the element t ∈ V , we get (28).  Corollary 4.3. Let {vi | i ∈ I} be a basis of the space V and uij = hvi , vj i ∈ Z(A), then the elements uij satisfy the Plücker relations uij = −uji , uij ukl + uik ulk + uil ujk = 0. (33) An example of a family of elements uij = −uji satisfying relations (33) may be obtained by taking in an associative commutative algebra K elements a1 , . . . , an and setting uij = ai − aj . Another example, which we will use later, is the coordinate algebra of grassmanian G2,n (see, for example, [14, vol.1, p.42]). Lemma 4.4.   Consider the algebra of polynomials F [x1 , . . . , xn ; y1 , . . . , yn ], and let αij = x y det i i ∈ F [x1 , . . . , xn ; y1 , . . . , yn ]. Then the elements αij = −αij satisfy relations xj yj (33). Moreover, the algebra F [αij | 1 ≤ i < j ≤ n] is a free algebra modulo relations (33). Proof: Firstly, one can easily check that the elements αij satisfy relations (33). Furthermore, it follows from the relation α12 αij + α1i αj2 + α1j α2i = 0, 12 −1 that αij for i, j > 2 lies in the algebra F [α1i , α2j , α12 ] ⊂ F (x1 , . . . , xn ; y1 , . . . , yn ). Therefore, −1 F [αij | 1 < i ≤ n, 2 < j ≤ n] ⊆ F [α12 , . . . , α1n ; α23 , . . . , α2n , α12 ]. (34) Observe that y2 = x11 α12 + xx2 y1 1 , hence F (x1 , x2 , y1 , y2 ) = F (x1 , x2 , y1 , α12 ). Similarly, resolving with respect to xn , yn the system α1n = x1 yn − y1 xn , α2n = x2 yn − y2 xn we get x2 α1n − x1 α2n , α12 y2 α1n − y1 α2n = ; α12 xn = yn hence xn , yn ∈ F (α1n , α2n , x1 , x2 , y1 , y2 ) = F (α1n , α2n , x1 , x2 , y1 , α12 ). Therefore, F (x1 , . . . , xn , y1 , . . . , yn ) = F (x1 , x2 , y1 , α12 , . . . , α1n , α23 , . . . , α2n ). and tr.deg F (x1 , x2 , y1 , α12 , . . . , α1n , α23 , . . . , α2n ) = 2n, which means that the elements α12 , . . . , α1n , α23 , . . . , α2n are algebraically independent. Now, let F [uij ] be a free algebra modulo relations (33). Consider the epimorphism π : F [uij ] −→ F [αij ]; uij 7−→ αij . We will prove that ker π = 0. Let f (u12 , . . . , u(n−1)n ) ∈ ker π, that is, f (α12 , . . . , α(n−1)n ) = 0. Inclusions (34) follow from relations (33), hence they are valid in the algebra F [uij ] as well. Therefore, there exists k such that uk12 f (u12 , . . . , u(n−1)n ) = g(u12 , . . . , u2n ) for some g(u12 , . . . , u2n ) ∈ F [u12 , . . . , u2n ]. Clearly, g(α12 , . . . , α2n ) = 0. Since the elements α12 , . . . , α2n are algebraically independent, we have g = 0. But the algebra F [uij ] is a domain (see, for example, [11, Chapter 8]), therefore f = 0, proving the lemma.  ∼ Recall that by Corollary 3.8 we have Aa = M2 (Za ), hence A = M2 (Za ) ⊕ V (1) ⊕ V (2). Proposition 4.5. Let  X, Y ∈  A, X = Xa + x(1) + y(2), Y = Ya + z(1) + t(2), where   e f a b , a, b, c, d, e, f, g, h ∈ Za , x, y, z, t ∈ V . Then the product , Ya = Xa = g h c d XY iz given by   −hx, ti −hy, ti + (az + ct + hx − gy)(1) + (bz + dt − f x + ey)(2). XY = Xa Ya + hx, zi hy, zi 13 Proof: The proof follows from identities (5), (29) - (32), and Lemma 3.7.  We can make the formula defining the product in A more transparent by using the following notation: for u, v ∈ V we denote (u, v) = u(1) + v(2). With this notation, using usual matrix multiplication and the fact that [Za , Vc ] = 0, we have for X = Xa + (x, y), Y = Ya + (z, t)   −hx, ti −hy, ti + (z, t)Xa + (x, y)(Ya )∗ , (35) XY = Xa Ya + hx, zi hy, zi ∗   a b d −b . = where c d −c a In the next section we will prove that Proposition 4.5 in fact describes all unital alternative extensions of the algebra M2 (F ).  5 The Main Theorem Let B be an associative unital algebra and V be a left B-module such that [B, B] annihilates V . Clearly, in this case V has a structure of a commutative B-bimodule with v · b = b · v, v ∈ V, b ∈ B. Assume that there exists a B-bilinear skew-symmetric mapping h, i : V 2 → B such that hV, V i ⊆ Z(B) and formula (27) holds for any u, v, w ∈ V . Let A = M2 (B) ⊕ V 2 , where V 2 = {(u, v) | u, v ∈ V } ∼ = V ⊕ V . Let X, Y ∈ A, X = Xa + (x, y), Y = Ya + (z, t), where Xa , Ya ∈ M2 (B) and (x, y), (z, t) ∈ V 2 . Define a product in A by formula (35):   −hx, ti −hy, ti + (z, t)Xa + (x, y)(Ya )∗ . XY = Xa Ya + hx, zi hy, zi Theorem 5.1. The algebra A with the product defined above is an alternative unital algebra containing M2 (F ) with the same unit. Conversely, every unital alternative algebra that contains the matrix algebra M2 (F ) with the same unit has this form. Proof: The second part of the theorem follows from Proposition 4.5 with B = Za . Let us now prove that A is alternative. Let us first prove that V 2 is a right alternative bimodule over M2 (B). Let A, B ∈ M2 (B), (x, y) ∈ V 2 . One can easily check that (x, y)((AB)∗ − B ∗ A∗ ) = 0 (since V [B, B] = 0). Therefore, ((x, y), A, A) = ((x, y)A∗ )A∗ − (x, y)(A2 )∗ = 0. 14 Furthermore, (A, (x, y), B) + (A, B, (x, y)) = = ((x, y)A)B ∗ − ((x, y)B ∗ )A + (x, y)(AB) − ((x, y)B ∗ )A∗ = (x, y)(AB ∗ − B ∗ A + AB − B ∗ A∗ ) = (x, y)[A, B + B ∗ ] = 0 2 2 since B +B ∗ = tr(B) commutes with  A on V . Therefore, V is a right alternative bimodule  a b with a, b, c, d ∈ B, consider over M2 (B). Now let A = c d   −hx, yi 0 (A, (x, y), (x, y)) = ((x, y)A) · (x, y) − A 0 hy, xi = (xa + yc, xb + yd) · (x, y) + hx, yiA   −hxa + yc, yi −hxb + yd, yi + hx, yiA = hxa + yc, xi hxb + yd, xi   −hxa, yi −hxb, yi + hx, yiA = −hx, yiA + hx, yiA = 0. = hyc, xi hyd, xi Furthermore, = = + = + = ((x, y), A, (u, v)) + ((x, y), (u, v), A) = ((x, y)A∗ ) · (u, v) − (x, y) · ((u, v)A) + ((x, y) · (u, v))A − (x, y) · ((u, v)A∗ ) (xd − yc, −xb + ya) · (u, v) − (x, y) · (ua + vc, ub + vd)   −hx, vi −hy, vi A − (x, y) · (ud − vc, −ub + va) hx, ui hy, ui     −hx, ub + vdi −hy, ub + vdi −hxd − yc, vi −h−xb + ya, vi − hx, ua + vci hy, ua + vci hxd − yc, ui h−xb + ya, ui     −hx, −ub + vai −hy, −ub + vai −hx, via − hy, vic −hx, vib − hy, vid − hx, ud − vci hy, ud − vci hx, uia + hy, uic hx, uib + hy, uid   X11 X12 . X21 X22 We have X11 = −hxd − yc, vi + hx, ub + vdi − hx, via − hy, vic + hx, −ub + vai = −hx, vid + hy, vic + hx, uib + hx, vid − hx, via − hy, vic − hx, uib + hx, via = 0, 15 and similarly X12 = −h−xb + ya, vi + hy, ub + vdi − hx, vib − hy, vid + hy, −ub + vai = hx, vib − hy, via + hy, uib + hy, vid − hx, vib − hy, vid − hy, uib + hy, via = 0, X21 = hxd − yc, ui − hx, ua + vci + hx, uia + hy, uic − hx, ud − vci = hx, uid − hy, uic − hx, uia − hx, vic + hx, uia + hy, uic − hx, uid + hx, vic = 0, X22 = h−xb + ya, ui − hy, ua + vci + hx, uib + hy, uid − hy, ud − vci = hx, uid − hy, uic − hx, uia − hx, vic + hx, uia + hy, uic − hx, uid + hx, vic = 0. Finally, ((x, y), (z, t), (z, t)) =     −hz, ti 0 −hx, ti −hy, ti · (z, t) − (x, y) · = 0 ht, zi hx, zi hy, zi   −hx, ti −hy, ti + hz, ti(x, y) = (z, t) hx, zi hy, zi = (−hx, tiz + hx, zit + hz, tix, −hy, tiz + hy, zit + hz, tiy) (27) = (ht, xiz + hx, zit + hz, tix, ht, yiz + hy, zit + hz, tiy) = (0, 0). Therefore, A is right alternative. Similarly, one can prove that A is left alternative.  6 Examples 6.1 Algebra of octonions. Let B be a unital associative commutative algebra and A = O(B) be a split octonion algebra over B. In this case, A = M2 (B) ⊕ vM2 (B) with v 2 = 1, Aa =M2 (B),  Ac = vM2 (B), 0 0 , (a, b)(2) = Za = Z(A) = B. Take V = B 2 = {(a, b) | a, b ∈ B}, (a, b)(1) = v −b a     a b b −a . , then we have A = M2 (B)⊕V (1)⊕V (2), with h(a, b), (c, d)i = − det v c d 0 0 In fact, by (26), h(a, b), (c, d)i = (a, b)(2) · (c, d)(1) − (a, b)(1) · (c, d)(2)         d −c 0 0 0 0 b −a ·v −v ·v = v 0 0 −b a −d c 0 0           a b a 0 d −c 0 a 0 0 . = − det · − · = c d b 0 0 0 0 b −d c 16 Now for any u = (a, b), v = (c, d), w = (e, f ) ∈ V we have hu, viw + hv, wiu + hw, uiv =       e f c d a b (c, d) (a, b) − det (e, f ) − det = − det a b e f c d = −(ad − bc)(e, f ) − (cf − de)(a, b) − (eb − f a)(c, d) = (0, 0), hence O(B) satisfies (27). The following Proposition gives conditions under which the algebra A from Theorem 5.1 is isomorphic to an octonion algebra O(B). Proposition 6.1. The unital algebra A = M2 (B) ⊕ V 2 from Theorem 5.1 is isomorphic to an octonion algebra O(B) if and only if there exist x, y ∈ V such that hx, yi = 1. Proof: We have already checked that the algebra O(B) has form M2 (B) ⊕ V 2 , it suffices to note that hx, yi = 1 for x = (1, 0), y = (0, −1) ∈ V . Let now A = M2 (B) ⊕ V 2 be such that there exist x, y ∈ V with hx, yi = 1. Observe first that for any u, v ∈ V, a, b ∈ B the following equality holds [a, b]hu, vi = 0. (36) In fact, we have abhu, vi = ahbu, vi = hbu, avi = bhu, avi = bahu, vi. For any a, b ∈ B we now have 0 = [a, b]hx, yi = [a, b], hence B is commutative. Consider C = M2 (F ) + Cay(x) + Cay(y). It follows from Proposition 4.1 and its proof that C is a subalgebra of A isomorphic to the split octonion algebra O(F ). Therefore, by KaplanskyJacobson Theorem, A ∼ = O(A) for some commutative associative algebra A. It follows from (27) that V = B · Cay(x) + B · Cay(y) and A = B.  6.2 Algebras obtained by (commutative) Cayley-Dickson process Note that if the mapping h, i : V 2 → Z(B) is trivial then the algebra A is just a split null extension of the algebra M2 (B) by a bimodule V 2 . In this case, V may be an arbitrary associative B-module, (annihilated by [B, B] if B is not commutative). For instance, when B = F and V = F we get in this way the algebra A = M2 (F ) ⊕ Cay. If the mapping h, i : V 2 → Z(B) is not trivial then by (27) the rank of V as a B-module is less than 3. Observe that the left side of (27) is B-multilinear and skew-symmetric on u, v, w. Therefore, it holds when Λ3 (VB ) = 0. In particular it holds if the rank of V is less or equal to 2. If V ⊆ B · x then the mapping h, i is trivial by skew-symmetry. Let us consider now the case when V is a 2-generated B-module. 17 Let A be an associative commutative algebra and α ∈ A. Denote by CD(M2 (A), α) the algebra M2 (A) ⊕ vM2 (A) with a product defined by the following analogue of (4): a · b = ab, a · vb = v(a∗ b), vb · a = v(ab), va · vb = α(ba∗ ), (37) where a, b ∈ M2 (A), a 7→ a∗ is the symplectic involution in M2 (A). The algebra CD(M2 (A), α) is an alternative algebra containing M2 (A) with the same unit. We will call it the algebra obtained from M2 (A) by the Cayley-Dickson process with a parameter α. The algebra CD(M2 (A), α) is an octonion algebra if and only if the parameter α is invertible in A. Theorem 6.2. Let B be a unital commutative algebra, V = B 2 and h, i : V 2 → B be a skew-symmetric B-bilinear mapping. Then the algebra A = M2 (B) ⊕ V 2 is isomorphic to an algebra CD(M2 (B), α) where α = −h(1, 0), (0, 1)i. Conversely, every algebra CD(M2 (A), α) has this form.   0 0 2 , Proof: Let A = CD(M2 (A), α). Take V = A = {(a, b) | a, b ∈ A}, (a, b)(1) = v −b a   b −a ∈ vM2 (A), then we have, as before, A = M2 (A) ⊕ V (1) ⊕ V (2), (a, b)(2) = v 0 0   a b . In particular, h(1, 0), (0, 1)i = −α. with h(a, b), (c, d)i = −α det c d Conversely, let A = M2 (B) ⊕ V 2 where V ∼ = B 2 and h(1, 0), (0, 1)i = −α. Define the mapping ϕ : V 2 = V (1) ⊕ V (2) → vM2 (B) ⊂ CD(M2 (B), α) by sending, for any a, b ∈ B     b −a 0 0 . , (a, b)(2) 7→ v (a, b)(1) 7→ v 0 0 −b a It is easy to see that ϕ is an isomorphism of alternative M2 (B)-bimodules. Furthermore, let x = (a, b), y = (c, d) ∈ V = B 2 , then we have hx, yi = h(a, b), (c, d)i = ha(1, 0) + b(0, 1), c(1, 0) + d(0, 1)i = (ad − bc)h(1, 0), (0, 1)i = −α(ad − bc). Let z = (e, f ), t = (g, h) ∈ V , then we have by (35)     −ah + bg −ch + dg −hx, ti −hy, ti . = −α (x, y)(z, t) = af − be cf − de hx, zi hy, zi On the other hand,    h −g ·v ϕ(x, y) · ϕ(z, t) = v −f e   ah − bg ch − dg . = α −af + be −cf + ed d −c −b a  =α  h −g −f e    a c · b d Therefore, the mapping id + ϕ : A = M2 (B) ⊕ V 2 → CD(M2 (B), α) = M2 (B) ⊕ vM2 (B) is an isomorfism.  18 6.3 Algebras obtained by noncommutative Cayley-Dickson process Let us now generalize the Cayley-Dickson process for non-commutative coefficient algebras. Let A be a unital associative algebra, not necessarily commutative, α ∈ A such that αA ⊆ Z(A). Denote N CD(M2 (A), α) = M2 (A) ⊕ vM2 (Ā), where Ā = A/[A, A]A, and define a product in it by setting a · b = ab, a · v b̄ = v(ā∗ b̄), v b̄ · a = v(āb̄), vā · v b̄ = α(b1 a∗1 ), (38) where a, b ∈ M2 (A), ā, b̄ are their images in M2 (Ā), ā 7→ ā∗ is the symplectic involution in M2 (Ā), and a∗1 , b1 ∈ M2 (A) are some pre-images of ā∗ , b̄ under the canonical epimorphism M2 (A) → M2 (Ā). Observe that the last product in (38) is defined correctly since α[A, A] = 0. Proposition 6.3. The algebra N CD(M2 (A), α) is a unital alternative algebra that contains M2 (A) with the same unit. Proof: Denote I = [A, A]A, then M2 (I) is an ideal of N CD(M2 (A), α) which annihilates vM2 (Ā) and is annihilated by α; moreover, N CD(M2 (A), α)/I ∼ = CD(M2 (Ā), α). Therefore, the M2 (A)-bimodule vM2 (Ā) is in fact an M2 (Ā)-bimodule, and since the algebra CD(M2 (Ā), α) is alternative, vM2 (Ā) is an alternative M2 (A)-bimodule. In this way, it suffices to check the alternativity identities only when we have at least two arguments belonging to vM2 (Ā). For any a, b ∈ M2 (A) we have (a, v b̄, v b̄) = (v(ā∗ b̄)) · v b̄ − a(αbb∗1 ) = α b(b∗1 a) − α a(bb∗1 ), where b∗1 = b̄∗ . Consider b(b∗1 a) − a(bb∗1 ) = b̄(b̄∗ ā) − ā(b̄b̄∗ ) = (det b̄)ā − ā(det b̄) = 0̄. Thus b(b∗1 a) − a(bb∗1 ) ∈ M2 (I) and α(b(b∗1 a) − a(bb∗1 )) = 0. Furthemore, (vā, v b̄, v b̄) = (αba∗1 ) · v b̄ − vā · (αbb∗1 ) = αv((āb̄∗ )b̄ − (b̄b̄∗ )ā) = 0. Finally, consider, for c ∈ M2 (A), (vā, v b̄, c) + (vā, c, v b̄) = α ba∗1 · c − vā · v(c̄b̄) + v(c̄ā) · v b̄ − vā · v(c̄∗ b̄) = α (ba∗1 · c − cb · a∗1 + b · a∗1 c∗1 − c∗1 b · a∗1 ) We have ba∗1 · c − cb · a∗1 + b · a∗1 c∗1 − c∗1 b · a∗1 = b̄ā∗ · c̄ − c̄b̄ · ā∗ + b̄ · ā∗ c̄∗ − c̄∗ b̄ · ā∗ = b̄ā∗ t(c̄) − t(c̄) b̄ā∗ = 0̄. 19 Hence ba∗1 · c − cb · a∗1 + b · a∗1 c∗1 − c∗1 b · a∗1 ∈ M2 (I) and α (ba∗1 · c − cb · a∗1 + b · a∗1 c∗1 − c∗1 b · a∗1 ) = 0. We have proved that the algebra N CD(M2 (A), α) is right alternative. Similarly, one can check that it is left alternative.  Now we can generalize Theorem 6.2 to the case when B is not commutative. 2 Theorem 6.4. Let B be a unital associative algebra, B = B/[B, B]B, V = B and h, i : V 2 → B be a skew-symmetric B-bilinear mapping. Then the algebra A = M2 (B) ⊕ V 2 is isomorphic to an algebra N CD(M2 (B), α) where α = −h(1, 0), (0, 1)i. Conversely, every algebra N CD(M2 (A), α) has this form. Proof: Let first A = N CD(M α). Denote Ā = A/[A, A]A and take V = Ā2 =  2 (A), b̄ −ā 0 0 , (ā, b̄)(2) = v (ā, b̄) | a, b ∈ A, (ā, b̄)(1) = v ∈ vM2 (Ā); then we have, 0 0 −b̄ ā ¯ = −α(ad − bc). In particular, as before, A = M2 (A) ⊕ V (1) ⊕ V (2), with h(ā, b̄), (c̄, d)i h(1̄, 0̄), (0̄, 1̄)i = −α. 2 Conversely, let A = M2 (B)⊕V 2 where V ∼ = B , B = B/[B, B]B and h(1̄, 0̄), (0̄, 1̄)i = −α. Define the mapping ϕ : V 2 = V (1) ⊕ V (2) → vM2 (B) ⊂ N CD(M2 (B), α) by sending, for any a, b ∈ B     b̄ −ā 0 0 . , (ā, b̄)(2) 7→ v (ā, b̄)(1) 7→ v 0 0 −b̄ ā Then, as in the proof of Theorem 6.2, one can easily see that the mapping id + ϕ : A = M2 (B) ⊕ V 2 → N CD(M2 (B), α) = M2 (B) ⊕ vM2 (B) is an isomorfism.  Algebras of type CD(M2 (A), α) can be constructed for any commutative algebra A and any α ∈ A. For algebras of type N CD(M2 (A), α) one have to check the condition [αA, A] = 0. For instance, one can take A = F hx, y | y[x, y] = [x, y]y = 0i, with α = y 2 . 6.4 The case when V is not 2-generated In all the examples considered above the B-module V was 2-generated. Here we will give an example when V is 3-generated. Let B be a commutative unital algebra, a, b, c ∈ B, V = B 3 /I where I = B · (a, b, c); denote e1 = (1, 0, 0) + I, e2 = (0, 1, 0) + I, e3 = (0, 0, 1) + I. Then we have V = B · e1 + B · e2 + B · e3 where a · e1 + b · e2 + c · e3 = 0. Define a B-bilinear skew-symmetric mapping h, i : V × V → B by setting he1 , e2 i = c, he2 , e3 i = a, he3 , e1 i = b. 20 One can easily check that the mapping h, i is defined correctly. Moreover, we have he1 , e2 ie3 + he2 , e3 ie1 + he3 , e1 ie2 = c · e3 + a · e1 + b · e2 = 0, that is, identity (27) is true for u = e1 , v = e2 , w = e3 . Since the left side of (27) is skewsymmetric and multilinear on u, v, w, it follows that (27) is valid in V . By Theorem 5.1, the algebra A = M2 (B) ⊕ V 2 is a unital alternative algebra containing M2 (B) as a unital subalgebra. Observe that taking here a = b = 0, we will get the algebra CD(B, c) from the Theorem 6.2. Moreover, following the scheme from the previous section, this construction can be extended for noncommutative algebra B. One has only to choose the elements a, b, c ∈ B such that aB + bB + cB ⊂ Z(B). 7 Open questions 1. The first natural question which we left open is the case when the algebra H is not split, that is, when H is a division algebra. This case complicated since while P4is more k k Cayi · Cayj = αij H, the product Reg i H · Reg j H = α H for some αij ∈ Z(A), k=1 ij k and instead of Plücker relations (33) the elements αij satisfy more complicated system of relations. We plan to consider this case in a forthcoming paper. 2. An interesting question is to study the alternative algebras that contain H (or Halgebras) from categorical point of view. Clearly, the class of H-algebras form a category, with morphisms being the homomorphisms acting identically on H. Given an H-bimodule V , the free H-algebra over V or tensor algebra H[V ] of the bimodule V plays a role of a free object in this category. When V = Va is associative, V = ⊕m i=1 Reg i H and the algebra H[V ] is associative and is isomorphic to H ⊗ F hx1 , . . . , xm i where F hx1 , . . . , xm i is the free associative algebra on m generators. When V = Vc is a Cayley H-bimodule, V = ⊕m i=1 Cayi , the situation is not so clear even in the split case. For m = 1, H[V ] = H⊕Cay with Cay2 = 0 is just a well known 6-dimensional subalgebra of a split Cayley-Dickson algebra; for n = 2, H[V ] ∼ = CD(M2 (F [α12 ]), α12 ), but for n ≥ 3 the structure of the algebra H[V ] is not known. The situation is even more complicated for the mixed case, when V = Va ⊕ Vc with Va , Vc 6= 0, again except some trivial cases. 8 Acknowledgements The paper is a part of the PhD-thesis of the first author, realized at the University of São Paulo with the support of the CAPES. The second author was partially supported by the CNPq grant 304313/2019-0 and by the FAPESP grant 2018/23690-6. 21 References [1] N. Jacobson, Structure and Representations of Jordan Algebras. AMS, Providence, RI, 1968. [2] N. Jacobson, A Kronecker factorization theorem for Cayley algebras and the exceptional simple Jordan algebra. Amer. J. Math. 76, (1954). 447-452. [3] N. Jacobson, Structure of alternative and Jordan bimodules. Osaka Math. J. 6, (1954). 1-71. [4] I. Kaplansky, Semi-simple alternative rings, Portugal. Math.10 (1966), 37-50. [5] E. N. Kuzmin, I. P. Shestakov, Nonassociative structures, VINITI, Itogi nauki i tekhniki, seria “Fundamental Branches”, v.57, 179-266, Moscow, 1990; English transl. in “Encyclopaedia of Math. Sciences, v.57, Algebra VI, 199-280”, edited by A.I.Kostrikin and I.R.Shafarevich, Springer-Verlag. [6] M. C. López-Díaz, I. P. Shestakov, Representations of exceptional simple Jordan superalgebras of characteristic 3. Comm. Algebra 33 (2005), no. 1, 331-337. [7] M. C. López-Díaz, I. P. Shestakov, Representations of exceptional simple alternative superalgebras of characteristic 3. Trans. Amer. Math. Soc. 354 (2002), no. 7, 2745-2758. [8] Consuelo Martínez, Ivan Shestakov, Efim Zelmanov, Jordan bimodules over the superalgebras P (n) and Q(n). Trans. Amer. Math. Soc. 362 (2010), no. 4, 2037-2051. [9] C. Martínez, E. Zelmanov, A Kronecker factorization theorem for the exceptional Jordan superalgebra. Journal of Pure and Applied Algebra 177 (2003) no.1, 71-78. [10] K. McCrimmon, Structure and representations of noncommutative Jordan algebras, Trans. Amer. Math. Soc. 121 (1966), 187–199. [11] S. Mukai, An Introduction to Invariants and Moduli. Cambridge University Press, 2003. [12] A. P. Pozhidaev, I. P. Shestakov, Noncommutative Jordan superalgebras of degree n > 2, Algebra i Logika. 49 (2010), no. 1, 18–42. [13] R. D. Schafer, An introduction to nonassociative algebras. Academic Press, N.Y., 1966. [14] I. R. Shafarevich, Basic Algebraic Geometry, 1, Springer-Verlag, 1994. [15] I. P. Shestakov, Prime Alternative superalgebras of arbitrary characteristic, Algebra i Logika. 36 (1997), no. 6, 675–716; English transl.: Algebra and Logic 36, no.6 (1997), 389–420. 22 [16] K. Zhevlakov, I. Shestakov, A. Slin’ko, A.Ṡhirshov, Rings that are nearly associative, Academic Press, N.Y., 1982. 23