A Note on the Real Division Algebras with Non-trivial Derivations

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1 International Journal of Algebra, Vol. 10, 2016, no. 1, 1-11 HIKARI Ltd, A Note on the Real Division Algebras with Non-trivial Derivations A. S. Diabang, O. Diankha, M. Ly Département de Mathématiques et Informatique Faculté des Sciences et Techniques Université Cheikh Anta Diop, Dakar, Senegal A. Rochdi Département de Mathématiques et Informatique Faculté des Sciences Ben M Sik Université Hassan II, 7955 Casablanca, Morocco Copyright c 2015 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We show that every 4-dimensional real division algebra having non-zero derivations is obtained from the real algebra C by a special kind of duplication process accompanied by an appropriate isotopy. Also this process produces new examples in dimension 8. Mathematics Subject Classification: 17A35, 17A36 Keywords: Division algebra, derivation, isotopy, duplication process 1. Introduction In order to study real division algebras Benkart and Osborn adopted an approach using derivations [4], [5]. Let A be such an algebra and let be an arbitrary derivation of A. For every σ in the field C of complex numbers, Benkart and Osborn considered the sub-space B σ = {x A : σi A σi A x = 0},

2 2 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi σ being the complex conjugate of σ. They showed that 1 A is the direct sum of those B σ such that σ is a characteristic root of [4, p. 1139]. 2 B σ B τ B σ+τ + B σ+τ, so B 0 is a sub-algebra of A [4, p. 1139]. 3 The roots of of are purely imaginary [4, Lemma 9]. 4 For each λ C, dimb λ is even [4, Corollary 10]. 5 B 0 0 and B 0 = {x A : x = 0} [4, Lemma 11]. Next, in turn of the celebrated {1, 2, 4, 8}-Theorem [6], [8] they gave all possibilities for the Lie algebra of derivations DerA [4]: Theorem 1. The following possibilities actually occurs for DerA : 1 dima = 1 or 2 implies DerA = 0. 2 dima = 4, implies DerA is su2 or dim DerA 1. 3 dima = 8, implies DerA is one of the following Lie algebras: a compact G 2, b su3, c su2 su2, d su2 N where N is an abelian ideal and dimn 1, e N where N is abelian and dimn 2. They then determine those real division algebra A, in dimension 4, when DerA is su2, and, in dimension 8, when DerA is compact G 2, su3 and su2 su2. However the problem remains still open for the other cases of the Lie algebra of derivations. In particular, the one where dim DerA = 1 although some particular cases were studied in dimension 4 [4, Theorem 7.7], [2, Corollary 8], [3]. With an appropriate adaptation of the so-called Cayley-Dickson process, we give here a new construction method of real division algebras whose Lie algebra of derivations has dimension 1. This will allow us to fully determine the algebras in dimension 4 Corollaries 1, 2 and some new ones in dimension 8 Corollary 3, Theorem 3, Remark Unit-duplication process All algebras will be of finite-dimension over the field R of real numbers. An algebra A is said to be 1 flexible if it satisfies xyx = xyx for all x, y in A, 2 quadratic if it contains an unit element e and for every x in A the three elements e, x, x 2 are linearly dependent,

3 Real division algebras with non-trivial derivations 3 3 a division algebra if the linear operators L a, R a of left and right multiplication by every non-zero element a in A are bijective. In this case the algebra A a having A as underlying space and product given by: x y = Ra xl a y, called Albert-isotope of A, is also a division algebra with unit element e = a 2 [1]. A linear mapping : A A is said to be a derivation of the algebra A if the equality xy = xy+x y holds for all x, y in A. We denore by DerA the well known Lie algebra of derivations of A [10]. Also Lin{x 1,..., x n } will denote the lineal hull spanned by x 1,..., x n A. Definition 1. Unit-duplication process Let B be a real algebra having an unit element e and let ρ, σ, φ, ψ : B B be linear mappings such that φe = ψe = e. We define on the space B B the produit: x, y x, y = xx + ρ σy y, yφx + y ψx. 2.1 The algebra resulting has an unit element e, 0 and contains B {0} as sub-algebra. It is said to be obtained from B and ρ, σ, φ, ψ by unit-duplication process and is denoted by UDP B ρ, σ, φ, ψ. This generalizes the classical Cayley-Dickson process as well as the process given in [11, p. 1] 3. Four-dimensional real division algebras with non-trivial derivations We have the following preliminary result taken from [4]: Lemma 1. Let A be a four-dimensional real division algebra admitting a nonzero derivation. Then, according to the notations in Section 1, we have: 1 B 0 has dimension 2, 2 There exists a non-zero purely imaginary σ = ζi, with ζ > 0, such that A = B 0 B σ, 3 B 2σ = {0} and B 2 σ = B 0. The following additional preliminary result, easy to show by taking into account [4, Theorem 6.3], will help us: Lemma 2. Let A be a four-dimensional real division algebra. Then the following are equivalent: 1 A contains an unit element and DerA = su2. 2 A is a quadratic and flexible algebra.

4 4 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi We state now the following result: Theorem 2. Let A be a four-dimensional real division algebra with unit element e. If A admits a non-zero derivation then there exists a basis {e, u, x 1, x 2 } of A for which the multiplication is given by the following table: e u x 1 x 2 e e u x 1 x 2 u u e αx 1 + βx 2 βx 1 + αx 2 x 1 x 1 γx 1 + δx 2 λe + µu ωe + θu x 2 x 2 δx 1 + γx 2 ωe θu λe + µu Table 1 for some scalars α, β, γ, δ, λ, µ, ω, θ. In addition, every algebra A whose multiplication is given by Table 1 admits a non-zero derivation. It is a division algebra if and only if the function Φα 1, α 2, α 3, α 4 = α1 2 + α2 2 α 1 + αα β 2 α2 2 + Qα 1, α 2 α3 2 + α4 2 +δλθ ωµα α where Qα 1, α 2 = γµ + δθ + λα1 2 + αγλ + δω µ + βδλ γω + θ α2 2 + αγµ + δθ + λ + βγθ δµ + ω + γλ + δω µ α 1 α 2 vanishes only at α 1, α 2, α 3, α 4 = 0, 0, 0, 0. In these conditions DerA = su2 if and only if α = γ = µ = ω = 0, λ < 0, β = δ 0 and θ = βλ 0. Proof. According to the notations of Lemma 1, we have A = B 0 B i. Moreover, B 0 contains e, which is the only non-zero idempotent of algebra A [12, Theorem 1], and is isomorphic to C [13, Corollary 1], [9]. So u 2 = e for some u in B 0. On the other hand there exists a basis {x 1, x 2 } of B i such that x 1 = x 2 and x 2 = x 1.

5 Real division algebras with non-trivial derivations 5 We obtain a basis B = {e, u, x 1, x 2 } of A. As u ker, the left and right multiplication operators L u and R u commute with. So Im is both L u - invariant and R u -invariant. Writing ux 1 = αx 1 + βx 2 and x 1 u = γx 1 + δx 2 where α, β, γ, δ R, we have ux 2 = u x 1 = ux 1 = αx 2 βx 1, x 2 u = x 1 u = x 1 u = γx 2 δx 1. Moreover, x 2 1 B 2 αi = B 0 and there are some scalars λ, ρ such that x 2 1 = λe + ρu. In the other hand: x 2 x 1 + x 1 x 2 = x 1 x 1 + x 1 x 1 = x 2 1 = 0 and x 2 2 x 2 1 = x 1 x 2 + x 1 x 2 = x 1 x 2 = 0. We obtain the desired multiplication Table 1. In order to establish the second assertion, note that the linear mapping A A whose matrix with respect to the basis {e, u, x 1, x 2 } is given by is a derivation Let now x = α 1 e + α 2 u + α 3 x 1 + α 4 x 2, y = β 1 e + β 2 u + β 3 x 1 + β 4 x 2 be arbitrary in A. A direct calculation gives xy = α 1 β 1 α 2 β 2 + λα 3 β 3 + ωα 3 β 4 ωα 4 β 3 + λα 4 β 4 e + α 1 β 2 + α 2 β 1 + µα 3 β 3 + θα 3 β 4 θα 4 β 3 + µα 4 β 4 u + α 1 β 3 + αα 2 β 3 βα 2 β 4 + α 3 β 1 + γα 3 β 2 δα 4 β 2 x 1 + α 1 β 4 + βα 2 β 3 + αα 2 β 4 + δα 3 β 2 + α 4 β 1 + γα 4 β 2 x 2. So xy = 0 α 1 α 2 λα 3 ωα 4 ωα 3 + λα 4 α 2 α 1 µα 3 θα 4 θα 3 + µα 4 α 3 γα 3 δα 4 α 1 + αα 2 βα 2 α 4 δα 3 + γα 4 βα 2 α 1 + αα 2 β 1 β 2 β 3 β 4 = A laborious calculation shows that the determinant of the above matrix is

6 6 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi α1 2 + α2 2 α 1 + αα β 2 α2 2 + Qα 1, α 2 α3 2 + α4 2 + δλθ ωµα3 2 + α This shows the third proposition of the theorem. Assume now that A is a division algebra and DerA = su2. Then according to Lemma 2 the algebra A is quadratic and flexible. Now x 2 1 Lin{e, x 1 }, u + x 1 2 Lin{e, u + x 1 } give µ = 0 and α + γ = β + δ = As A is a division algebra then clearly δ 0 and we deduce from µ = 0 that λ < 0 and θ 0. On the other hand, the equalities x 1 x 2 x 1 = x 1 x 2 x 1, x 1 ux 1 = x 1 ux 1, x 1 +ux 2 x1 +u = x 1 +u x 2 x 1 +u give 2ω + α + γθ = β + δθ = 0, 3.3 γ αλ β + δω = γ αµ β + δθ = 0, 3.4 θ = βλ. 3.5 Now, taking into account the equalities 3.2, 3.3, 3.4, we have: α = γ = µ = ω = 0, λ < 0 and δ = β, θ = βλ 0. The converse is obvious Corollary 1. Every four-dimensional real unital division algebra A having a non-trivial derivation is obtained from the real unital algebra C by unitduplication process. Proof. Let σ : x x be the standard involution of the real algebra C and let α, β, γ, δ, λ, µ, ω, θ be real numbers. We consider the R-linear mappings ρ, φ, ψ : C C whose matrices with respect to the canonical basis {1, i} are given, respectively, by: λ ω µ θ, 1 γ 0 δ, 1 α 0 β.

7 Real division algebras with non-trivial derivations 7 Now, the multiplication table of algebra UDP C ρ, σ, φ, ψ with respect to the basis {1, 0, i, 0, 0, 1, 0, i} is identical to Table 1. The study of non-unital case can be reduced to the study of unital one by using Albert-isotopy: Lemma 3. Let A be a division algebra of dimension 4 having a non-trivial derivation and let a be non-zero in the kernel of. Then is a derivation of the unital division algebra A a. Proof. We have [, L x ] = L x for all x A. In the other hand, the operators of left and right multiplication by any x in algebra A a are given, respectively, by L x = L R a x L a, Rx = R L a x R a. As a = 0, commutes with L a and we have: [, L x ] = L R a x L a = L R a x L a = [, L R a x ] L a = L L Ra x a = L R a x L a = L x. L R a x L a L R a x L a This shows that is a derivation of the algebra A a. Corollary 2. Every four-dimensional real division algebra whose Lie algebra of derivations has dimension one is obtained from the real algebra C by unitduplication process accompanied by an Albert-isotopy. 4. New examples in dimension 8 We start with the following useful definition: Definition 2. Let A be an algebra. 1 A linear mapping f : A A is said to be an involution if f 2 = I A the identity operator of A and fxy = fyfx for all x, y in A. 2 If A contains an unit element e and is provided with an involution σ A : A A x x, it is called a Cayley algebra if x + x := T x, xx := Nx Re for all x, y in A. Clearly σ A e = e.

8 8 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi It is well known that every Cayley algebra A is a quadratic algebra and has a decomposition A = Re ImA into a sum of sub-space of scalars Re and sub-space of imaginary elements ImA = { x A : x 2 Re, x / R {0} } [7]. For every x in A we will denote by sx, imx, respectively, the scalar and imaginary parts of x. Let now A be a Cayley algebra with involution σ A : x x and let κ be in A. We denote by E κ A the algebra having underlying space A A and product x, yx, y = This is the algebra UDP A L κ, σ A, σ A, I A. We have the following preliminary result: xx + κy y, yx + y x. Lemma 4. Let A be an associative Cayley algebra, of unit e and involution x x, and let κ be in A Re. Then E κ A is a division algebra if and only if A is a division algebra. In this case, the equation x, y 2 = e, 0 in E κ A is equivalent to x 2 = e and y = 0. Proof. It remains only to show the if part. Let x, y, x, y be arbitrary elements of A with x, y 0, 0 such that x, yx, y = 0, 0. We can assume, without lost of generality, that y 0. We have So x, yx, y = xx + κy y, yx + yx. This gives x, yx, y = 0, 0 { xx + κy y = 0 yx + y x = 0 0 = yx + y xx y xx + κy y = Nx y κy y = Nx sκny y imκy y and then Nx sκny y imκy = Multiplying equality 4.6 on the left by y and right by y we get:

9 If Ny 0 then Real division algebras with non-trivial derivations 9 Nx 0 = y sκny y imκy y Nx = Ny sκny Ny imκ. Nx sκny Ny imκ = 0. As Nx, sκny are scalars, imκ must vanish, that is, κ Re, which is absurd. Therefore y = 0 = x. Let now x, y be in A, we have x, y 2 = e, 0 { x 2 + Nyκ = e T xy = 0 because η 0. If y 0 then T x = 0, that is x ImA and κ = Ny e + x 2 must be a scalar, a contradiction. This shows the second proposition. Corollary 3. Let H be the quaternion algebra and let κ be in H R. Then 1 E κ H is a division algebra. 2 Every two-dimensional subalgebra of E κ H is contained in H {0}. Proof. The first proposition is given in Lemma 4. The second one is consequence of Lemma 4 by taking into account that every 2-dimensional subalgebra of E κ H contains the unit element 1, 0 [12, Theorem 1] and is isomorphic to C [13, Corollary 1], [9]. Theorem 3. Let H be the quaternion algebra and let κ be in H R. Then Der E κ H = su2 N where N is 1-dimensional Lie algebra. Proof. Corollary 3 shows that the equation x, y 2 =, 0 in algebra E κ H cannot have solutions outside a sub-space of dimension 3 unlike the case of algebras whose multiplications are given by tables 2.1, 4.2, 5.2 in [5]. Moreover, an algebra whose multiplication is given by 3.1 in [4] cannot have an unit-element. So Der E κ H cannot be equal to G 2 compact, su3 or su2 su2. On the other hand, a simple calculation shows that for every non-zero d in ImH the mapping d : E κ H E κ H x, y 0, dy

10 10 A. S. Diabang, O. Diankha, M. Ly and A. Rochdi is a derivation with kernel H {0}. So Der E κ H has dimension 3 and cannot be abelian. Moreover, for every non-zero derivation of H that vanish in κ the mapping is a derivation with kernel D : E κ H E κ H x, y x, y kerd = Lin{1, 0, κ, 0, 0, 1, 0, κ}. In particular, D cannot belong to { } d : d ImH. Thus Der E κ H has dimension 4 and coincides with su2 N where N is abelian of dimension 1. Remark 1. Let C be the class of eight-dimensional division algebras A with DerA = su2 N where N is a 1-dimensional Lie algebra. For every θ in ]0, π[ the algebra E κ H, where κ = cos θ + i sin θ, belongs to C. The multiplication table of algebra E κ H relative to the canonical table {1, 0, i, 0, j, 0, k, 0, 0, 1, 0, i, 0, j, 0, k}, which we will denote {1, i, j, k, f, if, jf, kf}, is given by: 1 i j k f if jf kf 1 1 i j k f if jf kf i i -1 k -j if -f -kf jf j j -k -1 i jf kf -f -if k k j -i -1 kf -jf if -f f f -if -jf -kf cos θ 1 sin θ i sin θ 1 + cos θ i cos θ j + sin θ k sin θ j + cos θ k if if f -kf jf sin θ 1 cos θ i cos θ 1 sin θ i sin θ j cos θ k cos θ j + sin θ k jf jf kf f -if cos θ j sin θ k sin θ j + cos θ k cos θ 1 sin θ i sin θ 1 cos θ i kf kf -jf if f sin θ j cos θ k cos θ j sin θ k sin θ 1 + cos θ i cos θ 1 sin θ i This example using the unit-duplication process is different from the one in [4, Theorem 6.9]. It illustrates the immensity of the class C. Acknowledgements. This paper has benefited from several remarks and suggestions by Professor Antonio Jesús Calderón. The authors are very grateful to him.

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