Some remarks and properties of algebras obtained by the Cayley-Dickson process
aa r X i v : . [ m a t h . R A ] J a n Some remarks and properties of algebras obtained by theCayley-Dickson process
Cristina FLAUT, Delia MUSTAC ˘A
Abstract.
Finding identities in nonassociative algebras plays an important rolein the study of properties of these algebras. In this paper, we present some identi-ties in alternative algebras and in algebras obtained by the Cayley-Dickson process.Moreover, the spectrum of matrices with coefficients in such algebras are investigated.
Keywords.
Cayley-Dickson process; alternative algebras; left and righteigenvalues. : 17A35, 17A45, 15A18.
1. Introduction
Quaternions, octonions and algebras obtained by the Cayley-Dickson processhave at present many applications(in physics, coding theory, computer vision,etc). For this reasons these algebras are intense studied. Since the algebras ob-tained by the Cayley-Dickson process are poor in properties when their dimen-sion increase, losing commutativity, associativity and alternativity, the study ofall kind of identities on these algebras is very useful for obtaining new propertiesand relations. Several papers are devoted to the study of these identities([Ra;88], [Is; 84], [He; 97], etc.). Therefore, it is very interesting to continue thestudy of these identities in algebras obtained by the Cayley-Dickson process,since these relations can be helpful to replace the missing commutativity, asso-ciativity and alternativity.In the following, we suppose that K is a commutative field with charK = 2and A is an algebra over the field K . An algebra A is called unitary if thisalgebra contains an identity element with respect to the algebra’s multiplication.We define the associator of three elements a, b, c ∈ A as below( a, b, c ) = ( ab ) c − a ( bc ) . In an arbitrary algebra over a field K , the following relation is true ([Zo; 40],(1 . ab, c, d ) − ( a, bc, d ) + ( a, b, cd ) = a ( b, c, d ) + ( a, b, c ) d. (1.1.)1e consider the sets N ( A ) = { x ∈ A / ( x, a, b ) = 0, for all a, b ∈ A} , called the nucleus of the algebra A and C ( A ) = { x ∈ A / [ a, x ] = 0, for all a ∈ A } . The center C ( A ) of an algebra A is the set of all elements c ∈ A whichcommute and associate with all elements x ∈ A , therefore C ( A ) = N ( A ) ∩ C ( A ) (1.2.)An algebra A is central if its center is equal with the ground field, C ( A ) = K .An algebra A is a simple algebra if A is not a zero algebra and { } and A are the only ideals of A . The algebra A is called central simple if thealgebra A F = F ⊗ K A is simple for every extension F of K . A central simplealgebra is a simple algebra. An algebra A is called alternative if x y = x ( xy )and xy = ( xy ) y, for all x, y ∈ A , flexible if x ( yx ) = ( xy ) x = xyx, for all x, y ∈ A and power associative if the subalgebra < x > of A generated by anyelement x ∈ A is associative. Each alternative algebra is a flexible algebraand a power associative algebra. A unitary algebra A 6 = K such that we have x + α x x + β x = 0 for each x ∈ A , with α x , β x ∈ K, is called a quadratic algebra .A finite-dimensional algebra A is a division algebra if and only if A does notcontain zero divisors. (See [Sc;66]) Artin’s Theorem. [Sc; 66]
The subalgebra generated by two arbitrary ele-ments x, y of an alternative algebra A is associative. In the following, we briefly present the
Cayley-Dickson process and the prop-erties of the algebras obtained. (see [Sc; 66] and [Sc; 54]).We consider A , a finite dimensional unitary algebra over a field K, with a scalar involution : A → A , a → a, which it is a linear map with the following properties ab = ba, a = a, and a + a, aa ∈ K · a, b ∈ A . An element a is called the conjugate of the element a, the linear form t : A → K , t ( a ) = a + a and the quadratic form n : A → K, n ( a ) = aa trace and the norm of the element a, respectively . Hence analgebra A with a scalar involution is quadratic.We consider γ ∈ K a fixed non-zero element. We define the followingalgebra multiplication on the vector space A ⊕ A : ( a , a ) ( b , b ) = (cid:0) a b + γb a , a b + b a (cid:1) . (1.3.)The obtained algebra structure over A⊕A , denoted by ( A , γ ) is called the algebraobtained from A by the Cayley-Dickson process. We have dim ( A , γ ) = 2 dim A .Let x ∈ ( A , γ ), x = ( a , a ). The map: ( A , γ ) → ( A , γ ) , x → ¯ x = ( a , - a ) , is a scalar involution of the algebra ( A , γ ), extending the involution of thealgebra A . Let t ( x ) = t ( a )and n ( x ) = n ( a ) − γ n ( a )be the trace and the norm of the element x ∈ ( A , γ ), respectively.If we consider A = K and we apply this process t times, t ≥
1, we obtainan algebra over K , A t = (cid:16) γ , ..., γ t K (cid:17) . (1.4.)Using induction in this algebra, the set { , f , ..., f n } , n = 2 t , generates abasis with the properties: f i = γ i , i ∈ K, γ i = 0 , i = 2 , ..., n (1.5.)and f i f j = − f j f i = β ij f k , β ij ∈ K, β ij = 0 , i = j, i, j = 2 , ...n, (1.6.) β ij and f k being uniquely determined by f i and f j . From [Sc; 54], Lemma 4, it results that in any algebra A t with the basis { , f , ..., f n } satisfying relations (1 .
5) and (1 . , we have: f i ( f i x ) = γ i x = ( xf i ) f i , (1.7.)for all i ∈ { , , ..., n } and for every x ∈ A t .Let x ∈ A t = (cid:0) γ ,...,γ t K (cid:1) . From the above, it results that x can be writtenunder the form x = x ′ + x ′′ f t − , (1.8.)where x ′ and x ′′ ∈ A t − = (cid:0) γ ,...,γ t − K (cid:1) .The field K is the center of the algebra A t , for t ≥
2. (See [Sc; 54]). Algebras A t of dimension 2 t obtained by the Cayley-Dickson process, described above,3re central-simple, flexible and power associative for all t ≥ t ≥
1. But there exist fields on which, if we applythe Cayley-Dickson process, the obtained algebras A t are division algebras forall t ≥
1. (See [Br; 67], [Fl; 13] ).For t = 2 , we obtain the generalized quaternion algebras and for t = 3, weobtain the generalized octonion algebras. The generalized quaternion algebras
We consider two elements α, β ∈ K and we define a generalized quaternionalgebra, denoted by H ( α, β ) = (cid:16) α,βK (cid:17) , with basis { , f , f , f } and multiplica-tion given in the following table: · f f f f f f f f α f αf f f − f β − βf f f - αf βf − αβ If a ∈ H ( α, β ) , a = a + a f + a f + a f , then ¯ a = a − a f − a f − a f iscalled the conjugate of the element a. For a ∈ H ( α, β ), we consider the followingelements: t ( a ) = a + a ∈ K and n ( a ) = aa = a − αa − βa + αβa ∈ K, called the trace , respectively, the norm of the element a ∈ H ( α, β ). It follows that a − t ( a ) a + n ( a ) = 0, ∀ a ∈ H ( α, β ), therefore the generalized quaternion algebrais a quadratic algebra .If, for x ∈ H ( α, β ) , the relation n ( x ) = 0 implies x = 0, then the algebra H ( α, β ) is a division algebra. A quaternion non-division algebra is called a split algebra.Using the above notations, we remark that H ( − , −
1) = (cid:0) − , − R (cid:1) is a divi-sion algebra. The generalized octonion algebras
A generalized octonion algebra over an arbitrary field K , with char K = 2,is an algebra of dimension 8, denoted O ( α, β, γ ), with basis { , f , ..., f } andmultiplication given in the following table:4 f f f f f f f f f f f f f f f f α f αf f αf − f − αf f f − f β − βf f f βf βf f f - αf βf − αβ f αf − βf − αβf f f − f − f − f γ − γf − γf − γf f f - αf − f - αf γf - αγ γf αγf f f f − βf βf γf − γf - βγ − βγf f f αf − βf αβf γf − αγf βγf αβγ The algebra O ( α, β, γ ) is a non-commutative and a non-associative algebra,but it is alternative , flexible and power-associative .If a ∈ O ( α, β, γ ) , a = a + a f + a f + a f + a f + a f + a f + a f , then¯ a = a − a f − a f − a f − a f − a f − a f − a f is called the conjugate of the element a . For a ∈ O ( α, β, γ ), we define the elements: t ( a ) = a + a ∈ K and n ( a ) = aa = a − αa − βa + αβa − γa + αγa + βγa − αβγa ∈ K. These elements are called the trace , respectively, the norm of the element a ∈ O ( α, β, γ ). It follows that a − t ( a ) a + n ( a ) = 0 , ∀ a ∈ a ∈ O ( α, β, γ ),therefore the generalized octonion algebra is a quadratic algebra.If, for x ∈ O ( α, β, γ ), the relation n ( x ) = 0 implies x = 0, then the algebra O ( α, β, γ ) is a division algebra.( see [Sc; 54] and [Sc; 66])
2. Some remarks regarding alternative algebras and algebras ob-tained by the Cayley-Dickson process
In the following we consider A an arbitrary algebra. In this section wepresent some new relations, properties and equations in the above presentedalgebras. Remark 2.1.
1) If a, b ∈ N ( A ), therefore ab ∈ N ( A ). Indeed, fromrelation (1 . ab, c, d ) − ( a, bc, d ) + ( a, b, cd ) = a ( b, c, d ) + ( a, b, c ) d andsince ( a, bc, d ) = ( a, b, cd ) = ( b, c, d ) = ( a, b, c ) = 0, it results ( ab, c, d ) = 0, then ab ∈ N ( A ).2) If A is an alternative algebra and x ∈ C ( A ), therefore x n ∈ C ( A ), for n a positive integer. Indeed, let y ∈ C ( A ). Since in an alternative algebra x and y generate an associative subalgebra, we have that x n y = yx n .
3) In an alternative algebra A , the associator alternates. That means, if σ is a permutation of degree three, we have that( x , x , x ) = sgn ( σ ) (cid:0) x σ (1) , x σ (2) , x σ (3) (cid:1) , x , x , x ∈ A. roposition 2.2. ([Sm; 47], Lemma 2) Let A be an alternative algebra and x ∈ C ( A ). Then we have that x ∈ C ( A ) = C ( A ) ∩ N ( A ). Proposition 2.3.
Let A be an alternative algebra and a ∈ C ( A ). It resultsthat a q ∈ C ( A ). Proof.
We have that a q ∈ N ( A ), with q a positive integer. Indeed, fromProposition 2.2, we have that a ∈ C ( A ) ∩N ( A ) and from Remark 2.1, it resultsthat a q ∈ C ( A ) ∩ N ( A ).If A is an alternative algebra over the field K , therefore the following iden-tities are true ((1 . , (1 .
8) from [Zo; 40]):( ab, b, c ) = b ( b, c, a ) , for all a, b, c ∈ A ; (2.1.)( ab, a, c ) + ( ba, a, c ) − (cid:0) b, a , c (cid:1) = 0 , for all a, b, c ∈ A . (2.2.) Proposition 2.4 . Let A be an alternative division algebra and x ∈ C ( A ) − K such that there is c ∈ C ( A ) with ( x + c ) ∈ N ( A ). Therefore , x ∈ N ( A ). Proof . Let a, b ∈ A . It results that 0 = (cid:16) a, b, ( x + c ) (cid:17) = (cid:0) a, b, x (cid:1) +( a, b, xc ) + ( a.b, cx ) . By using relation (2 . x + c ) a, x + c, b ) + ( a ( x + c ) , x + c, b ) + (cid:16) a, b, ( x + c ) (cid:17) = 0. Therefore, byusing (2 . (cid:16) a, b, ( x + c ) (cid:17) = − x + c ) ( x + c, b, a ) = − x + c ) ( x, b, a ) . We obtain that ( x, b, a ) = 0 and x ∈ N ( A ) since A is a division algebra. Proposition 2.5.
Let A be an alternative algebra over the field K , withchar K / ∈ { , } . For x, u ∈ C ( A ) , we have the following relations: (cid:0) xu (cid:1) ( vw ) = (cid:0)(cid:0) xu (cid:1) v (cid:1) w = v (cid:0) w (cid:0) xu (cid:1)(cid:1) , for all v, w ∈ A , (2.3.)and ( x u ) ( vw ) = (cid:0)(cid:0) x u (cid:1) v (cid:1) w = v (cid:0) w (cid:0) x u (cid:1)(cid:1) , for all v, w ∈ A . (2.4.) Proof.
From Proposition 2.2, we have that x y = yx and x ( yz ) = (cid:0) x y (cid:1) z = y (cid:0) zx (cid:1) , for all y, z ∈ A . Denoting T = x ( yz ) , T = (cid:0) x y (cid:1) z, T = y (cid:0) zx (cid:1) , we linearize these relations. For x + λu, y + λv, z + λw , with u ∈ C ( A ), v, w ∈ A , λ ∈ K , we obtain T = ( x + λu ) [( y + λv ) ( z + λw )] == (cid:0) x + 3 λx u + 3 λ xu + λ u (cid:1) (cid:2) yz + λ ( vz + yw ) + λ vw (cid:3) == λ u ( vw ) + λ (cid:2) (cid:0) xu (cid:1) ( vw ) + u ( vz + yw ) (cid:3) ++ λ [ u ( yz ) + 3( x u ) ( vw ) + 3 (cid:0) xu (cid:1) ( vz + yw )]++ λ [ x ( vw ) + 3 (cid:0) x u (cid:1) ( vz + yw ) + 3( xu ) ( yz )]++ λ (cid:2) x ( vz + yw ) + 3 (cid:0) x u (cid:1) ( yz ) (cid:3) + x ( yz ) . = [( x + λu ) ( y + λv )] ( z + λw ) == (cid:2)(cid:0) x + 3 λx u + 3 λ xu + λ u (cid:1) ( y + λv ) (cid:3) ( z + λw ) == [ x y + 3 λ ( (cid:0) x u (cid:1) y + x v ) + 3 λ (cid:0)(cid:0) xu (cid:1) y + (cid:0) x u (cid:1) v (cid:1) ++ λ (cid:0) u y + 3 (cid:0) xu (cid:1) v (cid:1) + λ u v ] ( z + λw ) == λ (cid:0) u v (cid:1) w + λ (cid:2)(cid:0) u v (cid:1) z + (cid:0) u y + 3 (cid:0) xu (cid:1) v (cid:1) w (cid:3) ++ λ (cid:2)(cid:0) u y + 3 (cid:0) xu (cid:1) v (cid:1) z + 3 (cid:0)(cid:0) xu (cid:1) y + (cid:0) x u (cid:1) v (cid:1) w (cid:3) ++ λ (cid:2) (cid:0)(cid:0) xu (cid:1) y + (cid:0) x u (cid:1) v (cid:1) z + 3( (cid:0) x u (cid:1) y + x v ) w (cid:3) ++ λ (cid:2) (cid:0) x u (cid:1) y + x v ) z + (cid:0) x y (cid:1) w (cid:3) + (cid:0) x y (cid:1) z.T = ( y + λv ) (cid:2) ( z + λw ) (cid:0) x + 3 λx u + 3 λ xu + λ u (cid:1)(cid:3) == ( y + λv ) [ λ wu + λ (cid:0) w (cid:0) xu (cid:1) + zu (cid:1) ++3 λ (cid:0) w (cid:0) x u (cid:1) + z (cid:0) xu (cid:1)(cid:1) + λ (cid:0) z (cid:0) x u (cid:1) + wx (cid:1) + zx ] == λ v (cid:0) wu (cid:1) + λ (cid:2) y (cid:0) wu (cid:1) + v (cid:0) w (cid:0) xu (cid:1) + zu (cid:1)(cid:3) ++ λ (cid:2) y (cid:0) w (cid:0) xu (cid:1) + zu (cid:1) + 3 v (cid:0) w (cid:0) x u (cid:1) + z (cid:0) xu (cid:1)(cid:1)(cid:3) ++ λ (cid:2) y (cid:0) w (cid:0) x u (cid:1) + z (cid:0) xu (cid:1)(cid:1) + v (cid:0) z (cid:0) x u (cid:1) + wx (cid:1)(cid:3) ++ λ (cid:2) y (cid:0) z (cid:0) x u (cid:1) + wx (cid:1) + v (cid:0) zx (cid:1)(cid:3) + y (cid:0) zx (cid:1) .Since the coefficients of λ, λ , λ , λ are equal in both members of the equality T = T = T , by taking the coefficients of λ and λ , we obtain relations (2 . . Remark 2.6 . For an alternative algebra, from relations (2 .
3) and (2 .
4) itresults that xu ∈ N ( A ), which implies that x u ∈ N ( A ), due to the commu-tativity property and replacing u with x .From the above, by using Remark 2.1 and Proposition 2.3, for x, u ∈ C ( A ), itresults that xu k +2 ∈ N ( A ) and x k +2 u ∈ N ( A ). Remark 2.7.
From relation (2 . u, x ∈ C ( A ), u, x = 0 , we have that0 = (cid:0) x u, u, v (cid:1) = u (cid:0) u, v, x (cid:1) . Therefore (cid:0) u, v, x (cid:1) = (cid:0) u, x , v (cid:1) = 0 for all v ∈ A , A an alternative division algebra. From relation (2 . xu, x, v ) +( ux, x, v ) − (cid:0) u, x , v (cid:1) = 0. It results, ( xu, x, v ) = ( ux, x, v ) = 0. But, from (2 . ux, x, v ) = x ( u, v, x ) = 0. We get ( u, v, x ) = 0, for all u, x ∈ C ( A ) , v ∈ A .Therefore ” · ” is associative on C ( A ). Moreover, we get that ux ∈ C ( A ). Indeed,for v ∈ A , we have ( ux ) v = u ( xv ) = u ( vx ) = ( uv ) x == ( vu ) x = v ( ux ). Therefore, C ( A ) is an associative algebra. Remark 2.8. ([BH; 01]) In all algebras A t , obtained by the Cayley-Dicksonprocess, the following identity is satisfied: x ( yx ) = (cid:0) x y (cid:1) x. Proposition 2.9.
In all algebras A t , obtained by the Cayley-Dickson pro-cess, the following identities are satisfied: (cid:0) w , v, x (cid:1) + ( xw, v, w ) + ( wx, v, w ) = 0 (2.5.) (cid:0) w , y, x (cid:1) + (cid:0) x , v, w (cid:1) +( xw, y, w )+( wx, y, w )+( wx, v, x )+( xw, v, x ) = 0 . (2.6.)7 roof. From the above remark, we have that x ( yx ) = (cid:0) x y (cid:1) x , for all x, y ∈ A t . Denoting T = x ( yx ) , T = (cid:0) x y (cid:1) x , we linearize these relations.For x + λw, y + λv , with v, w ∈ A t , λ ∈ K , we obtain T = ( x + λw ) (( y + λv ) ( x + λw )) = (( x + λw ) ( y + λv )) ( x + λw ) == (cid:0) x + λxw + λwx + λ w (cid:1) ( yx + λyw + λvx + λ vw ) == λ w ( vw ) + λ (cid:2) w ( yw + vx ) + ( xw + wx ) vw (cid:3) ++ λ (cid:2) w ( yx ) + x ( vw ) + ( xw + wx ) ( yw + vx ) (cid:3) ++ λ (cid:2) ( xw + wx ) yx + x ( yw + vx ) (cid:3) + x ( yx ) T = (cid:0)(cid:0) x + λxw + λwx + λ w (cid:1) ( y + λv ) (cid:1) ( x + λw ) == (cid:2) x y + λ (cid:0) x v + ( xw + wx ) y (cid:1) + λ (cid:0) w y + ( xw + wx ) v (cid:1) + λ w v (cid:3) ( x + λw ) == λ (cid:0) w v (cid:1) w + λ (cid:2)(cid:0) w v (cid:1) x + ( w y + ( xw + wx ) v ) w (cid:3) ++ λ (cid:2)(cid:0) x v + ( xw + wx ) y (cid:1) w + (cid:0) w y + ( xw + wx ) v (cid:1) x (cid:3) ++ λ (cid:2)(cid:0) x y (cid:1) w + (cid:0) x v + ( xw + wx ) y (cid:1) x (cid:3) + (cid:0) x y (cid:1) x Comparing the coefficients of λ , we obtain (cid:2) w ( yw + vx ) + ( xw + wx ) vw (cid:3) = (cid:2)(cid:0) w v (cid:1) x + (cid:0) w y + ( xw + wx ) v (cid:1) w (cid:3) w ( yw ) + w ( vx ) + ( xw ) ( vw ) + ( wx ) ( vw ) == (cid:0) w v (cid:1) x + (cid:0) w y (cid:1) w + (( xw ) v ) w + (( wx ) v ) w , therefore (cid:0) w , v, x (cid:1) + ( xw, v, w ) + ( wx, v, w ) = 0Comparing the coefficients of λ , we obtain (cid:2) w ( yx ) + x ( vw ) + ( xw + wx ) ( yw + vx ) (cid:3) == (cid:2)(cid:0) x v + ( xw + wx ) y (cid:1) w + (cid:0) w y + ( xw + wx ) v (cid:1) x (cid:3) w ( yx ) + x ( vw ) + ( xw ) ( yw ) + ( xw ) ( vx ) + ( wx ) ( yw ) + ( wx ) ( vx ) == (cid:0) x v (cid:1) w +(( xw ) y ) w +(( wx ) y ) w + (cid:0) w y (cid:1) x +(( xw ) v ) x +(( wx ) v ) x , therefore (cid:0) w , y, x (cid:1) + (cid:0) x , v, w (cid:1) + ( xw, y, w ) + ( wx, y, w ) + ( wx, v, x ) + ( xw, v, x ) = 0 . Comparing the coefficients of λ , we obtain (cid:2) ( xw + wx ) yx + x ( yw + vx ) (cid:3) = (cid:2)(cid:0) x y (cid:1) w + (cid:0) x v + ( xw + wx ) y (cid:1) x (cid:3) → ( xw ) ( yx ) + ( wx ) ( yx ) + x ( yw ) + x ( vx ) == (cid:0) x y (cid:1) w + (cid:0) x v (cid:1) x + (( xw ) y ) x + (( wx ) y ) x , therefore (cid:0) x , y, w (cid:1) + ( xw, y, x ) + ( wx, y, x ) = 0 , which is similar with relation (2 . Proposition 2.10.
In all algebras A t , obtained by the Cayley-Dickson pro-cess, the following identities are satisfied: (cid:0) x n , y, x n (cid:1) = 0 , n ∈ N and char K = 3;( x n , y, x n ) = 0 , n ∈ N ; (cid:0) x n , y, x n (cid:1) = 0 , n ∈ N ;8 x n , y, x n (cid:1) = 0 , n ∈ N . Proof.
We use relation (2 . (cid:0) w , v, x (cid:1) + ( xw, v, w ) + ( wx, v, w ) = 0We take w = x n , v = y, x = x n . We obtain: (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) = 0, therefore (cid:0) x n , y, x n (cid:1) = 0 if char K =3; We take w = x n , v = y, x = 1. We obtain: (cid:0) x n , y, (cid:1) + ( x n , y, x n ) + ( x n , y, x n ) = 0, therefore ( x n , y, x n ) = 0;We take w = x, v = y, x = x . We obtain: (cid:0) x , y, x (cid:1) + (cid:0) x , y, x (cid:1) + (cid:0) x , y, x (cid:1) = 0, therefore (cid:0) x , y, x (cid:1) = 0;We take w = x , v = y, x = x . We obtain: (cid:0) x , y, x (cid:1) + (cid:0) x , y, x (cid:1) + (cid:0) x , y, x (cid:1) = 0, therefore (cid:0) x , y, x (cid:1) = 0;We take w = x n , v = y, x = x n . We obtain: (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) = 0, therefore (cid:0) x n , y, x n (cid:1) = 0;We take w = x n , v = y, x = x n . We obtain: (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) + (cid:0) x n , y, x n (cid:1) = 0, therefore (cid:0) x n , y, x n (cid:1) = 0. Proposition 2.11.
Let A t be an algebra obtained by the Cayley-Dicksonprocess and a ∈ A t .1) If a ∈ A t − K, then the solution in A t of the equation x = a are thesolutions in K of the following system: (cid:26) x − n ( x ) = a x x i = a i , i ∈ { , , ..., t − } , where a = t − P i =0 a i f i , x = t − P i =0 x i f i , a i , x i ∈ K .2) If a ∈ K, then the solution in A t of the equation x = a are the solutionsin K of the following system: (cid:26) x − n ( x ) = a x x i = 0 , i ∈ { , , ..., t − } . Proof.
1) Let a ∈ A t − K , a = t − P i =0 a i f i . Since A t is a quadratic algebra,we have that x = 2 x x − n ( x ) = a , therefore 2 x − n ( x ) = a and 2 x x i = a i , i ∈ { , , ..., t − } .2) Let a ∈ K . It results x = 2 x x − n ( x ) = a . We get 2 x − n ( x ) = a and 2 x x i = 0 , i ∈ { , , ..., t − } . If x = 0 , then n ( x ) = − a . If x = 0 , then x i = 0 , for all i ∈ { , , ..., t − } , and x is the solution in K of the equation x = a . Remark 2.12.
Let a ∈ A t . We denote a = a + −→ a , a ∈ K . Therefore allelements b ∈ A t of the form b = τ + θ −→ a , τ, θ ∈ K commute with a .9 . Some remarks regarding the left and right spectrum of algebrasobtained by the Cayley-Dickson process Let A t be a division algebra obtained by the Cayley-Dickson process, and A ∈ M n ( A t ) , a matrix of order n . Due to noncommutativity of these algebras,we have two distinct notions: the left and right eigenvalues. An element λ ∈ A t is called a left eigenvalue for the matrix A if there is a nonzero matrix X ∈M n × ( A t ) such that AX = λX. The set of distinct left eigenvalues is called the left spectrum of the matrix A ,denoted σ L ( A ).An element λ ∈ A t is called a right eigenvalue for the matrix A if there is anonzero matrix X ∈ M n × ( A t ) such that AX = Xλ.
The set of distinct right eigenvalues is called the right spectrum of the matrix A , denoted σ R ( A ) . If a right eigenvalue λ is in K , therefore λ is a left eigenvalueand viceversa.There are many papers devoted to the study of the left or right spectrum forthe quaternionic and octonionic matrices, ( See [HS; 01], [Ti; 00]). In [Br; 51],Theorem 1, the author proved that every matrix with coefficients in H , the realdivision quaternion algebra, has at least a right eigenvalue. After more than 30years, in [Wo; 85], Wood has proved a similar result for the left eigenvalues.In the case of a generalized quaternion division algebra H K ( α, β ), the aboveresults are not always true. Example 3.1.
We consider the matrix A = (cid:18) ik (cid:19) , A ∈ M ( H ) . We compute the left and the right spectrum. For the left spectrum, we have (cid:18) ik (cid:19) (cid:18) x x (cid:19) = λ (cid:18) x x (cid:19) . From Proposition 2.11, it results the following equations: i x = λx and k x = λx . We obtain that x = − i λx and k x = − λ i λx . Therefore, k = − λ i λ and ik = − ( i λ ) . We obtain that j = ( i λ ) . Denoting y = j , we get y =2 y y − (cid:0) y + y + y + y (cid:1) = j , where y = y + y i + y j + y k . It results y − y − y − y = 0, y = 0 , y y = 1 and y = 0. We get the solutions y ∈ { √ (1 + j ) , − √ (1 + j ) } . Therefore, λ ∈ {− √ ( i + k ) , √ ( i + k ) } . (cid:18) ik (cid:19) (cid:18) x x (cid:19) = (cid:18) x x (cid:19) λ. From Proposition 2.11, it results the following equations: i x = x λ and k x = x λ . We obtain that x = − i x λ and k x = − i x λ . Therefore, − j x = x λ .We have − x − j x = λ and we obtain λ = (cid:0) − x − j x (cid:1) (cid:0) − x − j x (cid:1) , thus λ +1 =0. Denoting y = λ , the solutions in H of the equation y + 1 = 0are of the form y = α i + β j + γ k , with a, β, γ ∈ R , such that α + β + γ = 1.From Proposition 2.11, the solution in H of the equation z = α i + β j + γ k , where a, β, γ ∈ R , such that α + β + γ = 1 are of the form δ √ + δ √ α i + δ √ β j + δ √ γ k , or of the form δ √ + δ √ α i + δ √ β j + δ √ γ k , with δ , δ ∈ {− . } and δ δ = − , therefore an infinite number of solutions.If we consider A ∈ M ( H Q ( − , − H Q ( − , −
1) is the quaterniondivision algebra over the rational field Q , we have that σ L ( A ) = σ R ( A ) = ∅ .The right spectrum of a quaternionic matrix was studied more than leftspectrum. The left eigenvalues were not so studied since the left spectrum isnot always easy to found. It is clear that these two notions, left and rightspectrum, are different and a left eigenvalue is not always a right eigenvalue andvice-versa, as we can see in the below examples. Example 3.2.
The algebra A = (cid:16) − , − Q (cid:17) is a division quaternion algebraover Q . We consider the matrix A = (cid:18) j − j (cid:19) . We compute the left and the right spectrum. For the left spectrum, we have (cid:18) j − j (cid:19) (cid:18) x x (cid:19) = λ (cid:18) x x (cid:19) . It results the following equations j x = λx and − j x = λx , λ, x , x ∈ A .From here, we have that x = − j λx and − j x = − λ j λx , which implies4 j = λ j λ , therefore ( j λ ) = −
16. We denote y = j λ and we obtain equation y + 16 = 0. The solutions in A of the equation y + 16 = 0are of the form y = α i + β j + γ k , with a, β, γ ∈ Q , such that α + 4 β + 4 γ = 16.Since λ = − j y , it results that σ L ( A ) is infinite.11e compute now the right spectrum. From relation (cid:18) j − j (cid:19) (cid:18) x x (cid:19) = (cid:18) x x (cid:19) λ. It results the following relations j x = x λ and − j x = x λ, λ, x , x ∈ A .We have x = − j x λ and − j x = − j x λλ , therefore λ = 4, with λ ∈ A .Therefore, λ ∈ {− , } . (Theorem 2.4, [HS; 01]). We obtain σ R ( A ) = {− , } .It is clear that σ R ( A ) ⊂ σ L ( A ) .For the quaternion real division algebra the following result was proved. Theorem 3.3. (Theorem 4.5, [HS; 01]). If A ∈ M ( H ) and σ L ( A ) and σ R ( A ) are both finite, then σ L ( A ) = σ R ( A ).In the following, we will generalize this result. Let A t be a division algebraobtained by the Cayley-Dickson process, and A ∈ M n ( A t ). Proposition 3.4. If A ∈ M n ( K ), σ R ( A ) = ∅ and σ L ( A ) = ∅ , then, in A t , we have | σ L ( A ) | = | σ R ( A ) | . Proof.
Let A ∈ M n ( K ) and λ ∈ σ L ( A ). It results that there is a nonzeromatrix X ∈ M n × ( A t ) such that AX = λX . Taking conjugate, we obtain AX = Xλ, therefore λ is a right eigenvalue for the matrix A and conversely. Proposition 3.5.
Let A ∈ M ( K ). Therefore σ L ( A ) = σ R ( A ) in A t . Proof. If A ∈ M ( K ) , A = (cid:18) a bc d (cid:19) , A = O . We will compute theright and the left spectrum in the case of λ ∈ A t .For the right spectrum, let λ ∈ σ R ( A ). There is X ∈ M × ( A t ) such that AX = Xλ . We obtain the system: (cid:26) ax + bx = x λcx + dx = x λ . Supposing b = 0, it results x = x ( λ − a ) b − and cx + dx ( λ − a ) b − = x ( λ − a ) b − λ . We obtain the equation b − λ + (cid:0) − ab − − db − (cid:1) λ + dab − − c = 0 λ − ( a + d ) λ + ad − cb = 0 . If λ = λ + −→ λ , with λ ∈ K and (cid:16) −→ λ (cid:17) = θ ∈ K , we get λ = 2 λ λ − n ( λ ).Therefore, − ( a + d − λ ) λ + ad − cb − n ( λ ) = 0. Case 1. λ − λ ( a + d ) + ad − cb − n ( λ ) = 0 and a + d − λ = 0.If a + d − λ = 0, we have λ = a + d and n ( λ ) = λ − αλ − βλ + αβλ = ad − cb , for quaternions and n ( λ ) = λ − αλ − βλ + αβλ − γλ + αγλ + βγλ − αβγλ == ad − cb , for octonions. Therefore, λ = a + d + −→ λ , where n ( −→ λ ) = ad − cb − ( a + d ) .12 ase 2. λ − λ ( a + d ) + ad − cb − n ( λ ) = 0 , a + d − λ = 0 and −→ λ = 0.If −→ λ = 0 , therefore λ = λ , where λ is solution in K of the equation λ − λ ( a + d ) + ad − cb = 0.For the left spectrum, let λ ∈ σ L ( A ). There is X ∈ M × ( A ) such that AX = λX . We obtain the system (cid:26) ax + bx = λx cx + dx = λx . Supposing b = 0, itresults x = ( λ − a ) b − x and cx + d ( λ − a ) b − x = λ ( λ − a ) b − x . Weobtain the same equation and cases as above.The above result generalize Theorem 4.5 from [HS; 01]. Example 3.6.
Let A = (cid:18) − (cid:19) . We obtain the equation λ − λ +11 =0. The quaternionic case. Case 1. λ = and 2 λ − λ ( a + d ) + ad − cb − n ( λ ) = 0. For quaternions,we have λ + λ + λ = 11 − = .If we work on the quaternion division algebra over the field Q , we havean infinite number of solutions. Therefore, λ = + λ i + λ j + λ k , with λ , λ , λ ∈ Q , such that λ + λ + λ = .If we work on the quaternion division algebra over the field R , we haveinfinitely many solutions: λ = + λ i + λ j + λ k , with λ , λ , λ ∈ R , suchthat λ + λ + λ = . Case 2.
The equation λ − λ + 11 = 0 has no real solutions.The octonionic case. λ = and λ + λ + λ + λ + λ + λ + λ = . If we work on thequaternion division algebra over the the field Q , we have an infinite number ofsolutions of the above equation. Therefore, λ = + P i =1 λ i f i , with λ i ∈ Q , suchthat P i =1 λ i = .If we work on the quaternion division algebra over the field R , we haveinfinitely many solutions: λ = + P i =1 λ i f i , with λ , λ , λ ∈ R , such that P i =1 λ i = . Definition 3.7.
Let A t be a division algebra obtained by the Cayley-Dickson process and A ⊆ A t be an associative subalgebra of A t . An element λ ∈ A is called a local A - left eigenvalue for the matrix A if there is a nonzeromatrix X ∈ M n × ( A ) such that AX = λX. The set of distinct local A -left eigenvalues is called the local A - left spectrum ofthe matrix A in A , denoted σ A L ( A ). 13n element λ ∈ A is called a local A - right eigenvalue for the matrix A ifthere is a nonzero matrix X ∈ M n × ( A ) such that AX = Xλ.
The set of distinct right eigenvalues is called the local A - right spectrum ofthe matrix A , denoted σ A R ( A ). Example 3.8.
We consider the real division octonion algebra A = (cid:0) − , − , − R (cid:1) .In this algebra, the elements f , f , f associate, therefore generate an associ-atiative subalgebra of A , denoted by A . Let A = (cid:18) f f (cid:19) M n ( A ). Wecompute the local A -left and right spectrum of the matrix A .For the local A -left spectrum, we have (cid:18) f f (cid:19) (cid:18) x x (cid:19) = λ (cid:18) x x (cid:19) .We obtain the system (cid:26) f x = λx f x = λx . We have x = − f λx and f x = λx .It results f x = − λf λx , then f = − λf λ , therefore f f = − f λf λ .We denote y = f λ and we get y = f . It results 2 y y − n ( y ) = f → y − n ( y ) = 0 and 2 y y = 1 , y = 0 , y = 0, y − y = 0. We get y = y = √ or y = y = − √ . We obtain f λ = √ + √ f or f λ = − √ − √ f , therefore λ ∈ { √ f + √ f , − √ f − √ f } For the local A -right spectrum, we obtain the system (cid:26) f x = x λf x = x λ . Itresults, x = − f x λ and f x = − f x λ , therefore f x = − x λ . We get − x − f x = λ , then λ + 1 = 0.With the above notations we can generalize Theorem 3.3. Proposition 3.9.
Let A t be a division algebra obtained by the Cayley-Dickson process and A ⊆ A t be an associative subalgebra of A t . If A ∈ M n ( A ) and σ A R ( A ) is finite, then σ A R ( A ) ⊆ σ A L ( A ). Proof.
Supposing that λ ∈ σ A R ( A ), therefore there is a nonzero matrix X ∈ M n × ( A ) such that AX = Xλ . For y ∈ A , y = 0 , we obtain that AXy = Xyy − λy . Since A is associative, it results that A ( Xy ) = ( Xy )( y − λy ).From here, we get that y − λy ∈ σ A R ( A ). Therefore, < λ > ⊆ σ A R ( A ), where < λ > = { q ∈ A / q = w − λw, w ∈ A , w = 0 } . It results that, if λ / ∈ K , then < λ > * K and < λ > contains infinitely many distinct quaternions. Therefore, < λ > is an infinite set. If λ ∈ K , we have that < λ > = { λ } and σ A R ( A ) ⊂ K .From here, we get that σ A R ( A ) ⊆ σ A L ( A ). Conclusions.
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