TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

4.2. Jordanian double extension of a quadratic vector space
4.2 Jordanian double extension of a quadratic vector space
Let Cc be a one-dimensional Jordan algebra. If c 2 = 0 then c 2 = λc for some non-zero
λ ∈ C. Replace c by 1
λ c, we obtain c2 = c. Therefore, there exist only two one-dimensional
Jordan algebras: one Abelian and one simple. Next, we will study double extensions of a
quadratic vector space by these algebras.
Let us start with (q,Bq) a quadratic vector space. We consider (t = span{x1,y1},Bt) a 2dimensional
quadratic vector space with Bt defined by
Bt(x1,x1) = Bt(y1,y1) = 0, Bt(x1,y1) = 1.
Let C : q → q be a non-zero symmetric map and consider the vector space
equipped with a product defined by
J = q ⊥
⊕ t
(x + λx1 + µy1) (y + λ ′ x1 + µ ′ y1) =
µC(y) + µ ′ C(x) + Bq(C(x),y)x1 + ε λ µ ′ + λ ′ µ x1 + µµ ′
y1 ,
ε ∈ {0,1}, for all x,y ∈ q,λ,µ,λ ′ , µ ′ ∈ C.
Proposition 4.2.1. Keep the notation just above.
(1) Assume that ε = 0. Then J is a Jordan algebra if and only if C 3 = 0. In this case, we call
J a nilpotent double extension of q by C.
(2) Assume that ε = 1. Then J is a Jordan algebra if and only if 3C 2 = 2C 3 +C. Moreover, J is
pseudo-Euclidean with bilinear form B = Bq +Bt. In this case, we call J a diagonalizable
double extension of q by C.
Proof.
(1) Let x, y ∈ q, λ, µ, λ ′ , µ ′ ∈ C. One has
and
((x + λx1 + µy1)(y + λ ′ x1 + µ ′ y1))(x + λx1 + µy1) 2 = 2µBq(C 2 (µy + µ ′ x),C(x))x1
(x + λx1 + µy1)((y + λ ′ x1 + µ ′ y1)(x + λx1 + µy1) 2 ) = 2µ 2 µ ′ C 3 (x)
Therefore, J is a Jordan algebra if and only if C 3 = 0.
(2) The result is achieved by checking directly the equality (I) for J.
104
+2µµ ′ Bq(C(x),C 2 (x))x1.