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

4.3. Pseudo-Euclidean 2-step nilpotent Jordan algebras
However, two forms Iλ1 and Iλ2 where λ1,λ2 = 0 are equivalent by the isomorphism tA satisfying tA(Iλ1 ) = Iλ2 defined by:
t A(x ∗ ) = αy ∗ , t A(y ∗ ) = βx ∗
where α,β ∈ C such that α 3 = λ1λ 2 2 and β 3 = 1
λ 2 1 λ2
that there are only two i-isomorphic classes of T ∗ -extensions of a.
and satisfying tA(Iλ1 ) = Iλ2 . That implies
Example 4.3.19. Let J0 = span{x,y,e, f } be a T ∗ -extension of a 2-dimensional vector space a
by I0 = (x ∗ ) 2 y ∗ , with e = x ∗ and f = y ∗ , that means B(x,e) = B(y, f ) = 1, the other are zero. It
is easy to compute the product in J0 defined by x 2 = f , xy = e. Let I λ = x ∗ y ∗ (x ∗ + λy ∗ ), λ = 0
and J λ = span{x,y,e, f } be two T ∗ -extensions of the 2-dimensional vector space a by I λ . The
products on J λ are x 2 = f , xy = e+λ f and y 2 = λe. These two algebras are neither i-isomorphic
nor isomorphic. Indeed, if there is A : J0 → J λ an isomorphism. Assume A(y) = α1x + α2y +
α3e + α4 f then
0 = A(y 2 ) = (α1x + α2y + α3e + α4 f ) 2 = α 2 1x 2 + 2α1α2xy + α 2 2y 2 .
We obtain (λα 2 2 + 2α1α2)e + (2λα1α2 + α 2 1 ) f = 0. Hence, α1 = ±λα2. Both cases imply
α1 = α2 = 0 (a contradiction).
We can also conclude that there are only two isomorphic classes of T ∗ -extensions of a.
115