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4.3. Pseudo-Euclidean 2-step nilpotent Jordan algebras
4.3 Pseudo-Euclidean 2-step nilpotent Jordan algebras
4.3.1 2-step nilpotent Jordan algebras
In Chapter 2, 2-step nilpotent quadratic Lie algebras are characterized up to isometric isomorphisms
and up to isomorphisms (see also [Ova07]). There is a similar natural property in
the case of pseudo-Euclidean 2-step nilpotent Jordan algebras. Let us redefine 2-step nilpotent
Jordan algebras in a more convenient way:
Definition 4.3.1. An algebra J over C with a product (x,y) ↦→ xy is called a 2-step nilpotent
Jordan algebra if it satisfies xy = yx and (xy)z = 0 for all x,y,z ∈ J. Sometimes, we use 2SN-
Jordan algebra as an abbreviation.
The method of double extension is a fundamental tool used in describing algebras that are
endowed with an associative non-degenerate bilinear form. This method is based on two principal
notions: central extension and semi-direct product of two algebras. We have just seen it
in Chapter 2 for 2-step nilpotent quadratic Lie algebras. In the next part, we apply it again but
as we will see that the method of double extension is not quite effective for pseudo-Euclidean
Jordan algebras, even in the 2-step nilpotent case. With our attention we will recall some definitions
given in Section 3 of [BB] but with a restricting condition for pseudo-Euclidean 2-step
nilpotent Jordan algebras.
Proposition 4.3.2. Let J be a 2SN-Jordan algebra, V be a vector space, ϕ : J × J → V be a
bilinear map and π : J → End(V ) be a representation. Let
equipped with the following product:
J = J ⊕V
(x + u)(y + v) = xy + π(x)(v) + π(y)(u) + ϕ(x,y), ∀x,y ∈ J,u,v ∈ V.
Then J is a 2SN-Jordan algebra if and only if for all x,y,z ∈ J one has:
(1) ϕ is symmetric and ϕ(xy,z) + π(z)(ϕ(x,y)) = 0,
(2) π(xy) = π(x)π(y) = 0.
Definition 4.3.3. If π is the trivial representation in Proposition 4.3.2, the Jordan algebra J is
called the 2SN-central extension of J by V (by means of ϕ).
V .
Remark that in a 2SN-central extension J, the annihilator Ann(J) contains the vector space
Proposition 4.3.4. Let J be a 2SN-Jordan algebra. Then J is a 2SN-central extension of an
Abelian algebra.
Proof. Set h = J/J 2 and V = J 2 . Define ϕ : h × h → V by ϕ(p(x), p(y)) = xy, for all x,y ∈ J
where p : J → h is the canonical projection. Then h is an Abelian algebra and J h ⊕V is the
2SN-central extension of h by means of ϕ.
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