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4.4. Symmetric Novikov algebras
symmetric Novikov algebra of dimension 6 up to isomorphisms. However, there exist noncommutative
symmetric Novikov algebras that are not 2-step nilpotent [AB10]. For example,
⊥
let N = g6 ⊕ Cc where g6 is the 6-dimensional elementary quadratic Lie algebra in Proposition
2.2.29 and Cc is a pseudo-Euclidean simple Jordan algebra with the bilinear form Bc(c,c) = 1
(obviously, this algebra is a symmetric Novikov algebra and commutative). Then N becomes
a symmetric Novikov algebra with the bilinear form defined by B = Bg6 + Bc where Bg6 is
the bilinear form on g6. We can extend this example for the case N = g ⊥
⊕ J where g is a
2-step nilpotent quadratic Lie algebra and J is a symmetric Jordan-Novikov algebra defined
below. However, these algebras are decomposable. An example in the indecomposable case of
dimension 7 can be found in the last part of this section.
Proposition 4.4.12. Let N be a Novikov algebra. Assume that its product is commutative,
that means xy = yx, for all x,y ∈ N. Then the identities (III) and (IV) of Definition 4.4.1 are
equivalent to the only condition:
(x,y,z) = (xy)z − x(yz) = 0, ∀ x,y,z ∈ N.
That means that N is an associative algebra. Moreover, N is also a Jordan algebra. In this
case, we say that N is a Jordan-Novikov algebra. In addition, if N has a non-degenerate associative
symmetric bilinear form, then we say that N is a symmetric Jordan-Novikov algebra.
Proof. Assume N is a commutative Novikov algebra. By (1) of Proposition 4.4.6, the product
is also associative. Conversely, if one has the condition:
(xy)z − x(yz) = 0, ∀ x,y,z ∈ N
then (III) is zero and (IV) is obtained by (yx)z = y(xz), for all x,y,z ∈ N.
Example 4.4.13. Recall the pseudo-Euclidean Jordan algebra J in Example 4.2.10 spanned by
{x,x1,y1} where the commutative product on J is defined by:
y 2 1 = y1,y1x = x,y1x1 = x1,x 2 = x1.
It is easy to check that this product is also associative. Therefore, J is a symmetric Jordan-
Novikov algebra with the bilinear form B defined by B(x1,y1) = B(x,x) = 1 and the other zero.
Example 4.4.14. Pseudo-Euclidean 2-step nilpotent Jordan algebras are symmetric Jordan-
Novikov algebras.
Remark 4.4.15.
(1) By Lemma 4.4.8, if the symmetric Novikov algebra N has Ann(N) = {0} then [x,y] =
xy − yx = 0, for all x,y ∈ N. It implies that N is commutative and then N is a symmetric
Jordan-Novikov algebra.
(2) If the product on N is associative then it may not be commutative. An example can be
found in the next part.
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