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4.4. Symmetric Novikov algebras
Proposition 4.4.25. Let N be a symmetric Novikov algebra. If J(N) is a 2-step nilpotent Jordan
algebra then N is a 2-step nilpotent Novikov algebra.
Proof. Since (4) of Proposition 4.4.21, if x,y,z ∈ N then one has
[[x,y]+,z]+ = [x,y]+z + z[x,y]+ = 2[x,y]+z = 0.
It means [x,y]+ = xy + yx ∈ Ann(N). On the other hand, [x,y] = xy − yx ∈ Ann(N) then xy ∈
Ann(N), for all x,y ∈ N. Therefore, N is 2-step nilpotent.
By Proposition 4.4.11, since the lowest dimension of non-Abelian 2-step nilpotent quadratic
Lie algebras is six then examples of non-commutative symmetric Novikov algebras must be at
least six dimensional. One of those can be found in [ZC07] and it is also described in term of
double extension in [AB10]. We recall this algebra as follows:
Example 4.4.26. First, we define the character matrix of a Novikov algebra N = span{e1,...,en}
by ⎛
⎞
⎜
⎝
∑k c k 11 ek ... ∑k c k 1n ek
.
. ..
∑k c k n1 ek ... ∑k c k nnek
where ck i j are the structure constants of N, i. e. eie j = ∑k ck i jek. Now, let N6 be a 6-dimensional vector space spanned by {e1,...,e6} then N6 is a noncommutative
symmetric Novikov algebras with character matrix
⎛
⎞
0 0 0 0 0 0
⎜
⎜0
0 0 0 0 0 ⎟
⎜
⎜0
0 0 0 0 0 ⎟
⎜0
0 0 0 e3 0 ⎟
⎜
⎝
0 0 0 0 0 e1
0 0 0 e2 0 0
and the bilinear form B defined by:
⎛
0
⎜
⎜0
⎜
⎜0
⎜
⎜1
⎝0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
⎞
0
0 ⎟
1 ⎟
0⎟.
⎟
0⎠
0 0 1 0 0 0
Obviously, in this case, N6 is a 2-step nilpotent Novikov algebra with Ann(N) = N 2 . Moreover,
N6 is indecomposable since it is non-commutative and all of symmetric Novikov algebras
up to dimension 5 are commutative.
In fact, in the next proposition, we prove that all non-commutative symmetric Novikov
algebras of dimension 6 are 2-step nilpotent. We need the following lemma:
124
.
⎟
⎠,
⎟
⎠