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4.4. Symmetric Novikov algebras
Lemma 4.4.27. Let N be a non-Abelian symmetric Novikov algebra then N = z ⊥
⊕ l where
z ⊂ Ann(N) and l is a reduced symmetric Novikov algebra, that means l = {0} and Ann(l) ⊂ l 2 .
Proof. Let z0 = Ann(N) ∩ N 2 , z is a complementary subspace of z0 in Ann(N) and l = z ⊥ .
If x is an element in z such that B(x,z) = 0 then B(x,N 2 ) = 0 since Ann(N) = (N 2 ) ⊥ . As a
consequence, B(x,z0) = 0 and then B(x,Ann(N)) = 0. Hence, x must be in N 2 since Ann(N) =
(N 2 ) ⊥ . It shows that x = 0 and z is non-degenerate. By Lemma 4.4.5, l is a non-degenerate
ideal and N = z ⊥
⊕ l.
Since N is non-Abelian then l = {0}. Moreover, l 2 = N 2 implies z0 ⊂ l 2 . It is easy to see
that z0 = Ann(l) and the lemma is proved.
Proposition 4.4.28. Let N be a non-commutative symmetric Novikov algebras of dimension 6
then N is 2-step nilpotent.
Proof. Let N = span{x1,x2,x3,z1,z2,z3}. By Remark 2.4.21, there exists only one non-Abelian
2-step nilpotent quadratic Lie algebra of dimension 6 (up to isomorphisms) then g(N) = g6.
We can choose the basis such that [x1,x2] = z3, [x2,x3] = z1, [x3,x1] = z2 and the bilinear form
B(xi,zi) = 1, i = 1,2,3, the other are zero.
Recall that Z(N) = {x ∈ N | xy = yx, ∀ y ∈ N} then Z(N) = {x ∈ N | [x,y] = 0, ∀ y ∈
N}. Therefore, Z(N) = span{z1,z2,z3} and N2 ⊂ Z(N) by Lemma 4.4.24. Consequently,
dim(N2 ) ≤ 3.
By Lemma 4.4.27, if N is not reduced then N = z ⊥
⊕ l with z ⊂ Ann(N) is a non-degenerate
ideal and z = {0}. It implies that l is a symmetric Novikov algebra having dimension ≤ 5
and then l is commutative. This is a contradiction since N is non-commutative. Therefore,
N must be reduced and Ann(N) ⊂ N2 . Moreover, dim(N2 ) + dim(Ann(N)) = 6 so we have
N2 = Ann(N) = Z(N). It shows that N is 2-step nilpotent.
In this case, the character matrix of N in the basis {x1,x2,x3,z1,z2,z3} is given by:
A 0
,
0 0
where A is a 3 × 3-matrix defined by the structure constants xix j = ∑k ck i jzk, 1 ≤ i, j,k ≤ 3, and
B has the matrix: ⎛
⎞
0 0 0 1 0 0
⎜
⎜0
0 0 0 1 0 ⎟
⎜
⎜0
0 0 0 0 1 ⎟
⎜
⎜1
0 0 0 0 0⎟.
⎟
⎝0
1 0 0 0 0⎠
0 0 1 0 0 0
Since B(xix j,xk) = B(xi,xjxk) = B(x j,xkxi) then one has ck i j = ci j
jk = cki , 1 ≤ i, j,k ≤ 3.
Next, we give some simple properties for symmetric Novikov algebras as follows:
Proposition 4.4.29. Let N be a non-commutative symmetric Novikov algebra. If N is reduced
then
3 ≤ dim(Ann(N)) ≤ dim(N 2 ) ≤ dim(N) − 3.
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