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4.4. Symmetric Novikov algebras
4.4 Symmetric Novikov algebras
Definition 4.4.1. An algebra N over C with a bilinear product N×N → N, (x,y) ↦→ xy is called
a left-symmetric algebra if it satisfies the identity:
or in term of associators
It is called a Novikov algebra if in addition
(xy)z − x(yz) = (yx)z − y(xz), ∀ x,y,z ∈ N. (III)
(x,y,z) = (y,x,z), ∀ x,y,z ∈ N.
(xy)z = (xz)y (IV)
holds for all x,y,z ∈ N. In this case, the commutator [x,y] = xy − yx of N defines a Lie algebra,
denoted by g(N), which is called the sub-adjacent Lie algebra of N. It is known that g(N) is a
solvable Lie algebra [Bur06]. Conversely, let g be a Lie algebra with Lie bracket [ , ]. If there
exists a bilinear product g × g → g,(x,y) ↦→ xy that satisfies (III), (IV) and [x,y] = xy − yx, for
all x,y ∈ J then we say that g admits a Novikov structure.
Example 4.4.2. Every 2-step nilpotent algebra N satisfying (xy)z = x(yz) = 0 for all x,y,z ∈ N,
is a Novikov algebra.
For x ∈ N, denote by Lx and Rx respectively the left and right multiplication operators
Lx(y) = xy, Rx(y) = yx, for all y ∈ N. The condition (III) is equivalent to [Lx,Ly] = L [x,y] and
(IV) is equivalent to [Rx,Ry] = 0. In the other words, the left-operators form a Lie algebra and
the right-operators commute.
It is easy to check two Jacobi-type identities:
Proposition 4.4.3. Let N be a Novikov algebra then for all x,y,z ∈ N:
[x,y]z + [y,z]x + [z,x]y = 0,
x[y,z] + y[z,x] + z[x,y] = 0
Definition 4.4.4. Let N be a Novikov algebra. A bilinear form B : N × N → C is called associative
if
B(xy,z) = B(x,yz), ∀ x,y,z ∈ N.
We say that N is a symmetric Novikov algebra if it is endowed with a non-degenerate associative
symmetric bilinear form B.
Let (N,B) be a symmetric Novikov algebra and S be a subspace of N. Denote by S ⊥ the
set {x ∈ N | B(x,S) = 0}. If B|S×S is non-degenerate (resp. degenerate) then we say that S is
non-degenerate (resp. degenerate).
Lemma 4.4.5. Let (N,B) be a symmetric Novikov algebra and I be a two-sided ideal (or simply
an ideal) of N then
(1) I ⊥ is also a two-sided ideal of N and II ⊥ = I ⊥ I = {0}
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