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