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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4.1. Preliminaries<br />
T (x,y) = φ(x)(y), for all x,y ∈ J then T is non-degenerate. Moreover, it is easy to check that T<br />
is also associative since R and R ∗ are equivalent by φ. However, T is not necessarily symmetric.<br />
Then we need construct a symmetric bilinear form that is still non-degenerate and associative<br />
from T as follows. Define the symmetric (resp. skew-symmetric) part Ts (resp. Ta) by<br />
Ts(x,y) = 1<br />
2 (T (x,y) + T (y,x)) (resp. Ta(x,y) = 1<br />
(T (x,y) − T (y,x))), ∀ x,y ∈ J.<br />
2<br />
By straightforward checking, Ts and Ta are also associative. Consider subspaces<br />
Js = {x ∈ J | Ts(x,J) = 0} and Ja = {x ∈ J | Ta(x,J) = 0}.<br />
If x ∈ Js ∩ Ja then T (x,J) = Ts(x,J) + Ta(x,J) = 0. So x = 0 since T is non-degenerate. It<br />
means Js ∩ Ja = {0}. Moreover, Js and Ja are also ideals of J since Ts and Ta are associative.<br />
Now, for all x,y,z ∈ J, Ta(xy,z) = Ta(x,yz) = −Ta(xy,z). Therefore, J 2 ⊂ Ja and then J 2 s ⊂<br />
Js ∩ Ja = {0}. Let J = W ⊕ Js where W is a complemenary subspace of Js in J. It is obvious<br />
that Ja ⊂ W. Therefore, WW ⊂ J 2 ⊂ W and WJs = {0}. Consider F : Js × Js → C be a nondegenerate<br />
symmetric bilinear form on Js. Since J 2 s = {0} then F is associative. Finally, we<br />
define the bilinear form B : J × J → C by:<br />
B|W×W = Ts|W×W , B|Js×Js = F and B(W,Js) = B(Js,W) = 0.<br />
Let x = xw +xs,y = yw +ys,z = zw +zs ∈ W ⊕Js. Remark that Ts(Js,J) = 0 so if Ts(xw,W) =<br />
0 then Ts(xw,J) = 0. It implies xw ∈ Js so xw = 0. One has<br />
B(xw + xs,J) = 0 if and only if Ts(xw,W) = 0 and F(xs,Js) = 0.<br />
By preceding remark and F non-degenerate on Js then xw = xs = 0. It means B non-degenerate.<br />
It is easy to see that<br />
Hence, B is associative.<br />
B((xw + xs)(yw + ys),zw + zs) = Ts(xwyw,zw) = Ts(xw,ywzw)<br />
= B(xw + xs,(yw + ys)(zw + zs))<br />
We will need some special subspaces of an arbitrary algebra J:<br />
Definition 4.1.11. Let J be an algebra.<br />
(1) The subspace<br />
is the associator of J.<br />
(2) The subspaces<br />
(J,J,J) = span{(x,y,z) | x,y,z ∈ J}<br />
LAnn(J) = {x ∈ J | xJ = 0},<br />
RAnn(J) = {x ∈ J | Jx = 0} and<br />
Ann(J) = {x ∈ J | xJ = Jx = 0}<br />
are respectively the left-annihilator, the right-annihilator and the annihilator of J. Certainly,<br />
if J is commutative then these subspaces coincide.<br />
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