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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2.4. 2-step nilpotent quadratic Lie algebras<br />
Corollary 2.4.18. Let g and g ′ be T ∗ -extensions of h with respect to I1 and I2. Then g and<br />
g ′ are i-isomorphic if and only if there exists an isomorphism A of h such that I1(x,y,z) =<br />
I2(A(x),A(y),A(z)), for all x,y,z ∈ h.<br />
Remark 2.4.19. The element I in this case is exactly the 3-form associated to g in Definition<br />
2.2.2 and the above corollary is only a particular case of Lemma 2.2.8.<br />
Lemma 2.4.20. Let h be a vector space and I ∈ A 3 (h) satisfying ιx(I) = 0, ∀ x ∈ h \ {0}}.<br />
If there are nontrivial subspaces h1, h2 of h such that h = h1 ⊕ h2 and I is decomposed by<br />
I = I1 + I2 where I1 ∈ A 3 (h1), I2 ∈ A 3 (h2). Then the T ∗ -extension g of h with respect to I is<br />
decomposable.<br />
Proof. Let a = h1 ⊕ h ∗ 1 be the T ∗ -extension of h1 with respect to I1 then a is non-degenerate.<br />
We will show that a is an ideal of g. Indeed, one has:<br />
[h1,h1 ⊕ h2] = I(h1,h1 ⊕ h2,.) = I(h1,h1,.) + I(h1,h2,.).<br />
Since I(h1,h2,.) = 0 then [h1,h1 ⊕ h2] = I(h1,h1,.) = I1(h1,h1,.) ⊂ h∗ 1 . Therefore a is an ideal<br />
of g and then g is decomposable.<br />
Remark 2.4.21. Denote by N(2n) the set of i-isomorphism classes of 2n-dimensional reduced<br />
2SNQ-Lie algebras. It is obvious that N(2) = N(4) = /0 and N(6) has only an element (see<br />
Appendix C and Example 2.4.17, also in [PU07] or [Ova07]). By Appendix C and Lemma<br />
2.4.20, N(8) = /0, N(10) contains only an element and N(2n) = /0 if n ≥ 6.<br />
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