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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2.3. Quadratic dimension of quadratic Lie algebras<br />
2.3 Quadratic dimension of quadratic Lie algebras<br />
In this section, we will study an interesting characteristic of a quadratic Lie algebra g called<br />
the quadratic dimension. It is defined by the dimension of the space of invariant symmetric<br />
bilinear forms on g and denoted by dq(g). This notion involves special maps which commute<br />
with inner derivations of g. We call such maps centromorphisms. Some simple properties of a<br />
centromorphism are given in the first subsection. We calculate the formula dq(g) for reduced<br />
singular quadratic Lie algebras in the second subsection and use it to show that dup-number is<br />
invariant under isomorphisms. Finally, we study centromorphisms with respect to some extensions<br />
of a quadratic Lie algebra g.<br />
2.3.1 Centromorphisms of a quadratic Lie algebra<br />
Let (g,B) be a quadratic Lie algebra. To any symmetric bilinear form B ′ on g, there is an<br />
associated map D : g → g satisfying<br />
B ′ (X,Y ) = B(D(X),Y ), ∀ X,Y ∈ g.<br />
Since B and B ′ are symmetric, one has D symmetric (with respect to B), i.e. B(D(X),Y ) =<br />
B(X,D(X)) for all X, Y ∈ g.<br />
Lemma 2.3.1.<br />
(1) B ′ is invariant if and only if D satisfies<br />
D([X,Y ]) = [D(X),Y ] = [X,D(Y )], ∀ X,Y ∈ g. (II)<br />
(2) B ′ is non-degenerate if and only if D is invertible.<br />
Proof.<br />
(1) Let X,Y,Z ∈ g then B ′ ([X,Y ],Z) = B(D([X,Y ]),Z). Since B is invariant one has:<br />
B ′ (X,[Y,Z]) = B(D(X),[Y,Z]) = B([D(X),Y ],Z).<br />
Therefore B ′ is invariant if and only if D([X,Y ]) = [D(X),Y ] since B is non-degenerate.<br />
The Lie bracket anticommutative implies that D([X,Y ]) = [X,D(Y )].<br />
(2) Assume that B ′ is non-degenerate. If X is an element in g such that D(X) = 0 then<br />
B(D(X),g) = 0. Then one has B ′ (X,g) = 0. It implies that X = 0 and therefore D is<br />
invertible. Conversely, if D is invertible then B ′ (X,g) = 0 reduces to B(D(X),g) = 0.<br />
Since B is non-degenerate one has D(X) = 0. Thus X = 0 and B ′ is non-degenerate.<br />
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