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Double extensions of restricted Lie algebras. (arXiv:1810.03086v1 [math.RT])
来源于:arXiv
A double extension ($D$-extension) of a Lie algebra ${\mathfrak a}$ with a
non-degenerate invariant symmetric bilinear form $B_{\mathfrak a}$, briefly a
NIS Lie algebra, is an enlargement of ${\mathfrak a}$ by means of a central
extension and a derivation; the affine Kac-Moody algebras are the most known
examples.
Let ${\mathfrak a}$ be a restricted Lie algebra equipped with a NIS
$B_{\mathfrak a}$. Suppose $\mathfrak a$ has a restricted derivation $D$ such
that $B_{\mathfrak a}$ is $D$-invariant. We show that the double extension of
$\mathfrak a$ caries a $p$-mapping constructed by means of $B_{\mathfrak a}$
and $D$. We show that, the other way round, any restricted NIS Lie algebra can
be obtained as a $D$-extension of another restricted NIS Lie algebra of
codimension 2 provided that the center is not trivial together with an extra
condition pertaining to the central element.
We give examples of $D$-extensions of restricted Lie algebras in small
characteristic related with Manin tripl 查看全文>>