A double extension (D-extension) of a Lie (super)algebra a with a non-degenerate invariant symmetric bilinear form B, briefly, a NIS-(super)algebra, is an enlargement of a by means of a central extension and a derivation; the affine Kac–Moody algebras are the best known examples of double extensions of loops algebras. Let a be a restricted Lie (super)algebra with a NIS B. Suppose a has a restricted derivation D such that B is D-invariant. We show that the double extension of a constructed by means of B and D is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a D-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of D-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.
- Double extension
- Restricted Lie (super)algebra
- Vectorial Lie (super)algebra
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