VALUATIONS OF LIE ALGEBRAS

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We introduce the concept of valuations of Lie algebras and completions of valued Lie algebras. We show that if (L, | · |) is a valued Lie algebra then there is the unique completion (L , | · | ) up to isomorphism. We show that if a subalgebra H of L is a weak subideal of L (resp. a subideal of L, finitedimensional, soluble, nilpotent) then its topological closure H in (L , | · | ) is a weak subideal of L (resp. a subideal of L , finite-dimensional, soluble, nilpotent). We also introduce valuations |·| of the generalized Witt algebra WZ and present some properties of (W Z , | · |). Finally we illustrate the completion of (WZ, | · |) with a specific example.

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