Degenerations of central quotients (Q913928)

A theorem is proved and then used to answer some degeneration questions in the variety of nilpotent Lie algebras of dimension 8. The theorem states that if the Lie algebra \(L_ 1\) degenerates to \(L_ 0\) then the central quotient \(L_ 1/Z(L_ 1)\) also degenerates to a modified central quotient of \(L_ 0\) having the same dimension as \(L_ 1/Z(L_ 1)\). This other algebra is \(L_ 0/Z(L_ 0)\oplus (an\) abelian Lie algebra of dimension d), where \(d=\dim (Z(L_ 0))-\dim (Z(L_ 1))\). The proof uses the notion of a complete variety and the Iwasawa decomposition for a reductive algebraic group.