Some properties of the c-nilpotent multiplier of a pair of Lie algebras

The concept of the Schur multiplier of a Lie algebra was introduced by Batten et al. (1996). This concept was extended to that of a c-nilpotent multiplier of a Lie algebra, and then further extended to a theory of a c-nilpotent multiplier of a pair of Lie algebras, by the author and others. In this paper we present some new inequalities for the dimension of the c-nilpotent multiplier of a pair of Lie algebras. We also give a necessary and sufficient condition for the c-nilpotent multiplier of a pair of Lie algebras to embed into the c-nilpotent multiplier of their quotient. Finally we provide a sufficient condition for the c-nilpotent multiplier of a pair of Lie algebras to be finite dimensional.