Finite-dimensional representations for a class of generalized intersection matrix Lie algebras

In this paper, we study a class of generalized intersection matrix Lie algebras $gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{bf a}, {bf c},{bf d})$. Particularly, any finite dimensional irreducible $gim(M_n)$ module must be an irreducible module of $M(n,{bf a}, {bf c},{bf d})$ and any finite dimensional irreducible $M(n,{bf a}, {bf c},{bf d})$ module must be an irreducible module of $gim(M_n)$.