We construct highest weight unitary irreducible representations of a Lie algebra for open quantum matrix chains akin to quotients of Verma modules for simple finite-dimensional Lie algebras. Those representations resembling typical unitary irreducible representations ofgl(n)turn out to be tensor products of the defining representation. They can be physically identified as multiple meson states or multiple open string states. Other representations are intimately related to the Cuntz algebra. They may be related to novel bound states. ©2000 American Institute of Physics.