A review of symmetry algebras of quantum matrix models in the large N limit

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infini te-dimensional Lie algebras of quantum matrix models, one for the open stri ng sector and the other for the closed string sector. Physical observables of quantum matrix models in the large N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient alg ebras of a larger Lie algebra. We will also discuss some properties of thes e Lie algebras not published elsewhere yet, and briefly review their relati onship with well-known algebras like the Cuntz algebra, the Witt algebra an d the Virasoro algebra. We will also review how the Yang-Mills theory, vari ous low energy effective models of string theory, quantum gravity, string-b it models, and the quantum spin chain models can be formulated as quantum m atrix models. Studying these algebras thus help us understand the common sy mmetry of these physical systems.