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This is a review article in which we will introduce, in a unifying fashionand with more intermediate steps in some difficult calculations, twoinfinite-dimensional Lie algebras of quantum matrix models, one for the openstring sector and one for the closed string sector. Physical observables ofquantum matrix models in the large-N limit can be expressed as elements ofthese Lie algebras. We will see that both algebras arise as quotient algebrasof a larger Lie algebra. We will also discuss some properties of these Liealgebras not published elsewhere yet, and briefly review their relationshipwith well-known algebras like the Cuntz algebra, the Witt algebra and theVirasoro algebra. We will also review how Yang--Mills theory, various lowenergy effective models of string theory, quantum gravity, string-bit models,and quantum spin chain models can be formulated as quantum matrix models.Studying these algebras thus help us understand the common symmetry of thesephysical systems.Comment: 77 pages, 21 eps figures, 1 table, LaTeX2.09; an invited review  articl

A REVIEW OF SYMMETRY ALGEBRAS OF QUANTUM MATRIX MODELS IN THE LARGE N LIMIT