  Quantum matrix models in the large-N limit arise in many physical systems
like Yang-Mills theory with or without supersymmetry, quantum gravity,
string-bit models, various low energy effective models of string theory,
M(atrix) theory, quantum spin chain models, and strongly correlated electron
systems like the Hubbard model. We introduce, in a unifying fashion, a hierachy
of infinite-dimensional Lie superalgebras of quantum matrix models. One of
these superalgebras pertains to the open string sector and another one the
closed string sector. Physical observables of quantum matrix models like the
Hamiltonian can be expressed as elements of these Lie superalgebras. This
indicates the Lie superalgebras describe the symmetry of quantum matrix models.
We present the structure of these Lie superalgebras like their Cartan
subalgebras, root vectors, ideals and subalgebras. They are generalizations of
well-known algebras like the Cuntz algebra, the Virasoro algebra, the Toeplitz
algebra, the Witt algebra and the Onsager algebra. Just like we learnt a lot
about critical phenomena and string theory through their conformal symmetry
described by the Virasoro algebra, we may learn a lot about quantum
chromodynamics, quantum gravity and condensed matter physics through this
symmetry of quantum matrix models described by these Lie superalgebras.
