  We consider quantum dynamical systems whose degrees of freedom are described
by $N \times N$ matrices, in the planar limit $N \to \infty$. Examples are
gauge theoires and the M(atrix)-theory of strings. States invariant under U(N)
are `closed strings', modelled by traces of products of matrices. We have
discovered that the U(N)-invariant opertors acting on both open and closed
string states form a remarkable new Lie algebra which we will call the heterix
algebra. (The simplest special case, with one degree of freedom, is an
extension of the Virasoro algebra by the infinite-dimensional general linear
algebra.) Furthermore, these operators acting on closed string states only form
a quotient algebra of the heterix algebra. We will call this quotient algebra
the cyclix algebra. We express the Hamiltonian of some gauge field theories
(like those with adjoint matter fields and dimensionally reduced pure QCD
models) as elements of this Lie algebra. Finally, we apply this cyclix algebra
to establish an isomorphism between certain planar matrix models and quantum
spin chain systems. Thus we obtain some matrix models solvable in the planar
limit; e.g., matrix models associated with the Ising model, the XYZ model,
models satisfying the Dolan-Grady condition and the chiral Potts model. Thus
our cyclix Lie algebra described the dynamical symmetries of quantum spin chain
systems, large-N gauge field theories, and the M(atrix)-theory of strings.
