  In this paper we present a new algebraic structure (a super hyperbolic system
in our terminology) for finite quantum systems, which is a generalization of
the usual one in the two-level system.
  It fits into the so-called generalized Pauli matrices, so they play an
important role in the theory. Some deep relation to the modified Bessel
functions of integer order is pointed out.
  By taking a skillful limit finite quantum systems become quantum mechanics on
the circle developed by Ohnuki and Kitakado.
