  We introduce the concept of algebra eigenstates which are defined for an
arbitrary Lie group as eigenstates of elements of the corresponding complex Lie
algebra. We show that this concept unifies different definitions of coherent
states associated with a dynamical symmetry group. On the one hand, algebra
eigenstates include different sets of Perelomov's generalized coherent states.
On the other hand, intelligent states (which are squeezed states for a system
of general symmetry) also form a subset of algebra eigenstates. We develop the
general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups.
Complete solutions to the general eigenvalue problem are found in the both
cases, by a method that employs analytic representations of the algebra
eigenstates. This analytic method also enables us to obtain exact closed
expressions for quantum statistical properties of an arbitrary algebra
eigenstate. Important special cases such as standard coherent states and
intelligent states are examined and relations between them are studied by using
their analytic representations.
