  Usual quantum statistics is written in Fock space but it is not an algebraic
theory. We show that at a deeper level it can be algebraically formalized
defining the different statistics as (multi-mode) coherent states of the
appropriate (but different from the usual ones) Lie-Hopf groups. The
traditional connection between groups and statistics, established in vacuum, is
indeed subverted by the interaction with the thermal bath. We show indeed that
h(1), related in quantum field theory to bosons, must be used to define in
presence of a bath the Boltzmann statistics while, to build the Bose
statistics, we have to take into account su(1,1). Astonishing to describe
fermions we are forced to use not the superalgebra h(1|1) but su(2) in the
fundamental representation. Higher representations of su(2) allow also to give
a possible definition of anyon statistics with generalized Pauli principle.
Physical implications are discussed; the results is more general then the usual
on the discrete spectrum, but everything collapses to standard theory when the
continuum limit is performed.

Statistical Mechanics