  We analyze the structure of the group of (local) non-linear canonical
transformations that exist in a system with n fermionic modes. To perform our
study we develop an alternative framework to represent the generators of these
canonical transformations; indeed we show how their definition, understanding
and control is significantly improved using the Majorana fermion
representation. These canonical transformations have the structure of a Lie
group and we provide a representation for the elements of the Lie algebra that
is very convenient both conceptually and practically (computationally): indeed
our framework yields as side product an extremely effective tool to handle and
work with SU(2n) Lie groups and algebras. Granting an enhanced control of the
group of non-linear canonical transformation, our framework can be helpful in
the study of strongly correlated electron systems, since it allows to easily
identify fermionic degrees of freedom able to capture part of the correlations
and thus may give a simpler representation of the Hamiltonian. Thanks to our
analysis, also symmetry-based studies of the quantum Hamiltonians can be
improved, since the simple representation of the generators of the canonical
group permits to identify and understand otherwise hidden symmetries of
difficult interpretation. The main aim of this work is to provide a
comprehensive, general and scrupulous analysis of this framework, that we
already applied in some circumstances. Therefore specific applications will not
be presented in detail, since they can be found elsewhere, while only the
formal and conceptual aspects will be developed thoroughly.
