  Holonomies of the Ashtekar-Barbero connection can be considered as abstract
elements of a Lie group exponentially mapped from their connections
representation. This idea provides a possibility to compare the geometric and
algebraic properties of these objects. The result allows to identify the
next-to-the-leading-order terms in the geometric and algebraic expansion of a
holonomy. This identification leads to the verification of the related Hilbert
space formulation. If states are the representations of the holonomy's symmetry
group, they preserve gauge transformations according to Wigner's theorem. Thus,
the spin network in loop quantum gravity satisfies this theorem. Moreover, the
considered identification of the different expansions ensures the reality of
the Ashtekar connection. Only the holonomies of real connections lead to the
formulation of states that satisfy Wigner's theorem.
