  We define basic notions in the category of conic representations of a
topological group and prove elementary facts about them. We show that a conic
representation determines an ordinary dynamical system of the group together
with a multiplier, establishing facts and formulae connecting the two
categories. The topic is also closely related to the affine representations of
the group. The central goal was attaining a better understanding of irreducible
conic representations of a group, and - particularly - to determine whether
there is a phenomenon analogous to the existence of a universal irreducible
affine representation of a group in our category (the general answer is
negative). Then we inspect embeddings of irreducible conic representations of
semi-simple Lie groups in some "regular" conic representation they possess. We
conclude with what is known to us about the irreducible conic representations
of $SL_{2}\left(\mathbb{R}\right)$.
