# Conic Representations of Topological Groups

## TL;DR

This paper introduces the concept of conic representations of topological groups, explores their relation to dynamical systems and affine representations, and investigates irreducible cases, especially for semi-simple Lie groups.

## Contribution

It defines conic representations, connects them to dynamical systems and affine representations, and analyzes irreducible cases, including specific groups like SL(2,R).

## Key findings

- Conic representations determine associated dynamical systems with multipliers.
- No universal irreducible conic representation exists for all groups.
- Irreducible conic representations of semi-simple Lie groups embed into regular conic representations.

## Abstract

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 $SL_{2}\left(\mathbb{R}\right)$.