# A Construction of Representations of Loop Group and Affine Lie Algebra of $\mathfrak{sl}_n$

## TL;DR

This paper constructs a new representation of the loop group and affine Lie algebra of sl_n, providing explicit formulas and a realization of Whittaker functionals, advancing the understanding of their structure and representations.

## Contribution

It introduces a novel construction of loop group and affine Lie algebra representations with explicit formulas and realizations of Whittaker functionals.

## Key findings

- Explicit representation formulas for loop group and affine Lie algebra of sl_n.
- Concrete realization of Whittaker functionals in the dual representation.
- Advancement in understanding the structure of affine Lie algebra representations.

## Abstract

In this paper, we construct a representation of loop group and derive the formula of the corresponding representation of the affine Kac-Moody algebra with level 1. And we also provide a concrete realization of Whittaker functionals in the dual representation.