Principal $\hat{sl}(3)$ subspaces and quantum Toda Hamiltonian

TL;DR

This paper explores representations of a Lie algebra related to sl(3), deriving character formulas and connecting them to quantum Toda Hamiltonian eigenfunctions, revealing new algebraic and analytical insights.

Contribution

It introduces Weyl-type character formulas for specific Lie algebra representations and links them to quantum Toda eigenfunctions, advancing understanding of quantum algebraic structures.

Findings

Derived Weyl-type character formulas for sl(3) representations

Connected bosonic formulas to Whittaker vectors in quantum groups

Obtained fermionic formulas for quantum Toda Hamiltonian eigenfunctions

Abstract

We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.