Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$

TL;DR

This paper introduces the $rak{gl}_{k,m}$ Yangian, a new quantum algebra derived from involutive, non-combinatorial solutions of the braid equation, and explores its representations and connections to quantum spin chains.

Contribution

It defines the $rak{gl}_{k,m}$ Yangian as a novel algebraic structure from special braid solutions and constructs its highest-weight modules linked to quantum spin-chain Hamiltonians.

Findings

Identified $rak{gl}_{k,m}$ as a subalgebra of the Yangian.

Constructed highest-weight modules with eigenstates of spin-chain Hamiltonians.

Linked $rak{gl}_{1,1}$ representations to Young tableaux shapes.

Abstract

We investigate involutive, non-combinatorial solutions of the braid equation viewed as special deformations of the permutation map. By employing these solutions, we identify the associated quantum algebra, which we introduce as the $\mathfrak{gl}_{k,m}$ Yangian. The algebra $\mathfrak{gl}_{k,m}$ is also recognized as a subalgebra of the Yangian. Furthermore, we construct specific highest-weight modules of $\mathfrak{gl}_{k,m},$ which simultaneously yield the eigenstates of certain quantum spin-chain-like Hamiltonians. In the special case of the algebra $\mathfrak{gl}_{1,1}$ the spin chain Hamiltonian reduces to a variant of the Heisenberg XX model. Furthermore, we present a comprehensive analysis of combinatorial bases of highest weight representations of $\mathfrak{gl}_{1,1}$, explicitly linking them to specific shapes of Young tableaux.