Finite-Dimensional Irreducible Modules for the Three-Point 2 Loop Algebra

Abstract

Recently Brian Hartwig and the second author found a presentation for the three-point ₂ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point ₂ loop algebra. In this article, we describe the finite-dimensional irreducible ⊠-modules from multiple points of view.

Key Words:

• Kac–Moody algebra
• Onsager Lie algebra
• Tetrahedron Lie algebra
• Tridiagonal pair

2000 Mathematics Subject Classification:

• Primary 17B65
• Secondary 17B67

ACKNOWLEDGMENTS

The authors thank Brian Curtin, Eric Egge, Mark MacLean, Kazumasa Nomura, and the anonymous referee for giving this paper a close reading and offering many valuable suggestions.

Tatsuro Ito was supported in part by JSPS grant 18340022.

Notes

Communicated by A. Elduque.