Refining the grading of irreducible Lie colour algebra representations

• Published in: arXiv.org 2024-07
• Main Author: Mitchell, Ryan
• Format: Article
• Language: English
• Subjects: Algebra; Color; Hilbert space
• ISSN: 2331-8422

We apply the loop module construction of arXiv:1504.05114 in the context of Lie colour algebras. We construct a bijection between the equivalence classes of all finite-dimensional graded irreducible Lie colour algebra representations from the irreducible representations for Lie superalgebras. This bijection is obtained by applying the loop module construction iteratively to simple groups in the Jordan--H\"older decomposition of the grading group. Restricting to simple groups in this way greatly simplifies the construction. Despite the bijection between Lie colour algebra representations and Lie superalgebra representations, Lie colour algebras maintain a non-trivial representation theory distinct from that of Lie superalgebras. We demonstrate the applicability of the loop module construction to Lie colour algebras in two examples: a Hilbert space for a quantum mechanical model and representations of a colour version of \( \mathfrak{sl}_2 \).