Refining the grading of irreducible Lie colour algebra representations

2024/03/05 by Mitchell Ryan, Ryan, Mitchell
Physics and Astronomy · #(Primary) 17B10 (Secondary) #17B70 #17B75 #Color Science and Applications #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2403.02855

openalex publication_date 2024/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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--Hölder 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 \mathfraksl₂ .

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