arXiv subjects

Mitchell Ryan

Publications and source records attributed to Mitchell Ryan.

4 recordsLinked to original sources

arXiv2025

The Dirac oscillator, generalised parastatistics and colour Lie superalgebras

We study the Dirac oscillator in one, two and three spatial dimensions, showing that the corresponding ladder operators realise the $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded Lie superalgebras $ \mathfrak{pso}(3|2) $, $ \mathfrak{pso}(3|4) $ and $ \mathfrak{osp}_{01}(1|2) \oplus \mathfrak{sl}_{10}(1|1)$. These algebraic structures are related to parastatistics and their Fock spaces. We demonstrate that these colour algebras and Fock spaces are useful for analysing the Dirac oscillator and its eigenspaces, particularly in $ (1+3) $-dimensions. Apart from this current work, to our knowledge, the recent article by Ito and Nago (arXiv:2501.07311) is the only other such work that makes use of $ \mathbb{Z}_2\times \mathbb{Z}_2 $ graded colour Lie superalgebras in a relativistic setting.

math-ph↗

arXiv2024

Graded colour Lie superalgebras for solving L\'evy-Leblond equations

The L\'evy-Leblond equation with free potential admits a symmetry algebra that is a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra (see arXiv:1609.08224). We extend this result in two directions by considering a time-independent version of the L\'evy-Leblond equation. First, we construct a $ \mathbb{Z}_2^3 $-graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the L\'evy-Leblond equation admits colour Lie superalgebras with gradings higher than $ \mathbb{Z}_2\times\mathbb{Z}_2 $ and colour Lie superalgebras appear for potentials besides the free potential.

math-ph↗

arXiv2024

Refining the grading of irreducible Lie colour algebra representations

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 $.

math-ph↗

arXiv2022

Soliton cellular automata for the affine general linear Lie superalgebra

The box-ball system (BBS) is a cellular automaton that is an ultradiscrete analogue of the Korteweg--de Vries equation, a non-linear PDE used to model water waves. In 2001, Hikami and Inoue generalised the BBS to the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We further generalise the Hikami--Inoue BBS to column tableaux using the Kirillov--Reshetikhin crystals for $\hat{\mathfrak{gl}}{(m|n)}$ devised by Kwon and Okado (arXiv:1804.05456), where we find similar solitonic behaviour under certain conditions.

nlin.SI↗