Spin representations and binary numbers
Archivum mathematicum, Tome 60 (2024) no. 4, pp. 231-241
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We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a $\mathbb{Z}$-graded associative algebra (rather than the usual $\mathbb{N}$-filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some $n$. Additionally we can encode the spin representations combinatorially as (coloured) graphs.
We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a $\mathbb{Z}$-graded associative algebra (rather than the usual $\mathbb{N}$-filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some $n$. Additionally we can encode the spin representations combinatorially as (coloured) graphs.
DOI :
10.5817/AM2024-4-231
Classification :
22E46
Keywords: spin group; fundamental representations; spin matrices; binary numbers
Keywords: spin group; fundamental representations; spin matrices; binary numbers
@article{10_5817_AM2024_4_231,
author = {Winther, Henrik},
title = {Spin representations and binary numbers},
journal = {Archivum mathematicum},
pages = {231--241},
year = {2024},
volume = {60},
number = {4},
doi = {10.5817/AM2024-4-231},
mrnumber = {4833550},
zbl = {07980752},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2024-4-231/}
}
Winther, Henrik. Spin representations and binary numbers. Archivum mathematicum, Tome 60 (2024) no. 4, pp. 231-241. doi: 10.5817/AM2024-4-231