Spin Representations and Binary Numbers
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913251524083712 |
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| author | Winther, Henrik |
| author_facet | Winther, Henrik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00931 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spin Representations and Binary Numbers Winther, Henrik Representation Theory 22E46 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. |
| title | Spin Representations and Binary Numbers |
| topic | Representation Theory 22E46 |
| url | https://arxiv.org/abs/2403.00931 |