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Keywords:
spin group; fundamental representations; spin matrices; binary numbers
spin group; fundamental representations; spin matrices; binary numbers
Summary:
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.
References:
[1] Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds. Teubner Verlag Leipzig, Stuttgart, 1991. MR 1164864 | Zbl 0734.53003
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