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Spin Representations and Binary Numbers

Henrik Winther

Abstract

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.

Spin Representations and Binary Numbers

Abstract

We consider a construction of the fundamental spin representations of the simple Lie algebras in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a -graded associative algebra (rather than the usual -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 . Additionally we can encode the spin representations combinatorially as (coloured) graphs.
Paper Structure (9 sections, 13 theorems, 37 equations)

This paper contains 9 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Arithmetic operators
  4. The odd spin algebra
  5. The even spin algebra
  6. Graph algorithm
  7. Compact real form $\operatorname{Spin}(N)$
  8. Real structure
  9. Future directions

Key Result

Theorem 1

Let $N>0$. The fundamental spin representation $(\mathbb{S}_{2N+1}, \mathfrak{so}(2N+1,\mathbb{C}))$ is equivalent to $(\mathbb{C} \{1,0\}^N, \langle A^{\pm k}\circ A^{\mp (k-1)}| 0\le k<N \rangle)$ where $\mathbb{C} \{1,0\}^N$ is a free complex vector space generated by binary strings of fixed leng and adding or subtracting fractions $2^{-1}$ is simply disregarded. By $\langle,\rangle$ we mean th

Theorems & Definitions (29)

  • Theorem 1
  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 19 more