The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions

TL;DR

This paper reveals that certain Morita equivalences involving quaternions and Clifford algebras can be derived from the process of quantizing specific Hamiltonian reductions related to spin groups and exceptional Lie groups.

Contribution

It establishes a novel connection between quaternionic and Clifford algebra Morita equivalences and the geometric process of Hamiltonian reduction in a quantum setting.

Findings

Morita equivalence $ ext{Cliff}(4) o ext{H}$ from quantizing ${f R}^{0|4} // ext{Spin}(3)$

Morita equivalence $ ext{Cliff}(7) o ext{Cliff}(-1)$ from quantizing ${f R}^{0|7} // G_2$

Morita equivalence $ ext{Cliff}(8) o ext{R}$ from quantizing ${f R}^{0|8} // ext{Spin}(7)$

Abstract

We show that the Morita equivalences $\mathrm{Cliff}(4) \simeq {\mathbb H}$, $\mathrm{Cliff}(7) \simeq \mathrm{Cliff}(-1)$, and $\mathrm{Cliff}(8) \simeq {\mathbb R}$ arise from quantizing the Hamiltonian reductions ${\mathbb R}^{0|4} // \mathrm{Spin}(3)$, ${\mathbb R}^{0|7} // G_2$, and ${\mathbb R}^{0|8} // \mathrm{Spin}(7)$, respectively.