'Twisted duality' for Clifford Algebras

TL;DR

This paper explores the superalgebra structure of complex Clifford algebras and proves that the supercommutant of a subalgebra corresponds exactly to the Clifford algebra of the orthogonal complement, providing multiple proofs.

Contribution

It introduces several proofs of the 'twisted duality' property for Clifford algebras viewed as superalgebras, clarifying their supercommutant structure.

Findings

Supercommutant of C(W) is C(W^{ot}) for subspace W

Multiple proofs established for the duality property

Clarification of superalgebra structure in Clifford algebras

Abstract

Viewing the complex Clifford algebra $C(V)$ of a real inner product space $V$ as a superalgebra, we offer several proofs of the fact that if $W$ is a subspace of the complexification of $V$ then the supercommutant of the Clifford algebra $C(W)$ is precisely the Clifford algebra $C(W^{\perp})$.