An Elementary, First Principles Approach to the Indefinite Spin Groups

Abstract: In this work we provide an elementary derivation of the indefinite spin groups in low-dimensions. Our approach relies on the isomorphism of Cl(p+1, q+1) to the algebra 2x2 matrices with entries in Cl(p,q), simple properties of Kronecker products, characterization of when an even dimensional real (resp. complex) matrix represents a complex (resp. quaternionic) linear transformation, and basic aspects of the isomorphism between M(4, R) and the quaternion tensor product. Of these the last is arguably the most vital. Among other things it yields naturally, a surprisingly ubiquitous role for an alternative to the standard representation of the symplectic form in dimension four. Our approach has the benefit of identifying these spin groups as explicit groups of matrices within the same collection of matrices which define $Cl(p, q)$. In other words, we do not work in the algebra of matrices that the collection of even vectors in $Cl(p,q)$ is isomorphic to. This is crucial in some applications and also has the advantage of being didactically simple.