Isomorphisms with the even subalgebra of a Clifford algebra #
This file provides some notable isomorphisms regarding the even subalgebra,
Main definitions #
CliffordAlgebra.equivEven: Every Clifford algebra is isomorphic as an algebra to the even subalgebra of a Clifford algebra with one more dimension.
CliffordAlgebra.evenEquivEvenNeg: Every even subalgebra is isomorphic to the even subalgebra of the Clifford algebra with negated quadratic form.
CliffordAlgebra.evenToNeg: The simp-normal form of each direction of this isomorphism.
Main results #
The embedding from the even subalgebra with an extra dimension into the original algebra.
Any clifford algebra is isomorphic to the even subalgebra of a clifford algebra with an extra
dimension (that is, with vector space
M × R), with a quadratic form evaluating to
-1 on that new
The representation of the clifford conjugate (i.e. the reverse of the involute) in the even subalgebra is just the reverse of the representation.
One direction of
The even subalgebras of the algebras with quadratic form
-Q are isomorphic.
Stated another way,
𝒞ℓ⁺(q,p,r) are isomorphic.