Inverse of a Multivector
Heerak Sharma -- Indian Institute of Science Education and Research, Pune, India
We discuss the problem of finding multiplicative inverses of multivectors in non-degenerate Clifford algebras. We will start by considering well known examples of Clifford algebras such as complex numbers, quaternions and split complex numbers and notice a pattern which we get in the formulas for their inverses. This motivates the formulas for inverse of multivector in other Clifford algebras. We discuss known formulas for inverses in Cℓp,q with p + q ≤ 6 and their proofs using interesting techniques such as matrix representations and quaternion typification. We also discuss some special cases of inverses of paravectors, dual of a paravector and sum of a paravector and its dual which have a formula for their inverse independent of the values p and q. We will look at some isomorphisms between different Clifford algebras and use formula for inverse in a smaller dimension Clifford algebra to find inverses of multivectors in larger dimension Clifford algebras. We will also see why inverses of grade 4 elements are particularly important and existence of certain ’trouble causing’ subalgebras which do not allow for the possibility of the formula of inverse of a multivector being a single product in Cℓp,q with p + q ≥ 6.