Compositions of invertibility preserving maps for some monoids and their application to Clifford algebras

TL;DR

This paper develops methods for composing invertibility preserving maps in monoids and applies these to define determinants for Clifford algebras with up to five generators, clarifying their connection and limitations.

Contribution

It introduces a new approach to composing invertibility preserving maps and defines determinants for finite dimensional algebras, specifically applied to Clifford algebras.

Findings

Invertibility preserving maps for Clifford algebras into a field.

Determinant formulas for Clifford algebras with up to 5 generators.

No determinant formula exists for Clifford algebras with more than 5 generators.

Abstract

For some monoids, we give a method of composing invertibility preserving maps associated to "partial involutions." Also, we define the notion of "determinants for finite dimensional algebras over a field." As examples, we give invertibility preserving maps for Clifford algebras into a field and determinants for Clifford algebras into a field, where we assume that the algebras are generated by less than or equal to 5 generators over the field. On the other hand, "determinant formulas for Clifford algebras" are known. We understand these formulas as an expression that connects invertibility preserving maps for Clifford algebras and determinants for Clifford algebras. As a result, we have a better sense of determinant formulas. In addition, we show that there is not such a determinant formula for Clifford algebras generated by greater than 5 generators.