A Clifford module is a mathematical construct that arises in the context of Clifford algebras and serves as a way to represent these algebras in a structured manner. To understand Clifford modules, we first need to briefly cover some foundational concepts: ### Clifford Algebras Clifford algebras are algebraic structures that generalize the concept of complex numbers and quaternions. They are generated by a vector space equipped with a quadratic form.
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For a quadratic form on a vector space , a Clifford module is a module over the Clifford algebra . Equivalently it is a vector space with a linear assignment satisfyingOver a field of characteristic different from two, expanding gives , with the bilinear form associated to . The universal property of the Clifford algebra extends these generator actions to an action of the whole algebra over a field.
For over , the matrices and both square to and anticommute. They make a Clifford module. Such modules provide the algebraic setting for spinors; changing the sign convention for changes the required generator squares.