Clifford Algebras of Forms of Higher Degrees

Abstract

The aim of this lecture is to introduce Clifford algebras of polynomial forms of higher degrees. We recall that these algebras are in general of infinite dimension, and we give a basis depending on a given basis of the underlying vector space. We then show that, though they contain large free associative algebras, we may construct finite dimensional representations of these algebras, also called linearizations of the polynomial form. If the polynomial form is, in a certain sense, non degenerate, the dimensions of these representations are multiples of the degree of the form. In the end, we recall some results known for the special case of a binary cubic form with at least one simple zero, when explicit computations can be done: the Clifford algebra is an Azumaya algebra of rank 9 over its center, which is the algebra of functions over a cubic curve depending on the given cubic form.