On Commutative Analogues of Clifford Algebras and Their Decompositions

We investigate commutative analogues of Clifford algebras – algebras whose generators square to $\pm 1$ but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces – we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to `multi split-complex space’ (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques.