A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category

Abstract

It is well known that Clifford algebras are group algebras deformed by a 2-cocycle. Furthermore, these algebras, which are not commutative in the usual sense, can be viewed as commutative algebras in certain symmetric monoidal categories of graded vector spaces. In this note we invent a Clifford process for coalgebras that will allow us to show that Clifford algebras have also cocommutative coalgebra structures, and consequently commutative and cocommutative weak braided Hopf algebras structures, within the same symmetric monoidal categories where they lie as commutative algebras. Also, we will show that they are selfdual weak braided Hopf algebras, monoidal Frobenius algebras and monoidal coFrobenius coalgebras.