We prove that a commutative (resp. modified anticommutative) connected graded Hopf algebra over a commutative algebra over a field of characteristic zero is a symmetric algebra (resp. exterior algebra) if and only if it satisfies a certain “codistributive law”, expressed by a commutative diagram. Both results are obtained as a corollary of an analogous characterization of the symmetric superalgebra of a supermodule over a superring: The symmetric superalgebras over a superring R in characteristic zero are exactly the comodules over the symmetric superalgebra of R.
- . Superalgebra
- exterior algebra
- graded Hopf algebra. 1980 Mathematics subject classifications 16A24 15A75 15A78
- symmetric algebra