Hai JIN, Pu ZHANG
This paper is to look for bi-Frobenius algebra structures on quantum complete intersections over field $k$. We find a class of comultiplications, such that if $\sqrt{-1}\in k$, then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters $q_{ij} = \pm 1$. Also, it is proved that if $\sqrt{-1}\in k$ then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter $q = \pm 1$. While if $\sqrt{-1}\notin k$, then the exterior algebra with two variables admits no bi-Frobenius algebra structures. We prove that the quantum complete intersections admit a bialgebra structure if and only if it admits a Hopf algebra structure, if and only if it is commutative, the characteristic of $k$ is a prime $p$, and every $a_i$ a power of $p$. This also provides a large class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures.
