Bi-Frobenius Algebra Structures on Quantum Complete Intersections

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 \(\sqrt{-1}\in k\) $$\sqrt{-1}\in k$$ , then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters q_ij = ±1. Also, it is proved that if \(\sqrt{-1}\in k\) $$\sqrt{-1}\in k$$ then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter q = ±1. While if \(\sqrt{-1}\notin k\) $$\sqrt{-1}\notin k$$ , 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.