Bi-Frobenius algebra structure on quantum complete intersections

TL;DR

This paper investigates bi-Frobenius algebra structures on quantum complete intersections, identifying conditions under which these structures exist, especially relating to the parameters and field characteristics, and providing examples beyond bialgebras.

Contribution

It characterizes when quantum complete intersections and exterior algebras admit bi-Frobenius structures, highlighting cases where such structures differ from bialgebras.

Findings

Bi-Frobenius structures exist if all parameters are ±1 when √-1 is in the field.

Quantum exterior algebra in two variables admits bi-Frobenius structure iff q = ±1 if √-1 in the field.

Quantum complete intersections over characteristic zero do not admit bialgebra structures.

Abstract

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections. 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. Since a quantum complete intersection over a field of characteristic zero admits no bialgebra structures, this gives a class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). On the other hand, a quantum exterior algebra admits a bialgebra structure if and only if ${\rm char} \ k = 2$. In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures.