Bi-Frobenius quantum complete intersections with permutation antipodes

Abstract

Quantum complete intersections $A=A(\mathbf{q},\mathbf{a})$ are Frobenius algebras, but in the most cases they can not become Hopf algebras. This paper aims to find bi-Frobenius algebra structures on $A$. A key step is the construction of comultiplication, such that $A$ becomes a bi-Frobenius algebra. By introducing compatible permutation and permutation antipode, a necessary and sufficient condition is found, such that $A$ admits a bi-Frobenius algebra structure with permutation antipode; and if this is the case, then a concrete construction is explicitly given. Using this, intrinsic conditions only involving the structure coefficients $(\mathbf{q},\mathbf{a})$ of $A$ are obtained, for $A$ admitting a bi-Frobenius algebra structure with permutation antipode. When $A$ is symmetric, $A$ admits a bi-Frobenius algebra structure with permutation antipode if and only if there exists a compatible permutation $\ddot{I}€$ with $A$ such that $\ddot{I}{€}^2=\mathrm{Id}$.