Bi-Frobenius quantum complete intersections with permutation antipodes

TL;DR

This paper investigates conditions under which quantum complete intersections can be endowed with bi-Frobenius algebra structures using permutation antipodes, providing explicit constructions and intrinsic criteria based on algebra parameters.

Contribution

It introduces a method to construct bi-Frobenius structures on quantum complete intersections via permutation antipodes, with explicit criteria based on algebra parameters.

Findings

A necessary and sufficient condition for bi-Frobenius structure with permutation antipode.

Explicit construction of bi-Frobenius algebra structures when conditions are met.

Intrinsic conditions involving structure coefficients for symmetric cases.

Abstract

Quantum complete intersections $A= A({\bf q, a})$ 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 $({\bf q, a})$ 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 $\pi$ with $A$ such that $\pi^2 = {\rm Id}$.