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Bi-Frobenius quantum complete intersections with permutation antipodes

Hai Jin, Pu Zhang

TL;DR

This work addresses the lack of Hopf structures on quantum complete intersections by developing bi-Frobenius algebra structures via a carefully constructed comultiplication and a permutation antipode. It introduces the notion of a compatible permutation and derives a necessary-and-sufficient condition, expressed through a compatible $ ho$ with $ ho^2= ext{Id}$ and scalars $c_i$, along with the invariant $q_ ho$ and the product constraint $q_ hoig(ig( orall iig) c_i^{a_i-1}ig)=1$, to realize a bi-Frobenius structure. The authors provide explicit coalgebra data, including the constants $g_{f a-1-v, ho(v)}$ and the Δ-map, and prove that under these conditions the algebra $A({f q},{f a})$ becomes bi-Frobenius with a graded antipode $S$ that acts as a permutation on the standard basis. They further show intrinsic, coefficient-based criteria for the existence of such structures, including symmetric and characteristic-dependent cases, offering a large class of bi-Frobenius algebras not arising from Hopf algebras. The results extend the landscape of bi-Frobenius algebras by linking combinatorial permutation data to algebra/coalgebra compatibility in quantum complete intersections.

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 $π$ with $A$ such that $π^2 = {\rm Id}$.

Bi-Frobenius quantum complete intersections with permutation antipodes

TL;DR

This work addresses the lack of Hopf structures on quantum complete intersections by developing bi-Frobenius algebra structures via a carefully constructed comultiplication and a permutation antipode. It introduces the notion of a compatible permutation and derives a necessary-and-sufficient condition, expressed through a compatible with and scalars , along with the invariant and the product constraint , to realize a bi-Frobenius structure. The authors provide explicit coalgebra data, including the constants and the Δ-map, and prove that under these conditions the algebra becomes bi-Frobenius with a graded antipode that acts as a permutation on the standard basis. They further show intrinsic, coefficient-based criteria for the existence of such structures, including symmetric and characteristic-dependent cases, offering a large class of bi-Frobenius algebras not arising from Hopf algebras. The results extend the landscape of bi-Frobenius algebras by linking combinatorial permutation data to algebra/coalgebra compatibility in quantum complete intersections.

Abstract

Quantum complete intersections 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 key step is the construction of comultiplication, such that becomes a bi-Frobenius algebra. By introducing compatible permutation and permutation antipode, a necessary and sufficient condition is found, such that 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 of are obtained, for admitting a bi-Frobenius algebra structure with permutation antipode. When is symmetric, admits a bi-Frobenius algebra structure with permutation antipode if and only if there exists a compatible permutation with such that .
Paper Structure (18 sections, 14 theorems, 78 equations)

This paper contains 18 sections, 14 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frobenius algebras and Nakayama automorphisms
  4. Frobenius coalgebras
  5. Bi-Frobenius algebras
  6. The Nakayama automorphism and the $S^4$-formula for a Bi-Frobenius algebra
  7. Quantum complete intersections.
  8. Acting with $S_n$
  9. Structure coefficients under permutations
  10. Structure coefficients under compatible permutations
  11. Bi-Frobenius quantum complete intersections
  12. Quantum complete intersections with permutation antipodes
  13. Bi-Frobenius quantum complete intersections with permutation antipode
  14. Equivalent descriptions of a permutation antipode
  15. The case of $a_i\ge 3$
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $(A, \phi)$ be a Frobenius algebra, $\phi'\in A^*$. Then $\phi'$ is also a Frobenius form of $A$ if and only if there is $z_1\in U(A)$ such that $\phi' = z_1\rightharpoonup\phi$; if and only if there is $z_2\in U(A)$ such that $\phi' = \phi\leftharpoonup z_2.$ If this is the case, then the Nakay

Theorems & Definitions (30)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 3.1
  • Remark 3.2
  • ...and 20 more