Planar spider theorem and asymmetric Frobenius algebras

TL;DR

This paper extends the spider theorem to noncommutative and asymmetric Frobenius algebras, classifying connected diagrams and exploring associated invariants and moduli spaces.

Contribution

It generalizes the spider theorem to asymmetric Frobenius algebras and introduces the F-dimension Hilbert series for these structures.

Findings

Classified connected diagrams in noncommutative Frobenius algebras.

Defined and analyzed the F-dimension Hilbert series.

Explored moduli of asymmetric Frobenius structures.

Abstract

The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series $\dim_x=\sum_{j=0}^\infty x^j\dim_j$, where $\dim_j=\epsilon\circ B^j\circ 1$ are invariants of the Frobenius structure. We also study moduli of asymmetric quasispecial and `weakly symmetric' Frobenius structures and their F-dimensions. Examples include general Frobenius structures on matrix algebras $A=M_d(k)$ and on group algebras $k G$ as well as on $u_q(sl_2)$ at low roots of unity.