Ribbon Hopf algebras from group character rings

TL;DR

This paper constructs ribbon Hopf algebras from character rings of matrix groups, providing a new algebraic framework for knot invariants derived from group characters and their diagrammatic representations.

Contribution

It introduces a novel approach to forming ribbon Hopf algebras from character rings of classical groups and their subgroups, linking algebraic structures with knot theory.

Findings

Established that Char-H_ rings are ribbon Hopf algebras.

Constructed crossing tangles satisfying braid relations within these algebras.

Developed weak knot invariants based on group character manipulations.

Abstract

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group GL(n) in the inductive limit, realised as the ring of symmetric functions \Lambda(X) on countably many variables X = {x_1,x_2, ...}. Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of GL(n), the orthogonal and symplectic groups. We also analyse the formal character rings Char-H_\pi\ of algebraic subgroups of GL(n), comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type \pi, which have been introduced in [5] (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for each \pi\ a crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-H_\pi\ ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.