Algebra Seminar: Peter D Jarvis -- Ribbon Hopf algebras from group character rings

Ribbon Hopf algebras from group character rings

Peter Jarvis
School of Physical Sciences
University of Tasmania

Joint work with Bertfried Fauser and Ronald King

We study the diagram alphabet of knot moves associated with the character
rings of certain matrix groups. The primary object is the Hopf algebra
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, as well as 
the formal character rings of algebraic subgroups of $GL(n)$, comprised of 
matrix transformations leaving invariant a fixed but arbitrary tensor of 
Young symmetry type $\pi$, which include the orthogonal and symplectic groups 
as special cases. 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.