Classical Lie Algebra Weight Systems of Arrow Diagrams by Louis Leung A

Classical Lie Algebra Weight Systems of Arrow Diagrams Louis Leung Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 The notion of finite type invariants of virtual knots introduced in [GPV] leads to the study of ~ An, the space of diagrams with n directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation V we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures’ defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T ([BN1]). In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces ~ An for n ≤ 4, and our results suggest that in ~ A4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.