Some \(IHX\)-type relations on trivalent graphs and symplectic representation theory
The fundamental representation \(H\) of the Lie algebra of symplectic matrices of fixed order \(2g\), for arbitrary (fixed) \(g\) at least 6, and an irreducible \(\text{sp}_g\)-module \(U\) isomorphic to \(\Lambda^3H/H\) are studied. The author proves the existence of isomorphisms (there is an isomorphism in degree \(m\) for all \(g\geq 3m\)), \({\mathcal C}(\varphi)/(IH_0,\text{loop}) \to (\Lambda U/[2^2]_{\text{sp}})^{\text{sp}}\) and \({\mathcal A}(\varphi) \to(\Lambda (\text{Sym} ^3H)/[4]_{\text{sp}})^{\text{sp}}\), which multiply the degree by two. \({\mathcal C}(\varphi)\) denotes the commutative graded algebra of trivalent graphs under disjoint union and \({\mathcal A}(\varphi)\) the quotient by the relations \(AS\) and \(IHX\); \(IH_0\) is the relation \(I=H\) on graphs. The algebra \({\mathcal A}(\varphi)\) plays a role in the theory of Vassiliev invariants of knots and finite-type invariants of 3-manifolds. Some possible connections with the study of the mapping class group of a surface of genus \(g\) and with the theory of finite type invariants of 3-manifolds [see \textit{S. Garoufalidis} and \textit{J. P. Levine}, J. Differ. Geom. 47, 257--320 (1997; Zbl 0917.57009); Invent. Math. 131, 541--594 (1998; Zbl 0895.57004)] are indicated.
scientific article; zbMATH DE number 1223277