Some \(IHX\)-type relations on trivalent graphs and symplectic representation theory
type: Item
instance of: scholarly article
title: Some \(IHX\)-type relations on trivalent graphs and symplectic representation theory
zbMATH Open document ID: 1012.57032
author: Stavros Garoufalidis
author: Hiroaki Nakamura
published in: Mathematical Research Letters
publication date: 1999-02-18T00:00:00Z
review text: 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.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1223277
zbMATH Keywords: fundamental representation
zbMATH Keywords: symplectic Lie algebra
MaRDI profile type: Publication
full work available at URL: mrl.1998.v5.n3.a11
OpenAlex ID: W2059947720
DOI: 10.4310/MRL.1998.V5.N3.A11
description: scientific article; zbMATH DE number 1223277