Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

Abstract

We study a filtered vector space A d (n) over a field k of characteristic 0, which consists of Jacobi diagrams of degree d on n oriented arcs for each n, d ≥ 0. We consider an action of the automorphism group Aut(Fn) of the free group Fn of rank n on the space A d (n), which is induced by an action of handlebody groups on bottom tangles. The action of Aut(Fn) on A d (n) induces an action of the general linear group GL(n, k) on the associated graded vector space of A d (n), which is regarded as the vector space B d (n) consisting of open Jacobi diagrams. Moreover, the Aut(Fn)-action on A d (n) induces an action on B d (n) of the associated graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of Fn with respect to its lower central series. We use an irreducible decomposition of B d (n) and computation of the gr(IA(n))-action on B d (n) to study the Aut(Fn)-module structure of A d (n). In particular, we consider the case where d = 2 in detail and give an indecomposable decomposition of A 2 (n). We also consider a functor A d , which includes the Aut(Fn)-module structure of A d (n) for all n.Contents 30 References 31

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. II

“…In a previous paper [14], we observed that the Aut(F n )-action on A d (n) induces two actions on B d (n): an action of the general linear group GL(n; Z) and an action of the graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of F n associated with the lower central series. We used these two actions on B d (n) to study the Aut(F n )-module structure of A d (n) for d = 2.…”

“…where A d,k (n) is the subspace of A d (n) spanned by Jacobi diagrams with at least k trivalent vertices. The Aut(F n )-action on A d (n) that we considered in [14] can be naturally extended to an action of the endomorphism monoid End(F n ) of F n on A d (n). (See Section 4.)…”

“…(See Section 4.) Let V n be an n-dimensional k-vector space, which will be identified with the first cohomology of a handlebody of genus n. The associated graded vector space of A d (n) is isomorphic via the PBW map [3] to a graded vector space We defined in [14] a gr(IA(n))-action on B d (n) by using the bracket map…”

“…The GL(n; Z)-module structure of B d (n). The GL(n; Z)-action on B d (n) that is induced by the Aut(F n )-action on A d (n) naturally extends to a polynomial GL(V n )-action on B d (n) [14]. Therefore, the GL(V n )-module B d (n) can be decomposed into the direct sum of images of the Schur functors.…”

“…and we studied the cases where d = 1, 2 in [14]. Thus, we mainly consider the cases where d ≥ 3, n ≥ 1.…”