Abstract
Abstract
The automorphism group
$\operatorname {Aut}(F_n)$
of the free group
$F_n$
acts on a space
$A_d(n)$
of Jacobi diagrams of degree d on n oriented arcs. We study the
$\operatorname {Aut}(F_n)$
-module structure of
$A_d(n)$
by using two actions on the associated graded vector space of
$A_d(n)$
: an action of the general linear group
$\operatorname {GL}(n,\mathbb {Z})$
and an action of the graded Lie algebra
$\mathrm {gr}(\operatorname {IA}(n))$
of the IA-automorphism group
$\operatorname {IA}(n)$
of
$F_n$
associated with its lower central series. We extend the action of
$\mathrm {gr}(\operatorname {IA}(n))$
to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of
$F_n$
. By using this action, we study the
$\operatorname {Aut}(F_n)$
-module structure of
$A_d(n)$
. We obtain an indecomposable decomposition of
$A_d(n)$
as
$\operatorname {Aut}(F_n)$
-modules for
$n\geq 2d$
. Moreover, we obtain the radical filtration of
$A_d(n)$
for
$n\geq 2d$
and the socle of
$A_3(n)$
.
Publisher
Cambridge University Press (CUP)
Cited by
1 articles.
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