On Classification of Irreducible Quandle Modules over a Connected Quandle

概要

A quandle is an algebraic system given by an operation ▷ which generalizes the conjugation
operation of groups, and quandles play an important role in knot theory. The notion of a quandle
was first introduced by Joyce and Matveev in 1980s ([Joy], [Mat], see Definition 1.1). Just as
in the cases of other algebraic objects such as groups and rings, it is expected that the quandle
modules are important in studying quandles. The notion of a general quandle module was given by
Andruskiewitsch and Gra˜
na [AG] and Jackson [Jac] (Definition 1.12). As an example of application
of modules, homology of quandle modules is defined and some important homological invariants of
quandle modules are found.
As suggested above, every group can be regarded as quandles by the conjugation operation. For
a group G and g, h ∈ G, the operation g ▷ h = ghg −1 defines a quandle denoted by Conj(G), which
is called the conjugation quandle of G. In the converse direction, a quandle Q naturally induces a
group As(Q) called the associated group (Definition 1.4). These assignments give rise to functors
Conj : Grp → Qd and As : Qd → Grp where Grp and Qd denote the categories of groups and
quandles respectively, and these functors are adjoint to each other. A module over As(Q) naturally
defines a module over Q. Such a module will be called a module induced from an As(Q)-module.
However, there also exist quandle modules which are not induced from As(Q)-modules. This makes
the classification of quandle modules more interesting.
In this paper, we study the problem of classifying irreducible modules over connected quandles.
For a quandle Q, there is another group Inn(Q) called the inner automorphism group which is
generated by left multiplication actions on Q. A quandle Q is said to be connected if the action of
Inn(Q) on Q is transitive. Given a quandle module M, we first look at the inner automorphism
group Inn(M) of M regarded as a quandle. Then we can construct another quandle module I(M)
over Q from Inn(M), which is induced from an As(Q)-module, and a homomorphism iM : M →
I(M) of quandle modules over Q. In particular, if M is an irreducible quandle module, iM is
either injective or zero. The main result of this paper is the followings:
1991 Mathematics Subject Classification. 20N02, 20G40.
Key words and phrases. Quandle; Quandle module. ...