In mathematics, the trace field of a linear group is the field generated by the traces of its elements.
It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition.
Fuchsian groups are discrete subgroups of
is well-defined up to sign (by taking the trace of an arbitrary preimage in
The invariant trace field is equal to the trace field of the subgroup
generated by all squares of elements of
[1] The invariant trace field of Fuchsian groups is stable under taking commensurable groups.
generated by the preimages of elements of
is called the quaternion algebra of
is called the invariant quaternion algebra of
As for trace fields, the former is not the same for all groups in the same commensurability class but the latter is.
is an arithmetic Fuchsian group then
together are a number field and quaternion algebra from which a group commensurable to
[4] The theory for Kleinian groups (discrete subgroups of
) is mostly similar as that for Fuchsian groups.
[5] One big difference is that the trace field of a group of finite covolume is always a number field.
[6] When considering subgroups of general Lie groups (which are not necessarily defined as a matrix groups) one has to use a linear representation of the group to take traces of elements.
The most natural one is the adjoint representation.
It turns out that for applications it is better, even for groups which have a natural lower-dimensional linear representation (such as the special linear groups
), to always define the trace field using the adjoint representation.
Thus we have the following definition, originally due to Ernest Vinberg,[7] who used the terminology "field of definition".
are commensurable then they have the same trace field in this sense.
be a semisimple Lie group and
Then local rigidity implies the following result.
Furthermore, there exists an algebraic group
such that the group of real points
in the sense that it is a field of definition of its Zariski closure in the adjoint representation.
For Fuchsian groups the field
defined above is equal to its invariant trace field.
For Kleinian groups they are the same if we use the adjoint representation over the complex numbers.