Connected component of the identity
identity component, of a group
The largest connected subset of the topological (or algebraic) group
that contains the identity element of this group. The component
is a closed normal subgroup of
; the cosets with respect to
coincide with the connected components of
. The quotient group
is totally disconnected and Hausdorff, and
is the smallest among the normal subgroups
such that
is totally disconnected. If
is locally connected (for example, if
is a Lie group), then
is open in
and
is discrete.
In an arbitrary algebraic group the identity component
is also open and has finite index; also,
is the minimal closed subgroup of finite index in
. The connected components of an algebraic group
coincide with the irreducible components. For every polynomial homomorphism
of algebraic groups one has
. If
is defined over a field, then
is defined over this field.
If is an algebraic group over the field
, then its identity component
coincides with the identity component of
considered as a complex Lie group. If
is defined over
, then the group
of real points in
is not necessarily connected in the topology of the Lie group
, but the number of its connected components is finite. For example, the group
splits into two components, although
is connected. The pseudo-orthogonal unimodular group
, which can be regarded as the group of real points of the connected complex algebraic group
, is connected for
or
, and splits into two components for
. However, if the Lie group
is compact, then
is connected.
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[2] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104 |
[3] | S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101 |
[4] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Connected component of the identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connected_component_of_the_identity&oldid=21831