Identity component

In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element.

In point set topology, the identity component of a topological group G is the connected component G⁰ of G that contains the identity element of the group. The identity path component of a topological group G is the path component of G that contains the identity element of the group.

In algebraic geometry, the identity component of an algebraic group G over a field k is the identity component of the underlying topological space. The identity component of a group scheme G over a base scheme S is, roughly speaking, the group scheme G⁰ whose fiber over the point s of S is the connected component (G_s)⁰ of the fiber G_s, an algebraic group.

Properties

The identity component G⁰ of a topological or algebraic group G is a closed normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological or algebraic group are continuous maps by definition. Moreover, for any continuous automorphism a of G we have

a(G⁰) = G⁰.

Thus, G⁰ is a characteristic subgroup of G, so it is normal.

The identity component G⁰ of a topological group G need not be open in G. In fact, we may have G⁰ = {e}, in which case G is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set.

The identity path component of a topological group may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if G is locally path-connected.

Component group

The quotient group G/G⁰ is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G⁰ is a discrete group if and only if G⁰ is open. If G is an algebraic group of finite type, such as an affine algebraic group, then G/G⁰ is actually a finite group.

One may similarly define the path component group as the group of path components (quotient of G by the identity path component), and in general the component group is a quotient of the path component group, but if G is locally path connected these groups agree. The path component group can also be characterized as the zeroth homotopy group, $\pi _{0}(G,e).$

Examples

• The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is ({1,−1},•).
• Consider the group of units U in the ring of split-complex numbers. In the ordinary topology of the plane {z = x + j y : x, y ∈ R}, U is divided into four components by the lines y = x and y = − x where z has no inverse. Then U⁰ = { z : |y| < x } . In this case the group of components of U is isomorphic to the Klein four-group.
• The identity component of the additive group (Z_p,+) of p-adic integers is the singleton set {0}, since Z_p is totally disconnected.
• The Weyl group of a reductive algebraic group G is the components group of the normalizer group of a maximal torus of G.
• Consider the group scheme μ₂ = Spec(Z[x]/(x² - 1)) of second roots of unity defined over the base scheme Spec(Z). Topologically, μ_n consists of two copies of the curve Spec(Z) glued together at the point (that is, prime ideal) 2. Therefore, μ_n is connected as a topological space, hence as a scheme. However, μ₂ does not equal its identity component because the fiber over every point of Spec(Z) except 2 consists of two discrete points.

An algebraic group G over a topological field K admits two natural topologies, the Zariski topology and the topology inherited from K. The identity component of G often changes depending on the topology. For instance, the general linear group GL_n(R) is connected as an algebraic group but has two path components as a Lie group, the matrices of positive determinant and the matrices of negative determinant. Any connected algebraic group over a non-Archimedean local field K is totally disconnected in the K-topology and thus has trivial identity component in that topology.

References

• Lev Semenovich Pontryagin, Topological Groups, 1966.
• Demazure, Michel; Gabriel, Pierre (1970), Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Paris: Masson, ISBN 978-2225616662, MR 0302656