Identity component

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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

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.^[1]

Properties

The identity component G⁰ of a topological or algebraic group G is a

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

The identity component G⁰ of a topological group G need not be

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, ${\displaystyle \pi _{0}(G,e).}$

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.

note

References

• Demazure, M.; Grothendieck, A., Gille, P.; Polo, P. (eds.), Schémas en groupes (SGA 3), I: Propriétés Générales des Schémas en Groupes Revised and annotated edition of the 1970 original.