Identity component
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In mathematics, specifically group theory, the identity component of a group G (also known as its unity component) 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 (Gs)0 of the fiber Gs, an algebraic group.[1]
Properties[edit]
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[edit]
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,
Examples[edit]
- 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 (ℤp,+) of p-adic integers is the singleton set {0}, since ℤp 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
μ 2 = 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,μ 2 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 GLn(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[edit]
- ^ SGA 3, v. 1, Exposé VI, Définition 3.1
References[edit]
- 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
- Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151). Lecture Notes in Mathematics (in French). Vol. 151. Berlin; New York: Springer-Verlag. pp. xv+564. doi:10.1007/BFb0058993. ISBN 978-3-540-05179-4. MR 0274458.
External links[edit]
- 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.