Connected component of one

The connected component of one is a term from the theory of topological groups that is used in mathematics and physics, especially in the theory of Lie groups .

definition

Let be a topological group with a neutral element . Then designates the connected component of the fuel , so that associated component of which contains the neutral element. ${\displaystyle G}$ ${\displaystyle 1\in G}$ ${\displaystyle G^{0}}$ ${\displaystyle G}$

properties

• ${\displaystyle G^{0}}$is a closed subset of . ${\displaystyle G}$
• ${\displaystyle G^{0}}$is a characteristic subgroup of and in particular a normal subgroup . ${\displaystyle G}$
• The factor group is a totally disjointed Hausdorff topological group. It is referred to as the component group of , its elements correspond to the connected components of . ${\displaystyle G/G^{0}}$ ${\displaystyle G}$ ${\displaystyle G}$
• If is locally path-related ( e.g. a Lie group), then is open. ${\displaystyle G}$ ${\displaystyle G^{0}}$
• If it's open, then it's discreet . ${\displaystyle G^{0}}$ ${\displaystyle G/G^{0}}$
• If is an algebraic group , then is finite. ${\displaystyle G}$ ${\displaystyle G/G^{0}}$

Examples

• For the general linear group is the subgroup of matrices with positive determinant . The component group is isomorphic to the cyclic group . ${\displaystyle G=GL(n,\mathbb {R} )}$ ${\displaystyle G^{0}=GL^{+}(n,\mathbb {R} )}$ ${\displaystyle G/G^{0}}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$
• For is . ${\displaystyle G=GL(n,\mathbb {C} )}$ ${\displaystyle G^{0}=G}$
• For a totally disjointed group it is . ${\displaystyle G}$ ${\displaystyle G^{0}=\left\{1\right\}}$

literature

• Armand Borel : Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. ISBN 0-387-97370-2
• Lev Pontryagin : Topological groups. Translated from the second Russian edition by Arlen Brown Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966.
• Sigurdur Helgason : Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34th American Mathematical Society, Providence, RI, 2001. ISBN 0-8218-2848-7
• Igor Schafarewitsch : Basic algebraic geometry. Translated from the Russian by KA Hirsch. Revised printing of Grundlehren der Mathematischen Wissenschaften, Vol. 213, 1974. Springer Study Edition. Springer-Verlag, Berlin-New York, 1977.

Web links

• Connected component of the identity (Encyclopedia of Mathematics)