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.
properties
- is a closed subset of .
- is a characteristic subgroup of and in particular a normal subgroup .
- 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 .
- If is locally path-related ( e.g. a Lie group), then is open.
- If it's open, then it's discreet .
- If is an algebraic group , then is finite.
Examples
- For the general linear group is the subgroup of matrices with positive determinant . The component group is isomorphic to the cyclic group .
- For is .
- For a totally disjointed group it is .
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)