Connected algebraic group: Difference between revisions
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* As for any [[ | * As for any [[semitopological group]], we can talk of the [[connected component of identity]]. | ||
* Over a [[finite field]], the only connected algebraic group is the trivial group. | * Over a [[finite field]], the only connected algebraic group is the trivial group. | ||
* Over an infinite field, both the [[additive group of a field|additive group]] and [[multiplicative group of a field|multiplicative group]] are connected. | * Over an infinite field, both the [[additive group of a field|additive group]] and [[multiplicative group of a field|multiplicative group]] are connected. | ||
* Over the [[field of real numbers]], [[field of complex numbers]], [[field of p-adic numbers]], or other field admitting an analytic structure, an algebraic group also becomes a [[Lie group]] (suitably interpreted). However, the Zariski topology is considerably coarser (fewer open subsets) than the topology arising from the analytic structure. The upshot is that it is possible to be a [[connected algebraic group]] and yet not be a [[connected Lie group]]. {{further|[[connected algebraic group need not be connected as a Lie group]]}} | * Over the [[field of real numbers]], [[field of complex numbers]], [[field of p-adic numbers]], or other field admitting an analytic structure, an algebraic group also becomes a [[Lie group]] (suitably interpreted). However, the Zariski topology is considerably coarser (fewer open subsets) than the topology arising from the analytic structure. The upshot is that it is possible to be a [[connected algebraic group]] and yet not be a [[connected Lie group]]. {{further|[[connected algebraic group need not be connected as a Lie group]]}} | ||
Latest revision as of 23:19, 14 January 2012
This article defines a property that can be evaluated for an algebraic group. it is probably not a property that can directly be evaluated, or make sense, for an abstract group|View other properties of algebraic groups
Definition
An algebraic group over a field is said to be connected if it satisfies the following equivalent conditions:
- It is connected as a semitopological group in the Zariski topology.
- It has no proper open subgroup. Note that whether or not the group is connected depends only on the underlying algebraic variety.
Facts
- As for any semitopological group, we can talk of the connected component of identity.
- Over a finite field, the only connected algebraic group is the trivial group.
- Over an infinite field, both the additive group and multiplicative group are connected.
- Over the field of real numbers, field of complex numbers, field of p-adic numbers, or other field admitting an analytic structure, an algebraic group also becomes a Lie group (suitably interpreted). However, the Zariski topology is considerably coarser (fewer open subsets) than the topology arising from the analytic structure. The upshot is that it is possible to be a connected algebraic group and yet not be a connected Lie group. Further information: connected algebraic group need not be connected as a Lie group