Simply connected algebraic group

In mathematics , simply connected algebraic groups are a term from algebraic geometry .

definition

A semi-simple algebraic group over a field is called simply connected if every isogeny ${\displaystyle G}$ ${\displaystyle k}$

${\displaystyle f\colon H\to G}$

of a related algebraic group is an isomorphism . ${\displaystyle H}$ ${\displaystyle G}$

Examples

• The special linear group is simply connected. ${\displaystyle SL_{n}}$
• The symplectic group is simply connected. ${\displaystyle Sp_{2n}}$
• A semisimple algebraic group over is simply connected if and only if the topological group is a singly connected space . ${\displaystyle G}$ ${\displaystyle k=\mathbb {C} }$ ${\displaystyle G(\mathbb {C} )}$

literature

• G. Hochschild, "The structure of Lie groups", Holden-Day (1965)
• R. Hermann, "Lie groups for physicists", Benjamin (1966)
• JE Humphreys, "Linear algebraic groups", Springer (1975)

Web links

• Simply-connected Group (Encyclopedia of Mathematics)