View source for Simply-connected group

A [[Topological group|topological group]] (in particular, a [[Lie group|Lie group]]) for which the underlying topological space is simply-connected. The significance of simply-connected groups in the theory of Lie groups is explained by the following theorems.

1) Every connected Lie group $G$ is isomorphic to the quotient group of a certain simply-connected group (called the universal covering of $G$) by a discrete central subgroup isomorphic to $\pi_1(G)$.

2) Two simply-connected Lie groups are isomorphic if and only if their Lie algebras are isomorphic; furthermore, every homomorphism of the Lie algebra of a simply-connected group $G_1$ into the Lie algebra of an arbitrary Lie group $G_2$ is the derivation of a (uniquely defined) homomorphism of $G_1$ into $G_2$.

The centre $Z$ of a simply-connected semi-simple compact or complex Lie group $G$ is finite. It is given in the following table for the various kinds of simple Lie groups.
$$
\begin{array}{||c||c|c|c|c|c|c|c|c|c|c||}
\hline \\
G & A_n & B_n & C_n & D_{2n} & D_{2n+1} & E_6 & E_7 & E_8 & F_4 & G_2 \\ 
\hline
Z & \mathbf{Z}_{n+1} &  \mathbf{Z}_2 & \mathbf{Z}_2 & \mathbf{Z}_2 \times  \mathbf{Z}_2 & \mathbf{Z}_4 & \mathbf{Z}_3 & \mathbf{Z}_2 & \{e\} & \{e\} & \{e\} \\
\hline
\end{array}
$$

In the theory of [[algebraic group]]s, a simply-connected group is a connected algebraic group $G$ not admitting any non-trivial [[isogeny]] $\phi : \tilde G \rightarrow G$, where $\tilde G$ is also a connected algebraic group. For semi-simple algebraic groups over the field of complex numbers this definition is equivalent to that given above.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) {{MR|0207883}} {{ZBL|0131.02702}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hermann, "Lie groups for physicists" , Benjamin (1966) {{MR|0213463}} {{ZBL|0135.06901}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR>
</table>

{{TEX|done}}