# Simply-connected group

A topological group (in particular, a 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 groups, 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

[a1] | G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) MR0207883 Zbl 0131.02702 |

[a2] | R. Hermann, "Lie groups for physicists" , Benjamin (1966) MR0213463 Zbl 0135.06901 |

[a3] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039 |

**How to Cite This Entry:**

Simply-connected group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Simply-connected_group&oldid=53628