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Simply-connected group

From Encyclopedia of Mathematics
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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.

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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.


Comments

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://www.encyclopediaofmath.org/index.php?title=Simply-connected_group&oldid=39777
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article