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| | 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. | | 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. |
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| − | 1) Every connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854301.png" /> is isomorphic to the quotient group of a certain simply-connected group (called the universal covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854302.png" />) by a discrete central subgroup isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854303.png" />. | + | 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 $\pi_1(G)$. |
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| − | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854304.png" /> into the Lie algebra of an arbitrary Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854305.png" /> is the derivation of a (uniquely defined) homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854306.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854307.png" />. | + | 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 $G_2$. |
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| − | The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854308.png" /> of a simply-connected semi-simple compact or complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s0854309.png" /> is finite. It is given in the following table for the various kinds of simple Lie groups.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543010.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543011.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543012.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543013.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543014.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543015.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543016.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543017.png" /></td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543018.png" /></td> <td colname="10" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543019.png" /></td> <td colname="11" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543020.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543021.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543022.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543023.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543024.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543025.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543026.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543027.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543028.png" /></td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543029.png" /></td> <td colname="10" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543030.png" /></td> <td colname="11" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543031.png" /></td> </tr> </tbody> </table> | + | 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. |
| | + | $$ |
| | + | \begin{array}{||c||c|c|c|c|c|c|c|c|c|c||} |
| | + | \hline \\ |
| | + | G & A_n & B_n & C_n & D_{2n} & D_{2n+1} & E_6 & E_7 & E_8 & F_4 & G_2 \\ |
| | + | \hline |
| | + | Z & \mathbf{Z}_{n+1} & \mathbf{Z}_2 & \mathbf{Z}_2 & \mathbf{Z}_2 \times \mathbf{Z}_2 & \mathbf{Z}_4 & \mathbf{Z}_3 & \mathbf{Z}_2 & \{e\} & \{e\} & \{e\} \\ |
| | + | \hline |
| | + | \end{array} |
| | + | $$ |
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| − | </td></tr> </table>
| + | 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. |
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| − | In the theory of algebraic groups (cf. [[Algebraic group|Algebraic group]]), a simply-connected group is a connected algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543032.png" /> not admitting any non-trivial [[Isogeny|isogeny]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085430/s08543034.png" /> 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> |
| − | ====Comments====
| + | <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}} |
| − | ====References====
| |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Hochschild, "The structure of Lie groups" , Holden-Day (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hermann, "Lie groups for physicists" , Benjamin (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1</TD></TR></table>
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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.
$$
\begin{array}{||c||c|c|c|c|c|c|c|c|c|c||}
\hline \\
G & A_n & B_n & C_n & D_{2n} & D_{2n+1} & E_6 & E_7 & E_8 & F_4 & G_2 \\
\hline
Z & \mathbf{Z}_{n+1} & \mathbf{Z}_2 & \mathbf{Z}_2 & \mathbf{Z}_2 \times \mathbf{Z}_2 & \mathbf{Z}_4 & \mathbf{Z}_3 & \mathbf{Z}_2 & \{e\} & \{e\} & \{e\} \\
\hline
\end{array}
$$
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