nLab algebra extension

For extension of morphisms in the sense dual to lift see at extension.

Idea

Given any kind of object $A$ in algebra, such as an associative algebra or a group or a Lie algebra, etc., then an extension of $A$ is an epimorphism

$$\widehat A \overset{p}{\longrightarrow} A$$

Typically the kernel of an epimorphism will exist in the given category, leading to a short exact sequence

$$ker(p) \longrightarrow \widehat A \overset{p}{\longrightarrow} A \,.$$

Then one says that $\widehat A$ is an extension by $ker(p)$ of $A$.

Examples

• group extension

• Lie algebra extension

• infinitesimal extension

• central extension

Related concepts

• higher extension?