Lie algebra isomorphism

Lie algebra isomorphism theorems

The isomorphism theorems for Lie algebras are expressed as follows.

First isomorphism theorem

Let 𝜑 :𝔤 →𝔥 be a Lie algebra homomorphism. Then the quotient by the kernel is isomorphic to the image: lie

𝔤ker⁡𝜑≅im⁡𝜑≤𝔥

Second isomorphism theorem

Let 𝔞,𝔟 ⊴𝔤. Then lie

𝔞+𝔟𝔟≅𝔞𝔞∩𝔟

Third isomorphism theorem

Let 𝔟 ⊴𝔞 ⊴𝔤 be nested ideals. Then 𝔞/𝔟 ⊴𝔤/𝔞 and lie

𝔤/𝔞𝔞/𝔟≅𝔤𝔟

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