description/proof of that for Lie algebra, ideal is kernel of canonical Lie algebra homomorphism onto quotient Lie algebra of Lie algebra by ideal
Topics
About: Lie algebra
The table of contents of this article
Starting Context
- The reader knows a definition of %structure kind name% homomorphism.
- The reader knows a definition of kernel of linear map.
- The reader knows a definition of quotient Lie algebra of Lie algebra by ideal of Lie algebra.
Target Context
- The reader will have a description and a proof of the proposition that for any Lie algebra, any ideal is the kernel of the canonical Lie algebra homomorphism onto the quotient Lie algebra of the Lie algebra by the ideal.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(V'\): \(\in \{\text{ the Lie algebras }\}\)
\(V\): \(\in \{\text{ the ideals of } V'\}\)
\(V' / V\): \(= \text{ the quotient Lie algebra }\)
\(f\): \(V' \to V' / V, v' \mapsto [v']\)
//
Statements:
\(f \in \{\text{ the Lie algebra homomorphisms }\}\)
\(\land\)
\(V = \text{ the kernel of } f\)
//
2: Note
By the proposition that the kernel of any Lie algebra homomorphism is an ideal of the domain, the kernel of any Lie algebra homomorphism is an ideal of the domain; by this proposition, any ideal of any Lie algebra is the kernel of a Lie algebra homomorphism.
So, the ideals are the kernels of the Lie algebra homomorphisms.
3: Proof
Whole Strategy: Step 1: see that \(f\) is a Lie algebra homomorphism; Step 2: see that \(V\) is the kernel of \(f\).
Step 1:
Let us see that \(f\) is a Lie algebra homomorphism.
Let \(v'_1, v'_2 \in V'\) and \(r_1, r_2 \in F\) be any.
\(f (r_1 v'_1 + r_2 v'_2) = [r_1 v'_1 + r_2 v'_2] = r_1 [v'_1] + r_2 [v'_2] = r_1 f (v'_1) + r_2 f (v'_2)\).
\(f ([v'_1, v'_2]) = [[v'_1, v'_2]] = [[v'_1], [v'_2]] = [f (v'_1), f (v'_2)]\).
So, \(f\) is a Lie algebra homomorphism.
Step 2:
Let us see that \(V\) is the kernel of \(f\).
\(f (v') = [0]\) means that \([v'] = [0]\), which means that \(v' \in V\).
On the other hand, for each \(v \in V\), \(f (v) = [v] = [0]\), so, \(v \in Ker (f)\).
So, \(V = Ker (f)\).
