A definition of general linear Lie algebra, \(\mathfrak{gl} (V)\)
Topics
About: Lie algebra
The table of contents of this article
Starting Context
- The reader knows a definition of Lie algebra.
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of %structure kind name% endomorphism.
Target Context
- The reader will have a definition of general linear Lie algebra, \(\mathfrak{gl} (V)\).
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: Definition
For any vectors space, \(V\), the Lie algebra whose elements are all the vectors space endomorphisms on \(V\) with the canonical vectors space structure (scalar multiplication and addition are possible, because the codomain is a vectors space) and bracket as the commutator of the argument endomorphisms, which means that \([f_1, f_2] = f_1 \circ f_2 - f_2 \circ f_1\)
