Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra  g {\displaystyle {\mathfrak {g}}} of the Lie algebra g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation.

Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of  g {\displaystyle {\mathfrak {g}}} (in fact, on a finite-dimensional vector space by Ado's theorem if  g {\displaystyle {\mathfrak {g}}} is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and  g {\displaystyle {\mathfrak {g}}} a subalgebra of  g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} . Then V is semisimple as a module over  g {\displaystyle {\mathfrak {g}}} if and only if (i) it is a direct sum of the center and a semisimple ideal and (ii) the elements of the center are diagonalizable (over some extension field).[1]

## Notes[edit]

1. ^ Jacobson 1979, Ch III, Theorem 10

## References[edit]

* Jacobson, Nathan (1979) [1962]. Lie algebras. New York: Dover Publications, Inc. ISBN 978-0-486-13679-0. OCLC 867771145.