Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra ${\displaystyle \mathfrak{𝔤}}$ of the Lie algebra ${\displaystyle \mathfrak{𝔤}\mathfrak{𝔩}(V)}$ 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 ${\displaystyle \mathfrak{𝔤}}$ (in fact, on a finite-dimensional vector space by Ado's theorem if ${\displaystyle \mathfrak{𝔤}}$ is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and ${\displaystyle \mathfrak{𝔤}}$ a subalgebra of ${\displaystyle \mathfrak{𝔤}\mathfrak{𝔩}(V)}$. Then V is semisimple as a module over ${\displaystyle \mathfrak{𝔤}}$ 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]

• Jacobson, Nathan (1979). Lie algebras. New York: Dover Publications, Inc.. ISBN 978-0-486-13679-0. OCLC 867771145. http://www.freading.com/ebooks/details/r:download/ZnJlYWQ6OTc4MDQ4NjEzNjc5MDpl. 

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