Linear Lie algebra

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

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

Jacobson 1962, Ch III, Theorem 10

References

Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4

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[1] Jacobson 1962, Ch III, Theorem 10