Linear Lie algebra

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

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

1. ↑ Jacobson 1979, Ch III, Theorem 10

## References

* 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Linear Lie algebra. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Linear_Lie_algebra&oldid=4157512"