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Unformatted text preview: or space over F . We deﬁne gl(V ) to be the
Lie algebra of all F -linear endomorphisms of V under the Lie bracket operation. A Lie
subalgebra of gl(V ) is called a linear Lie algebra.
Deﬁnition 1.2.3. A representation of the Lie algebra L is deﬁned to be a Lie algebra
homomorphism L → gl(V ) for some vector space V . The representation is called faithful
if this homomorphism is injective: L → gl(V ).
1.2.1. linear Lie algebras. There is a well-known theorem (due to Ado in characteristic 0
and Iwasawa in characteristic p) what every ﬁnite dimensional Lie algebra has a faithful
ﬁnite dimensional representation. I.e., it is isomorphic to a linear Lie algebra...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11