Pham, Khoa2014-08-282014-08-2820142014http://hdl.handle.net/10393/31502http://dx.doi.org/10.20381/ruor-6577The trace form gives a connection between the representation ring and the space of invariant bilinear forms of a Lie algebra $L$. This thesis reviews the definition of the trace of an endomorphism of a finitely generated projective module over a commutative ring $R$. We then use this to look at the trace form of a finitely generated projective representation of a Lie algebra $L$ over $R$ and its representation ring. While doing so, we prove a few trace formulas which are useful in the theory of the Dynkin index, an invariant introduced by Dynkin in 1952 to study homomorphisms between simple Lie algebras.enTrace formulaDynkin indexInvariant bilinear formThesis