Serre type relations for complex semisimple Lie algebras associated to positive definite quasi-Cartan matrices

Cartan matrices play an important role in the classification of complex semi-simple Lie algebras. The well-known Serre's Theorem states that every finite dimensional complex semisimple Lie algebra g can be constructed from a Cartan matrix A by using generators and relations. We generalize Serre's Theorem by associating to each positive definite quasi-Cartan matrix a complex semi-simple Lie algebra, and we prove that two positive definite quasi-Cartan matrices are equivalent if and only if its corresponding Lie algebras are isomorphic. This work complements the results obtained by Barot and Rivera in [1].