• Green
• Hall algebras
• Hall polynomials
• quivers
• Ringel

Ringel Hall algebra of an abelian category A encodes the structure of the extensions’ space between objects in A. More precisely, we take the k-vector space generated by the elements uM where M lies in the isomorphism classes of A and define a multiplication between two elements uM and uN as a combination of uP where P runs through the set of extensions of M by N. For hereditary categories, i.e. categories with global dimension equal or less to one, Green discovered in [Gre95] that it is possible to define a coproduct ∆∶H(A)→H(A)⊗H(A)compatible with the algebra structure after an opportune twist. Finally Xiao showed in [Xia97] that H(A) admits also an Hopf structure. One of the most important category which satisfies the above conditions is the category ofnilpotent representations of quivers over a finite field Fq.
The connection between Ringel Hall algebras and the various topics in mathematics came out with Ringel's Theorem. He showed that the Hall algebra of category of representations of quivers over a finite field Fq of finite type is isomorphic to the positive part of the quantum universal enveloping algebra of the simple complex Lie algebra associated to the same Dynkin diagram. From here it was discovered a natural generalization of this theorem for Kac-Moody Lie algebras and turns out a lot of consequences in the study of quantum enveloping algebra.