Hall algebras and quantum groups
Let \(R\) be a finite dimensional representation-finite hereditary algebra of type \(\Delta\) over a field. Then there is a bijection between the isoclasses of \(R\)-modules of finite length and the functions \(a: \Phi^ +\to N_ 0\), where \(\Phi^ +\) is the set of all positive roots of \(\Delta\). For a commutative ring \(\Lambda\) and \(q\in\Lambda\), the author defines the Hall algebra \({\mathcal H}(R,\Lambda,q)\) which is a free \(\Lambda\)-module with basis formed by the isoclasses of \(R\)-modules of finite length and multiplication given by Hall polynomials counting some filtrations of such modules. The author also defines some canonical derivations \(\delta_ 1,\dots,\delta_ n\) (where \(n\) is the number of vertices of \(\Delta\)) from \({\mathcal H}(R,\Lambda,q)\) to itself, and then the skew polynomial ring \({\mathcal H}'(R,\Lambda,q)={\mathcal H}(R,\Lambda,q)[T_ i,\delta_ i]\). Then \({\mathcal H}'(R,\mathbb C,1)\) is isomorphic to the enveloping algebra \(U(\mathfrak b_ +)\), where \(\mathfrak b_ +\) is the Borel algebra of the semisimple Lie algebra \(\mathfrak g\) of type \(\Delta\). Finally, the author defines the completion \(\widehat{{\mathcal H}'(R)}=\displaystyle\lim_{{\longleftarrow\atop m}}{\mathcal H}'(R,\mathbb C[q]/(q-1)^ m,q)\) of the generic algebra \({\mathcal H}'(R,\mathbb C[q],q)\). The main results of the paper describe the algebra \(\widehat{{\mathcal H}'(R)}\) by generators and relations. As a consequence one obtains that \(\widehat{{\mathcal H}'(R)}\) is precisely Drinfeld's quantization \(U_ h({\mathfrak b}_ +)\) of \(U(\mathfrak b_ +)\). In particular, \(\widehat{{\mathcal H}'(R)}\) is a Hopf algebra.
finite dimensional representation-finite hereditary algebra
scientific article; zbMATH DE number 19935