ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 16, 5, 18, 81 Zhang, Pu, Univ. of Sci. and Tech. of China Ringel-Hall algebras and quantum groups Let A be a finite-dimensional hereditary algebra over a finite field. The Ringel - Hall algebra H(A) is a ℚ(v)-vector space with basis the set of isoclasses [M] of finite A-modules, with product [M] · [N] = v^{⟨M,N⟩} [_{L}] → ∑ g_{M,N}^{L}[L], where g_{M,N}^{L} is the filtration number of L by M and N, ⟨−, −⟩ is the Euler form. Consider the composition algebra C(A), the subalgebra of H(A) generated by all simple modules. Works of C. M. Ringel and J. A. Green show that the generic version of C(A) is isomorphic to the positive quantum group U^{+}. If A is representation-finite (equivalently, C(A) = H(A)), then this isomorphism provides a basis of U^{+}. In affine case, properties of C(A) via the classification of representations will be presented.