ar X iv : m at h / 05 05 14 8 v 1 [ m at h . A G ] 9 M ay 2 00 5 ON THE HALL ALGEBRA OF AN ELLIPTIC CURVE , I

These rings admit numerous algebraic and geometric realizations, but one of the historically first constructions, which dates back to the work of Steinitz in 1900 and later completed by Hall, was given in terms of what is now called the classical Hall algebra H (see [Ma], Chapter III ). This algebra has a basis consisting of isomorphism classes of abelian q-groups, where q is a fixed prime power, and the structure constants are defined by counting extensions between such abelian groups. In fact, these structure constants are polynomials in q, and we can therefore consider H as a C[q±1]-algebra. A theorem of Steinitz and Hall provides an isomorphismH ≃ Λq = C[q ±1][x1, x2, . . .]∞ . Under this isomorphism, the natural basis of H (resp. the natural scalar product) is mapped to the basis of Hall-Littlewood polynomials (resp. the Hall-Littlewood scalar product). In addition, Zelevinsky [Z] endowed Λq with a structure of a cocommutative Hopf algebra and the whole algebra Λq = Λ ⊗ C[q±1] can be recovered from Λq by the Drinfeld double construction. This Hopf algebra structure is also intrinsically defined by means of the Hall algebra.