The Hall algebra of the category of coherent sheaves on the projective line
In important work of \textit{C. M. Ringel, J. A. Green} and others, Hall algebras of representations of quivers have been shown to be closely related with and very useful for the study of quantized enveloping algebras of Kac-Moody algebras.NEWLINENEWLINENEWLINEIn the last few years, several authors started to study Hall algebras for other types of categories. \textit{L. Peng} and \textit{J. Xiao} used derived categories of quiver representations [J. Algebra 198, No. 1, 19-56 (1997; Zbl 0893.16007) and Invent. Math. 140, No. 3, 563-603 (2000; Zbl 0966.16006)]. \textit{M. M. Kapranov} looked at derived categories of coherent sheaves of smooth projective curves over finite fields [J. Math. Sci., New York 84, No. 5, 1311-1360 (1997; Zbl 0929.11015), see also J. Algebra 202, No. 2, 712-744 (1998; Zbl 0910.18005)]. Kapranov's results gave very interesting analogies between certain relations in quantum affine algebras and the behaviour of certain automorphic forms. NEWLINENEWLINENEWLINEBy a classical result of \textit{A. A. Beilinson} [Funct. Anal. Appl. 12, 214-216 (1979; Zbl 0424.14003)], in the special case of the curve being the projective line, coherent sheaves are closely related to representations of the Kronecker quiver. The authors work out this example in great detail, thus illustrating both the situation for coherent sheaves and (implicitly) for Kronecker modules (where everything is known already from Ringel's work). They do not use any of the above references, except for definitions; instead they work out the structure of this Hall algebra from first principles.
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