Shuffle algebra realization of Hall algebras of curves

Shuffle algebras are algebras of symmetric functions in infinitely many variables, with multiplication defined by a twisted version of symmetrization, where the twist is determined by a fixed function. The classical Hall algebra of a fixed abelian category (satisfying certain finite conditions) is an associative algebra. Its underlying vector space is generated by isomorphism classes of objects of the category, and the multiplication depends on the space of extensions between objects. The thesis will delve into the classical Hall algebra of the category of coherent sheaves on a smooth projective curve over a finite field, examining its combinatorial realization as a shuffle algebra. In cases where the curve has genus one, a more detailed characterization of the corresponding shuffle algebra will be analyzed. Prospective students should possess a background in the theory of abelian categories, Lie theory, and complex analysis, as well as an interest in the combinatorial realizations of associative algebras.

Prerequisites: Familiarity with complex analysis, sheaf theory, and theory of abelian categories.

References: Chapters 1, 2, and 4 of Schiffmann’s lecture notes on Hall algebras, Neguţ’s paper, and Neguţ-Sala-Schiffmann’s paper.