The first connection between the representation theory of finite-dimensional algebras and Lie theory was found by Gabriel [36], who linked quivers of finite representation type with simply-laced Dynkin diagrams. Gabriel's result was later generalized to tame representation type by Nazarova [ 68], Donovan-Freislich [26], and Dlab-Ringel [27 ] and to wild representation type by Kac [50]. In Gabriel's theorem and its generalizations, the indecomposable representations of a quiver Q&ar; correspond to the positive roots of the graph Q . Ringel [71] extended this correspondence by using the Hall algebra of the category of representations of Q&ar; to realize the quantum enveloping algebra Uq n associated to Q . Ringel's result has led to substantial activity in representation theory and has been generalized in a number of different directions. In this thesis, we prove two fundamental results in this setting:;• We show that the Drinfeld double [29] of the (extended) Hall algebra of a finitary hereditary category is invariant under derived equivalences. This answers a question of Ringel [71] on how to obtain the full quantum group Uq g most naturally from Rep( Q&ar; ). Given two derived equivalent categories A and B , we provide an explicit isomorphism between the double Hall algebras DHA and DHB . This extends the results of Sevenhant-Van den Bergh [78], Xiao-Yang [85], and Burban-Schiffmann [16, 17]. As shown by Burban-Schiffmann [19], our theorem can be used to give a categorification of the Drinfeld-Beck [28, 5] homomorphism Uqsl2 &d14; → UqLsl2 when applied to Beilinson's [8] derived equivalence Db(Rep(K)) → Db( Coh( P1 )). Our theorem has also been applied to Geigle and Lenzing's equivalences [37] to give a new description of quantum affine algebras Uqg&d14; of simply-laced type (see [18]).;• Using a modified version of the Hall algebra, we define a Lie algebra structure on the semistable irreducible components in the Lusztig nilpotent varieties La , [63] associated to an affine quiver Q&ar; . This Lie algebra is isomorphic to the positive part n of the Kac-Moody algebra associated to Q . This confirms (after a slight modification) a conjecture of Frenkel, Malkin, and Vybornov [35]. As a refinement of Kac's theorem [ 50], Kac conjectured [51] that the constant term of the polynomial aaq counting absolutely indecomposable representations of a quiver Q&ar; in dimension α is equal to the multiplicity of the root α. Because the constant term as aa0 counts the semistable irreducible components in La (see [24]), our construction establishes a strengthened version of Kac's conjecture for affine quivers of types A1 1 , A1 2 , A1 2,2 , and D1 4 .;Recently, Hall algebras have been found to provide a framework for the study of Donaldson-Thomas invariants and their generalizations [49, 60, 61]. Both of our results have important implications in this context as well. The behavior of Hall algebras under derived equivalences appears in Donaldson-Thomas theory in connection with wall-crossing, homological mirror symmetry, and the crepant resolution conjecture [15]. Furthermore, our second result provides a mathematical basis for the "algebra of BPS states" of Harvey-Moore et al. [43, 32, 35], which is an antecedent of the motivic and cohomological Hall algebras of Joyce-Song [ 49] and Kontsevich-Soibelman [60, 61].
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