A note on Bridgeland Hall algebras

Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras

“…where the notations E M , F M , K α and K * α are the same as those in [16]. Hence, Theorem 4.3 establishes a relation between the Bridgeland Hall algebra of 2-cyclic complexes and that of m-cyclic complexes.…”

“…This provides a beautiful realization of the full quantized enveloping algebra by Hall algebras. Afterwards, Yanagida [15] (see also [16]) showed that the Bridgeland Hall algebra of 2-cyclic complexes of a hereditary algebra is isomorphic to the Drinfeld double of its extended Ringel-Hall algebras. Inspired by the work of Bridgeland, Chen and Deng [2] introduced the Bridgeland Hall algebra DH m (A) of m-cyclic complexes of a hereditary algebra A for each nonnegative integer m = 1.…”

PBW Basis of the Modified Ringel–Hall Algebra of Type A n via Gröbner–Shirshov Basis

“…In [11], Bridgeland also stated that the Drinfeld double of the extended Ringel-Hall algebra of a fnitedimensional hereditary algebra A is isomorphic to its Bridgeland Hall algebra but did not prove it. Later, Yanagita [12] and Zhang [13] proved the result in diferent ways. In [14], the authors defned the modifed Ringel-Hall algebras and got two main results about it.…”

Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

“…This provides a beautiful realization of the full quantum enveloping algebra by Hall algebras. Bridgeland [2] made a statement without proofs that the Bridgeland Hall algebra of A is isomorphic to the Drinfeld double of its extended Ringel-Hall algebra, which is later proved by Yanagida in [19] (see also [20]). Inspired by Bridgeland's work, for each positive integer m ≥ 2, Chen and Deng [4] considered the Hall algebra of m-cyclic complexes of projective A-modules.…”