Derived Hall algebras and lattice algebras (Q2909083)

The authors compare Kapranov's lattice algebra with Toën's derived Hall algebra. They prove that the derived Hall algebra can be identified with the lattice algebra by the ``twist and extend'' procedure, with a suitable subalgebra closely related to the Heisenberg double.NEWLINENEWLINENEWLINENEWLINELet \(k\) be a finite field and \(\mathcal{A}\) be a hereditary abelian \(k\)-category. Following \textit{C. M. Ringel} [Invent. Math. 101, No. 3, 583--591 (1990; Zbl 0735.16009)], one can associate to \(\mathcal{A}\) the Ringel-Hall algebra \(H(\mathcal{A})\). When \(\mathcal{A}\) is the module category of a finite dimensional hereditary \(k\)-algebra, \textit{C. M. Ringel} and \textit{J. A. Green} [Invent. Math. 120, No. 2, 361--377 (1995; Zbl 0836.16021)] have shown that the twisted Ringel-Hall algebra \(\mathbb{H}(\mathcal{A})\) gives a realization of the positive part of quantum group of the corresponding Kac-Moody algebra. Ringel has also shown that after adding a torus to \(\mathbb{H}(\mathcal{A})\), the extended twisted Hall algebra \(\mathcal{H}(\mathcal{A})\) gives a realization of the Borel part of quantum group.NEWLINENEWLINENEWLINENEWLINEIt remains a question to recover the whole quantum group naturally, while extending the Hall algebra formalism to the derived category \(D^{b}(\mathcal{A})\) seems hopeful. \textit{M. Kapranov} [J. Algebra 202, No. 2, 712--744 (1998; Zbl 0910.18005)] defined two associative algebras associated to the derived category. The first one is the Heisenberg double \(Heis(\mathcal{A})\), which is a double construction of \(\mathcal{H}(\mathcal{A})\) and can be viewed as a ``Hall algebra'' of \(D^{[-1,0]}(\mathcal{A})\), i.e., the subcategory of complexes situated in degrees \(0\) and \(-1\). The second one is the lattice algebra \(L(\mathcal{A})\) associated to \(D^{b}(\mathcal{A})\), which is invariant under derived equivalence. Recently, \textit{B. Toën} [Duke Math. J. 135, No. 3, 587--615 (2006; Zbl 1117.18011)] found the derived Hall algebra corresponding to a dg category by defining derived Hall numbers analogous to Hall numbers. This was simplified by \textit{J. Xiao} and \textit{F. Xu} [Duke Math. J. 143, No. 2, 357--373 (2008; Zbl 1168.18006)].NEWLINENEWLINENEWLINENEWLINEIn this paper, the authors compare their constructions. Following Ringel's methods, they first twist the multiplication in the derived Hall algebra of a hereditary abelian category \(\mathcal{A}\) denoted by \(DH(\mathcal{A})\), then add a torus to it. They prove that the extended twisted derived Hall algebra \(\mathcal{DH}(\mathcal{A})\) is exactly the lattice algebra \(L(\mathcal{A})\). Moreover, the extend (\(2\)-tori added) twisted form of \(DH^{[-1,0]}(\mathcal{A})\), the subalgebra of \(DH(\mathcal{A})\) corresponding to \(D^{[-1,0]}(\mathcal{A})\), is just the Heisenberg double.