Heisenberg doubles and derived categories
The paper addresses the problem of defining a Hall algebra of the derived category \(D^b(\mathcal A)\) of the abelian category \(\mathcal A\). One assumes that \(\mathcal A\) has finite homological dimension, every object \(A\) of \(\mathcal A\) has a finite number of subobjects, and all the groups \(\text{Ext}_{\mathcal A}^i(A,B)\) are finite and zero for almost all \(i\) and for all \(i>0\). This problem is related to \textit{C. M. Ringel}'s Hall algebra construction [Invent. Math. 101, No. 3, 583-591 (1990; Zbl 0735.16009)], associating to \(\mathcal A\) an algebra \(R(\mathcal A)\) (related in a particular case to the quantized enveloping algebra \(U_q({\mathfrak g})\) of a certain semisimple Lie algebra \({\mathfrak g}\)). Extending this construction to triangulated categories such as \(D^b(\mathcal A)\) appears to be difficult. In this paper the author constructs an algebra \(L(\mathcal A)\), called the lattice algebra of \(\mathcal A\), using the Heisenberg double as an essential tool. The paper is divided into four sections. Section 1 discusses the Heisenberg double Heis\((\mathcal A)=HD(B(\mathcal A),B(\mathcal A),\phi)\) of the extended Ringel algebra \(B(\mathcal A)={\mathbb{C}}[\mathcal K_0\mathcal A]\otimes R(\mathcal A)\), where \(\phi\) is a Hopf pairing on \(B(\mathcal A)\). The main result of Section 2 states that if \(F\:D^b(\mathcal A) \to D^b(\mathcal B)\) is a tilting (that is, an equivalence such that \(F(\mathcal A)\subset D^{[0,1]}(\mathcal B)\)), then there is an injective homomorphism of algebras \(F_*\:R(\mathcal A)\to\text{Heis}(\mathcal B)\). In Section 3 the lattice algebra \(L(\mathcal A)\) of \(\mathcal A\) is defined by taking infinitely many copies of \(R(\mathcal A)\), one for each site of an infinite lattice, and by imposing Heisenberg-like commutation relations between algebras at adjacent sites of the lattice. Several examples and related constructions are discussed in the last section. Some technical points of the proofs rely on another recent paper of the author [\textit{M. M. Kapranov}, J. Math. Sci. 84, No. 5, 1311-1360 (1997)].
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