Heisenberg doubles and derived categories
type: Item
instance of: scholarly article
title: Heisenberg doubles and derived categories
zbMATH Open document ID: 0910.18005
published in: Journal of Algebra
publication date: 1999-04-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/q-alg/9701009
review text: 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 o…
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1203880
zbMATH Keywords: abelian categories
zbMATH Keywords: derived categories
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Hall-Ringel algebra
zbMATH Keywords: Heisenberg doubles
zbMATH Keywords: lattice algebra
zbMATH Keywords: tilting
author: Mikhail Kapranov
reviewed by: Andrei Marcus
MaRDI profile type: Publication
OpenAlex ID: W2025283953
arXiv ID: q-alg/9701009
cites work: \((T^*G)_ t\): a toy model for conformal field theory
cites work: Hidden quantum groups inside Kac-Moody algebras
cites work: Q4321790
cites work: Q4889850
cites work: Hall algebras, hereditary algebras and quantum groups
cites work: Q3994341
cites work: Q4317871
cites work: Eisenstein series and quantum affine algebras
cites work: Q3127984
cites work: Q3999005
cites work: Introduction to quantum groups
cites work: Canonical Bases Arising from Quantized Enveloping Algebras
cites work: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
cites work: Hall algebras and quantum groups
cites work: The Composition Algebra of a Cyclic Quiver
cites work: Q4880625
cites work: Q4314350
cites work: Drinfeld double and Ringel-Green theory of Hall algebras
DOI: 10.1006/JABR.1997.7323
description: scientific article