Modified Ringel-Hall algebras, naive lattice algebras and lattice algebras (Q2309651)

In the paper under review, the author studies the relations between three associative algebras one can associate to a \(k\)-linear hereditary abelian category \(\mathcal{A}\) satisfying the finiteness conditions: \begin{enumerate} \item \(k\) is a finite field, \item \(\mathcal{A}\) is essentially small, \item The \(k\) vector spaces \(\mathrm{Hom}_{\mathcal{A}}(M,N), \mathrm{Ext}_{\mathcal{A}}^1(M,N)\) are finite dimensional. \end{enumerate} The isomorphisms provided are obtained as consequences of descriptions by generators and relations of the given algebras. The algebras under consideration are: the naive lattice algebra \(\mathcal{N}(\mathcal{A})\), the componentwise twisted Ringel-Hall algebra \(\mathcal{MH}_{\mathrm{ctw}}(\mathcal{A})\), the relative twisted Ringel-Hall algebra \(\mathcal{MH}_{\mathrm{rtw}}(\mathcal{A})\), the extended twisted Ringel-Hall algebra, \(\mathcal{H}_{\mathrm{tw}}^{e}(\mathcal{A})\), the lattice algebra \(\mathcal{L}(\mathcal{A})\), the Drinfeld-dual lattice algebra \(\mathcal{L}_*(\mathcal{A})\), the relative twisted modified Ringel-Hall algebra \(\mathcal{MH}_{\mathrm{rtw}}(\mathcal{A})\), and the derived versions, the derived Hall algebra \(\mathcal{DH}(\mathcal{A})\), the twisted derived Hall algebra \(\mathcal{DH}_{\mathrm{tw}}(\mathcal{A})\), the extended twisted derived Hall algebra \(\mathcal{DH}_{\mathrm{tw}}^e(\mathcal{A})\) and the completely extended twisted derived Hall algebra \(\mathcal{DH}_{\mathrm{tw}}^{\mathrm{ce}}(\mathcal{A})\). The extended twisted Ringel-Hall algebra possesses a Hopf bilinear form \(\phi:\mathcal{H}_{\mathrm{tw}}^e(\mathcal{A})\otimes\mathcal{H}_{\mathrm{tw}}^e(\mathcal{A})\rightarrow \mathbb{Q}(v)\). The author proves the existence of an isomorphism \(\mathcal{N}(\mathcal{A})\simeq \mathcal{MH}_{\mathrm{ctw}}(\mathcal{A})\), an epimorphism \(\mathcal{MH}_{\mathrm{rtw}}(\mathcal{A})\rightarrow \mathcal{L}_*(\mathcal{A})\) whose kernel is described explicitly and that the naive lattice algebra is invariant under derived equivalences. The Ringel-Hall algebra has been first defined by \textit{C. M. Ringel} [Invent. Math. 101, No. 3, 583--591 (1990; Zbl 0735.16009)]. This is the algebra whose underlying vector space has the set of isomorphism classes of objects of \(\mathcal{A}\) as basis and the structure constants of the multiplication count the number of extensions between two given objects with fixed central term. The twisted Ringel-Hall algebra has the same underlying vector space but the multiplication is twisted by the Euler form of the category. The extended twisted Ringel-Hall algebra is built from the twisted Ringel-Hall algebra by adding a Cartan part. The product and coproduct are slightly modified, which makes the extended twisted Ringel-Hall algebra a genuine Hopf algebra. The modified Ringel-Hall algebra is contructed from the Ringel-Hall algebra \(\mathcal{MH}(\mathcal{A})\) of the category of bounded complexes in \(\mathcal{A}\), \(\mathcal{C}^b(\mathcal{A})\), by first quotienting out by the relations \([L]=[K\oplus M]\) if \(L\) is an extension of a complex \(M\) by an acyclic complex \(K\) and then inverting bounded acyclic complex (right localization). Generators and relations are provided for the algebra \(\mathcal{MH}(\mathcal{A})\). The naive lattice algebra \(\mathcal{N}=\mathcal{N}(\Xi_m,\phi_m)\), defined by \textit{M. Kapranov} in [J. Algebra 202, No. 2, 712--744 (1998; Zbl 0910.18005)] is an algebra constructed from a collection of Hopf algebras \(\Xi_m\), \(m\in\mathbb{Z}\), together with Hopf pairings \(\phi_m:\Xi_{m+1}\times\Xi_m\rightarrow\mathbb{Q}(v)\). The algebra \(\mathcal{N}\) is generated by the algebras \(\Xi_m\), \(m\in\mathbb{Z}\), \(\Xi_m\) and \(\Xi_n\) commute if \(\left|m-n\right|>1\) and \[ \xi_m\xi_{m+1}=(\mathrm{Id}\otimes \phi_m\otimes\mathrm{Id})(\Delta_{\Xi_{m+1}}(\xi_{m+1})\otimes \Delta_{\Xi_m}(\xi_m)) \] for \(\xi_m\in\Xi_m, \xi_{m+1}\in\Xi_{m+1}\), where we denoted suggestively the comultiplications of \(\Xi_{m}\), \(\Xi_{m+1}\). The naive lattice algebra of the category \(\mathcal{A}\) is the naive lattice algebra associated to the collection \(\Xi_m=\mathcal{H}_{\mathrm{tw}}^{e}(\mathcal{A})\), \(m\in\mathbb{Z}\) and \(\phi_m=\phi\). Generators and relations are provided for the naive lattice algebra \(\mathcal{N}(\mathcal{A})\). The author defines the componentwise twisted Ringel-Hall algebra by twisting the multiplication by the Euler form of the category \(\mathcal{C}^b(\mathcal{A})\). This allows the author to give an explicit isomorphism between the naive lattice algebra \(\mathcal{N}(\mathcal{A})\) and the componentwise twisted Ringel-Hall algebra. The definition of the derived Hall algebra \(\mathcal{DH}(\mathcal{A})\) is recalled, and generators and relations are provided for its extended twisted version \(\mathcal{DH}_{\mathrm{tw}}^e(\mathcal{A})\). The lattice algebra, \(\mathcal{L}(\mathcal{A})\), defined by Kapranov [loc. cit.] is introduced. By a result of \textit{J. Sheng} and \textit{F. Xu} [Algebra Colloq. 19, No. 3, 533--538 (2012; Zbl 1250.18013)], we have an isomorphism \(\mathcal{DH}_{\mathrm{tw}}^e(\mathcal{A})\simeq \mathcal{L}(\mathcal{A})\). The twisted modified Ringel-Hall algebra \(\mathcal{MH}_{\mathrm{tw}}(\mathcal{A})\) is constructed from the modified Ringel-Hall algebra by twisting the multiplication using the bilinear form \[ \langle[M],[N]\rangle=\prod_{p}\left|\mathrm{Ext}^p_{\mathcal{C}^b(\mathcal{A})}(M,N)\right|^{(-1)^p} \] while the relative twisted Ringel-Hall algebra is defined using the relative Euler form (the author denotes \(\hat{M}\) the class of \(M\) in the category of which \(M\) is an object): \[ \langle[M],[N]\rangle_r=\sqrt{\sum_{i,j}\langle\hat{M}^i,\hat{N}^j\rangle^{(-1)^{j-i+1}(j-i+1)}}. \] Again, the relative twisted modified Ringel-Hall algebra is described by generators and relations. This presentation allows the author to define a surjective morphism \(\mathcal{MH}_{\mathrm{rtw}}(\mathcal{A})\rightarrow\mathcal{L}_*(\mathcal{A})\). In a quite technical proof, the kernel of this morphism is described. In the last section of the paper, the author takes advantage of his isomorphism between \(\mathcal{N}(\mathcal{A})\) and \(\mathcal{NH}_{\mathrm{ctw}}(\mathcal{A})\) to prove the derived invariance of the naive lattice algebra \(\mathcal{N}(\mathcal{A})\). Along the way, the author gives an explicit injection \(\mathcal{DH}_{\mathrm{tw}}(\mathcal{A})\hookrightarrow \mathcal{MH}_{\mathrm{ctw}}(\mathcal{A})\) which can be extended into an isomorphism by replacing \(\mathcal{DH}_{\mathrm{tw}}(\mathcal{A})\) by the completely extended twisted derived Hall algebra \(\mathcal{DH}_{\mathrm{tw}}^{\mathrm{ce}}(\mathcal{A})\).