We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement (A, B, C) of extriangulated categories, we show that cotorsion pairs in A and C induce cotorsion pairs in B under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomenon when it is applied to abelian categories.

Let A be the path algebra of a finite acyclic quiver Q over a finite field. We realize the quantum cluster algebra with principal coefficients associated to Q as a sub-quotient of a certain Hall algebra involving the category of morphisms between projective A-modules.

Abstract. Let A be a finite dimensional hereditary algebra over a finite field, and let m be a fixed integer such that m = 0 or m > 2. In the present paper, we first define an algebra L m (A) associated to A, called the m-periodic lattice algebra of A, and then prove that it is isomorphic to Bridgeland's Hall algebra DH m (A) of m-cyclic complexes over projective A-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of A into DH m (A).

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Let A be the path algebra of a Dynkin quiver Q over a finite field, and P be the category of projective A-modules. Denote by C 1 (P) the category of 1-cyclic complexes over P, and ñ+ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in C 1 (P). In this paper, we prove the existence of Hall polynomials in C 1 (P), and then establish a relationship between the Hall numbers for indecomposable objects therein and those for A-modules. Using Hall polynomials evaluated at 1, we define a Lie bracket in ñ+ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if Q is bipartite, ñ+ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type A n , ñ+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different ñ+ . We also characterize the Lie algebras ñ+ by generators and relations. When Q is of type A, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable A-module.