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We prove that Hall polynomial exists for each triple of decomposition sequences associated with tame quivers with automorphism (or, equivalently, for all tame hereditary algebras over finite fields). This generalizes the results of Hubery (2010) [18] and Deng and Ruan (2017) [9]. [ABSTRACT FROM AUTHOR]

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel–Hall algebra D ( Q ) of a quiver Q as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation L ( Λ 0 ) of D ( Δ n ) of the cyclic quiver Δ n provides a realization of the q -deformed Fock space ⋀ ∞ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of Δ n , we obtain a decomposition of the basic representation L ( Λ 0 ) which induces the Kashiwara–Miwa–Stern decomposition of ⋀ ∞ and a construction of the canonical basis of ⋀ ∞ defined by Leclerc and Thibon in terms of certain monomial basis elements in D ( Δ n ) . [ABSTRACT FROM AUTHOR]

In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel–Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein. [ABSTRACT FROM AUTHOR]

Abstract: We generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized Cartan matrices and a monomial basis property of those associated with symmetric (classical) Cartan matrices to their respective symmetrizable case. We then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word. As an application, we develop an algorithm to compute all tight monomials in the rank 2 Dynkin case. [Copyright &y& Elsevier]

Abstract: In the present paper, we show that the Ringel–Hall algebra of a finitary algebra over a finite field satisfies fundamental relations in a more general setting. By twisting the multiplication, we obtain the quantum Serre relations. Finally, certain relations between Ringel–Hall algebras of an algebra and its factor algebras are discussed. [Copyright &y& Elsevier]

Let D(Λ) be the double Ringel–Hall algebra of a finite dimensional hereditary algebra Λ. The present paper shows that the BGP-reflection operators of D(Λ) coincide with Lusztig''s symmetries (up to canonical isomorphisms) and satisfy the braid group relations on the whole double Ringel–Hall algebra D(Λ). This answers a question of Sevenhant and Van den Bergh [J. Algebra 221 (1999) 135–160]. [Copyright &y& Elsevier]