McKay quivers and absolute n-complete algebras

In this paper, we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group. Using this and our results on “returning arrows” in McKay quiver, we give an algorithm to construct the McKay quiver of a finite abelian group. Using this construction, we show how the cone and cylinder of an (n?1)-Auslander absolute n-complete algebra are truncated from the McKay quivers of abelian groups.     

Multiplicative preprojective algebras of Dynkin quivers

For a commutative ring R and an ADE Dynkin quiver Q, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as R-algebras if and only if the bad primes for Q – 2 in type D, 2 and 3 for $Q=E_6$, $E_7$ and 2, 3 and 5 for $Q=E_8$ – are invertible in R. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2,1/3]$ for $Q=E_6$, $E_7$ and over $\mathbb{Z}[1/2,1/3,1/5]$ for $Q=E_8$. Conversely, if some bad prime is not invertible in R, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest.In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q\ne A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q=D_{2n}$, $n\ge 2$ and $Q=E_7$, $E_8$.     

Construct non-graded bi-Frobenius algebras via quivers

Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.     

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.     

Presenting degenerate Ringel-Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel-Hall algebra H_q(n) of a cyclic quiver Δ(n), we give a presentation for the degenerate algebra H₀(n). This is achieved by establishing a presentation for the generic extension monoid algebra of Δ(n). As an application, we show that both the degenerate Ringel-Hall algebra and the degenerate quantum affine sl_n admit multiplicative bases.     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U ⁺ of the quantum group U _q (g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U ⁺, respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toën. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.