Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Semi-invariants of canonical algebras

We describe the algebras of semi-invariants on the varieties of regular representations of canonical algebras. In particular, we show that these algebras are polynomial algebras or complete intersections. Received: 29 March 1999     

Indecomposable modules for domestic canonical algebras

We show that-up to precisely one-each exceptional module over a domestic canonical algebra of quiver type over a field k can be represented by matrices whose entries are just 0 and 1. In the case we calculate the matrices of these representations explicitly.     

Extended canonical algebras and Fuchsian singularities

The authors introduce a new class of finite dimensional algebras called extended canonical, and determine the shape of their derived categories. Extended canonical algebras arise from a canonical algebra ?? by onepoint extension or coextension by an indecomposable projective module. Our main results concern the case of negative Euler characteristic of the corresponding weighted projective line ${\mathbb{X}}$ ; more specifically we establish, for a base field of arbitrary characteristic, a link to the Fuchsian singularity R of ${\mathbb{X}}$ which for the base field of complex numbers is isomorphic to an algebra of automorphic forms. By means of a recent result of Orlov we show that the triangulated category of the graded singularities of R (in the sense of Buchweitz and Orlov) admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra. Of particular interest are those cases where the attached Coxeter transformation has spectral radius one. A K-theoretic analysis then shows that this happens exactly for 38 cases including Arnold??s 14 exceptional unimodal singularities. The paper is related to recent independent work by Kajiura, Saito and Takahashi.     

Geometry of regular modules over canonical algebras

We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

     

Wild radical square zero algebras

It is shown that a radical square zero algebra is wild, if and only if it is of Corner’s type, and it is strictly wild if and only if it is Endo-wild. This gives a negative answer to a problem posed by Simson.     

Wild automorphisms of nilpotent-by-abelian Lie algebras

This research was carried out when the author was an Alexander von Humboldt fellow in the University of Bochum, Germany     

Wild automorphisms of free metabelian algebras

We establish a necessary condition for the invertibility of an endomorphism of a free associative algebra. As an application, we offer examples of wild automorphisms of certain free metabelian algebras.     

A characterization of selfinjective algebras of strictly canonical type

We prove that the class of selfinjective algebras of strictly canonical type, investigated in Kwiecień and Skowroński (2009) [27], Kwiecień and Skowroński (2009) [28], coincides with the class of selfinjective algebras having triangular Galois coverings with infinite cyclic group and the Auslander–Reiten quiver with quasi-tubes maximally saturated by simple and projective modules, satisfying natural conditions.     

Wild cluster tilted algebras of rank 3

Let H be a connected wild hereditary algebra of rank 3, T a square-free tilting H-module and the endomorphism ring of T in the cluster category C_H of H. It is shown that the Loewy length of the cluster tilted algebra Γ is 3+r, where r denotes the number of regular indecomposable direct summands of T.     

Graded quiver varieties,quantum cluster algebras and dual canonical basis

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.     

An algorithm for the construction of exceptional modules over tubular canonical algebras

Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λ-modules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length, the output are explicit matrices for the module with the data above.     

Decomposition of some pointed Hopf algebras given by the canonical Nakayama automorphism

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.     

Some of our primal algebras are missing: the canonical primal theory

It has been proven elsewhere that every variety has associated with it a unique canonical theory, where idempotent morphisms split. This article exhibits models of the canonical theory associated with any primal variety, for example, Boolean algebras. One such variety of models is generated by the several-sorted algebra with carriers of all prime cardinalities and with a clone of all finitary operations ω on and between carriers. This primal algebra was unknown. There are more. Presented by R. McKenzie. Received December 20, 2005; accepted in final form May 2, 2006.     

Geometry and the zero sets of semi-invariants for homogeneous modules over canonical algebras

We characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections (respectively, normal). We also investigate the sets of common zeros of semi-invariants of non-zero degree in important cases. In particular, we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components.     

A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras

This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety A, the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any A∈A have truly topological interpretations. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra nA(A) whose underlying algebra belongs to A. Furthermore, each of the operations of nA(A) coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in particular, the variety A is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general.     

On the zero set of semi-invariants for regular modules over tame canonical algebras

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete intersections.