For an Artinian (n− 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of modΛ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in modΛ is at most n− 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal s...

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving op...

The subject of this paper are general properties of direct sum decompositions of automata. Using certain properties of the lattice Sub(A) of subautomata of an automaton A and its Boolean part, lattices of direct sum congruences and direct sum decompositions of A are characterized. We show that every automaton A can be represented as a direct sum of direct sum indecomposable automata, and that t...

1.1 Let A = kG, the group algebra of some finite group where the characteristic of the field k divides |G|. In contrast to working over the complex field, the kG-modules are not usually semisimple. If a Sylow p-subgroup of G is not cyclic then there are infinitely many indecomposable kG-modules, and we usually enjoy little control over the category of such modules. It is therefore instructive t...

We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version quasi-resolutions introduced by Qin-Wang-Zhang. prove that noetherian bounded below and locally finite graded algebra with injective dimension 2 are always Morita equivalent. When we restrict to quadric hypersurfaces, hypersurface, singularity, admits pre-res...

Exceptional cycles in a triangulated category T with Serre duality, introduced by N. Broomhead, D. Pauksztello, and Ploog, have notable impact on the global structure of T. In this paper we show that if is homotopy-like, then any exceptional 1-cycle indecomposable at mouth (i.e., middle term Auslander-Reiten triangle ending it indecomposable); object an n-cycle n?3 mouth. Let A be gentle k-alge...

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where is an n-tuple of automorphisms, t elements in the center R. We show that, for fixed σ, there natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} ,t^{\prime })\). This gives tensor product operation on modules, inducing structure direct sum (over all t) Grothen...

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms Kronecker modules associated with pairs bilinear forms. particular, describe the complex and real case indecomposable Heisenberg as generalization classical $(2n+1)-$dimensional Lie algebra $\mathfrak{h}_{2n+1}$. Then use - local racks correspondence proposed by S. Covez to show that have always glob...