Nakayama Algebra Research Articles

An elementary construction of tilting complexes

Let A be an artin algebra and e∈ A an idempotent with add( eA A )=add( D( A Ae)). Then a projective resolution of Ae eAe gives rise to tilting complexes {P(l) •} l⩾1 for A, where P( l) • is of term length l+1. In particular, if A is self-injective, then End K( Mod-A) (P(l) •) is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes { T(2 l) •} l⩾1 for A, where T(2 l) • is of term length 2 l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes { T( l) •} l⩾1 for A, where T( l) • is of term length l+1.

Representation Dimension and Quasi-hereditary Algebras

We study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra A is stably hereditary, then the representation dimension of A is at most 3. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension.

The stable group of a selfinjective nakayama algebra

The stable group of a selfinjective nakayama algebra

A Note on Stable Equivalence and Nakayama Algebras

Two artin algebras $\Lambda$ and $\Lambda ’$ are said to be stably equivalent if the categories of finitely generated modules modulo projective for $\Lambda$ and $\Lambda ’$ are equivalent categories. If $\Lambda ’$ is stably equivalent to a Nakayama (i.e. generalized uniserial) algebra $\Lambda$, we prove that $\Lambda$ and $\Lambda ’$ have the same number of nonprojective simple modules. And if $\Lambda$ and $\Lambda ’$ are stably equivalent indecomposable Nakayama algebras where each indecomposable projective module has length at least 3, then $\Lambda$ and $\Lambda ’$ have the same admissible sequences.