Nakayama algebras are among the best understood representation-finite algebras. They are defined as those algebras such that each indecomposable projective and each indecomposable injective module admits a unique composition series. An equivalent characterisation is that τjS is simple (or zero) for all j ∈ ℤ and every simple module S. Here, τ denotes the Auslander–Reiten translation. Nakayama algebras can be classified by the sequence of lengths of their indecomposable projective modules, called the Kupisch series.
In this talk, we introduce a higher analogue of a Nakayama algebra for each Kupisch series 𝓁 in the sense of Iyama's higher Auslander–Reiten theory. More precisely, (in type A) the higher Nakayama algebra A𝓁(d) is a quotient of the higher Auslander algebra An(d) of type A, constructed by Iyama and studied extensively by Oppermann and Thomas. In type ̃A, one has to use an infinite version of An(d). The higher Nakayama algebra has a d-cluster-tilting module, i.e. a module M with
add(M) = {N | Exti(M,N) = 0 ∀i = 1, . . . , d−1 } = {N | Exti(N,M) = 0 ∀i = 1, . . . , d−1 }.
There are n simple modules in add(M) and they satisfy that τdjS is simple for all j ∈ ℤ and every simple module S in add(M), where τd = τΩd−1 is Iyama's higher Auslander–Reiten translation.
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