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Abstract
We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in Erdmann and Skowroński (Algebras Represent Theory 22:387–406, 2019), and hence that it is a tame symmetric periodic algebra of period 4.
Highlights
Introduction and main resultsThroughout this paper, K will denote a fixed algebraically closed field
We introduce and study higher spherical algebras, which are “higher analogs” of the non-singular spherical algebras introduced in [11], and provide a new exotic family of tame symmetric periodic algebras of period 4
It follows from the above theorem that the higher spherical algebras S(m, λ), m ≥ 2, λ ∈ K∗, form an exotic family of algebras of generalized quaternion type whose Gabriel quiver is not 2-regular
Summary
Introduction and main resultsThroughout this paper, K will denote a fixed algebraically closed field. The relations: βνδ = βγσ + λ(βγσα)m−1βγσ, νδα = γσα + λ(γσαβ)m−1γσα, σ ω = σαβ + λ(σαβγ)m−1σαβ, ωγ = αβγ + λ(αβγσ)m−1αβγ, (αβγσ)mα = 0, αβν = ων, δαβ = δ ω, ωγσ = ωνδ, γσ = νδ , (γσαβ)mγ = 0. The following theorem describes basic properties of higher spherical algebras.
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