Semilinear clannish algebras

Semilinear clannish algebras arising from surfaces with orbifold points

“…The concept of semilinear path algebra just defined forms a particular instance of the concept of path algebra defined in §2.1.4. This is not the case for the more general concept of semilinear path algebra that Bennett-Tennenhaus-Crawley-Boevey work with in [3]: they allow K to be a division ring, and do not require it to be finite-dimensional over some particular field.…”

“…As in §2.1.5 (and less generally than [3]), let ( Q, d) be a weighted quiver all of whose weights are the same positive integer d, K/F be a degree-d cyclic Galois field extension, and σ :…”

“…Remark 2.12. The semilinear clannish algebras introduced in [3] are far more general: K is allowed to be a division ring, not necessarily finite-dimensional over any field, and for each the arrow b ∈ Q 1 , σ b is allowed to be any ring automorphism of K.…”

“…Clannish algebras, defined by Crawley-Boevey over thirty years ago [9], form a class of tame algebras whose indecomposable modules enjoy explicit parameterizations in terms of strings and bands that generalize the familiar parameterizations of indecomposables for gentle algebras. Very recently, Bennett-Tennenhaus-Crawley-Boevey [3], have introduced semilinear clannish algebras, a more general class of algebras where the action of an arrow of the quiver on a representation allows the scalars to "come out" up to the application of Given Σ ω := (Σ, M, O, ω) and a triangulation τ of Σ = (Σ, M, O), in Subsection 6.1 we associate to (τ, ω) a loop-free weighted quiver (Q(τ, ω), d(τ, ω)), that is, a pair consisting of a loop-free quiver Q(τ, ω) and a tuple d(τ, ω)) = (d(τ, ω) k ) k∈τ of positive integers. The quiver Q(τ, ω) is a modification of the quiver Q(τ ) that defines the 1-skeleton of X(τ ).…”

“…Section 8 is devoted to recalling the parameterizations of the indecomposable modules over a semilinear clannish algebra in terms of symmetric and asymmetric strings and bands given in [3], to providing a full list of strings and bands for the semilinear clannish blocks from Subsection 4.2, and to illustrating the construction of symmetric-string modules by means of an explicit example, and the representation of the Jacobian algebra corresponding to it under Morita equivalence.…”