Domestic trivial extensions of simply connected algebras

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Let A be a finite-dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field. It is called simply connected it is triagular and, for any presentation of A as a bound quiver algebra, the fundamental group of its bound quiver is trivial. Let T(A) denote the trivial extension of A by its minimal injective cogenerator. We show that, if A is simply connected, then the following conditions are equivalent: (i) T(Z) is representation-infinite and domestic, (ii) T(A) is 2-parametric, (iii) there exists a representation-infinite tilted algebra β of Euclidean type Dn of Ep such that T(A)→T(B), (iv) A is an iterated tilted algebra of type Dn or Ep.

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