Semi-invariants of canonical algebras
Let \(K\) be an algebraically closed field, and let \(A\) be a finite dimensional associative \(K\)-algebra with an identity which is assumed to be basic and connected. There is an isomorphism \(A\cong KQ/I\), where \(KQ\) is the path algebra of the Gabriel quiver \(Q=Q_A\) of \(A\) and \(I\) is an admissible ideal of \(KQ\) generated by a finite system of forms (so called \(K\)-linear relations). Let \(n\geq 3\), \({\mathbf m}=(m_1,\ldots,m_n)\in\mathbb{Z}^n\), \(m_i\geq 2\), \({\mathbf x}=(x_3,\ldots,x_n)\in(K\setminus 0)^{n-2}\), \(x_3=1\), \(x_i\neq x_j\) for all \(i\neq j\). Denote by \(\Lambda({\mathbf m},{\mathbf x})=K\Delta(m_1,\ldots,m_n)/I(x_3,\ldots,x_n)\) the canonical algebra associated with the pair \(({\mathbf m},{\mathbf x})\) [\textit{C. M. Ringel}, Tame algebras and integral quadratic forms. Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. The paper is devoted to the study of the algebras of semi-invariants on the varieties of regular representations of canonical algebras \(\Lambda({\mathbf m},{\mathbf x})\). In particular, when \(\text{char }K=0\) the authors obtain an explicit description of such algebras and prove that these algebras are complete intersections or polynomial algebras. The proof is based on the techniques developed in their forthcoming paper [\textit{A. Skowroński, J. Weyman}, The algebras of semi-invariants of quivers. Transform. Groups (to appear)].
scientific article; zbMATH DE number 1385141