Characterizations of strongly simply connected polynomial growth algebras
Let \(K\) be an algebraically closed field and let \(A\) be a connected basic finite dimensional \(K\)-algebra viewed as a quotient \(KQ/I\) of a path algebra \(KQ\) of a quiver \(Q=(Q_0,Q_1)\) (i.e. a finite oriented graph) modulo an admissible ideal \(I\). Here \(Q_0\) is the set of vertices of \(Q\) and \(Q_1\) is the set of arrows of \(Q\). The algebra \(A\) is said to be strongly simply connected if \(Q\) has no oriented cycles and the first Hochschild cohomology space \(H^1(C,C)\) vanishes for every full convex subcategory \(C\) of \(A\). The algebra \(A\) is said to be tame of polynomial growth if there exists an integer \(m\) such that the indecomposable \(A\)-modules occur, in each dimension \(d\), in a finite number of discrete and at most \((d+1)^m\) one-parameter families. The main aim of the paper is to give a characterization of strongly simply connected tame algebras of polynomial growth in terms of an integral Tits quadratic form \(q_A:\mathbb{Z}^{Q_0}\to\mathbb{Z}\) associated with \(A\). Given a finite dimensional right \(A\)-module \(X\) we denote by \(\mathbf{dim} X\in\mathbb{Z}^{Q_0}\) the dimension vector of \(X\). By a support algebra of \(X\) we shall mean the full subcategory \(B(X)\) of \(A\) formed by the vertices of the support \(\text{supp}(X)=\{i\in Q_0;\;X(i)\neq 0\}\). We shall also use the number \(m(X)=\min\{\dim_KX(i);\;i\in\text{supp}(X)\}\). The main results of the paper assert that if \(A\) is a finite dimensional strongly simply connected algebra then the following conditions are equivalent: (1) \(A\) is tame of polynomial growth, (b) the integral Tits quadratic form \(q_A\) is weakly non-negative (that is, \(q_A(v)\geq 0\) for all \(v\in\mathbb{N}^{Q_0}\)) and \(m(X)=1\) for every indecomposable \(A\)-module \(X\) with \(q_A(\mathbf{dim} X)\geq 2\), (3) \(q_A(\mathbf{dim} X)\in\{0,1\}\) for every indecomposable \(A\)-module \(X\) with \(m(X)\geq 2\), (4) for every indecomposable \(A\)-module \(X\) with \(m(X)\geq 3\) and \(q_A(\mathbf{dim} X)=1\) the support algebra \(B(X)\) is either a representation-infinite tilted algebra of Euclidean type or a tubular algebra, (5) for every indecomposable \(A\)-module \(X\) with \(q_A(\mathbf{dim} X)\leq 0\) the support algebra \(B(X)\) is convex in \(A\) and is a tame concealed algebra or a tubular algebra.
connected basic finite dimensional algebras
simply connected tame algebras of polynomial growth
scientific article; zbMATH DE number 937027