Algebras with finitely many orbits.
Let \(K\) be a field and let \(R\) be a \(K\)-algebra. Denote by \(U(R)\) the group of units of \(R\). By a \(U(A)\)-orbit of \(R\) the authors mean the double cosets \(U(A)\cdot r \cdot U(A)\) of the elements \(r\in R\). Obviously the \(U(A)\)-orbits of \(R\) are in bijective correspondence with the conjugacy classes of left principal ideals of \(R\). Denote by \({\mathcal L}(R)\) the set of left ideals of \(R\) equipped with the usual multiplication. If \(R\) is a finite-dimensional \(K\)-algebra the authors study the semigroup \(C(R) = {\mathcal L}(R)/U(R)\) of \(U(R)\)-conjugacy classes on \({\mathcal L}(R)\). It is observed in the paper that a finite dimensional algebra \(R\) over an infinite field \(K\) is of finite representation type if and only if, for each \(m\geq 1\), the semigroup \(C({\mathbb M}_n(R))\) of the full matrix algebra \({\mathbb M}_n(R)\) is finite. Moreover, for any finite-dimensional algebra \(R\) over an arbitrary field \(K\), the following implications hold: (a)\( \Rightarrow\)(b)\(\Rightarrow\)(c)\(\Rightarrow\)(d), where (a)--(d) are the following conditions: (a) \(R\) is of finite representation type; (b) The semigroup \(C(R)\) is finite; (c) \(R\) has finitely many \(U(R)\)-orbits; (d) The semigroup \( {\mathcal I}(R)\) of all two-sided ideals of \(R\) is a distributive lattice. It is also shown that any unitary algebra such that the semigroup \(C(R)\) is finite is left Artinian. One of the main results of the paper asserts that if the field \(K\) is algebraically closed, \(R\) is a finite-dimensional \(K\)-algebra \(R\) such that the lattice \( {\mathcal I}(R)\) of two-sided ideals of \(R\) is distributive and the square of the Jacobson radical \(J(R)\) of \(R\) is zero, then \(R\) has finitely many \(U(R)\)-orbits if and only if the contour of \(J(R)\) (associated to \(J(R)\) in the paper) has no cycles and is a union of staircases, triples or flat contours. If, in addition, the algebra \(R\) is basic, then the semigroup \(C(A)\) is finite if and only if the separated graph \(\Gamma^s(R)\) of \(R\) has no cycles and \(\dim_K\,eJ(R)\leq 3\), for every primitive idempotent \(e\) of \(R\).
scientific article; zbMATH DE number 1956746