Algebras with finitely many orbits.
type: Item
instance of: scholarly article
title: Algebras with finitely many orbits.
zbMATH Open document ID: 1039.16011
DOI: 10.1016/S0021-8693(03)00129-7
author: Lex E. Renner
author: Jan Okniński
published in: Journal of Algebra
publication date: 2003-07-30T00:00:00Z
review text: 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 \(C(A)\) i…
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1956746
zbMATH Keywords: finite representation type
zbMATH Keywords: group of units
zbMATH Keywords: orbits
zbMATH Keywords: distributive lattice
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956746