Relative invariants for representations of finite dimensional algebras
The author considers a finite dimensional basic algebra \(\Lambda\) over a field \(K\) of characteristic 0. For a fixed dimension vector \(\beta\) one defines the representation space \(R(\Lambda,\beta)\) and a group \(\text{GL}(\beta)\) such that the \(\text{GL}(\beta)\)-orbits in \(R(\Lambda,\beta)\) correspond to the isomorphy classes of \(\beta\)-dimensional modules. A relative invariant is a polynomial function on \(R(\Lambda,\beta)\) on which \(\text{GL}(\beta)\) acts by a character. Using a construction of \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)] one can associate relative invariants to finite dimensional \(\Lambda\)-modules if \(\Lambda\) is a path algebra of a quiver. \textit{J. Weyman} and the reviewer [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)] showed that Schofield's relative invariants span the space of relative invariants for quivers without oriented cycles. \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] generalized this result to arbitrary quivers. In this paper, the author generalizes Schofield's construction to arbitrary finite dimensional basic algebras. The relative invariants obtained this way will again span the space of relative invariants. To get algebra generators of the algebra of relative invariants it suffices to take the relative invariants corresponding to indecomposable representations. A similar result was obtained by \textit{J. Weyman} and the reviewer [J. Algebra 258, No. 1, 216-227 (2002; Zbl 1048.16005)] in the case where \(\Lambda\) is the quotient of a finite dimensional path algebra.
scientific article; zbMATH DE number 1801621