Invariant algebras and completely reducible representations
Assume that \(K\) is a field. The authors present a construction of a large class of affine noetherian \(K\)-algebras \(A\) satisfying the following condition: (FCR) Every finite dimensional representation of \(A\) is completely reducible and the intersection of the kernels of all finite dimensional representations of \(A\) is zero. In particular it is shown that if \(B\) is a noetherian algebra satisfying the condition (BCR) and \(H\) is a finite group of automorphisms of \(B\) such that \((|H|,\text{char}(K))=1\) then the algebra \(B^H\) of \(H\)-invariants of \(B\) also satisfies (BCR). If \(K\) is algebraically closed of characteristic zero, \(\mathfrak g\) is a semisimple Lie \(K\)-algebra, \(B={\mathfrak U}({\mathfrak g})\) is the universal enveloping algebra of \(\mathfrak g\) and \(H\) is a finite group of automorphisms of \(\mathfrak g\) then \(B\) is affine, prime, left and right noetherian and the algebras \(B\), \(B^H\) satisfy the condition (FCR). It is also proved that a \(K\)-algebra \(A\) satisfies (BCR) if and only if \(\bigcap_{m\geq 0}I_m(A)=(0)\) and the factor algebra \(A/I_m(A)\) is finite dimensional and semisimple for every \(m\geq 0\), where \(I_m(A)\subseteq A\) is the ideal of \(A\) of polynomial identities of degree \(m\).
scientific article; zbMATH DE number 817696