Hereditary properties of direct summands of algebras. (English) Zbl 0952.16005
For an algebra, \(R\), the authors investigate the property, which they call FCR, saying that any finite-dimensional representation is completely reducible and that each element of the algebra acts non-zero on some finite-dimensional representation. As an example of such algebra one can take the universal enveloping algebra of a semisimple finite-dimensional complex Lie algebra. Let \(R=A\oplus V\), where \(A\) is a subalgebra such that \(R\) is finitely generated as an \(A\)-left module and \(V\) is an \(A\)-bimodule. The authors show that if \(R\) satisfies FCR, then so does \(A\).
This result is an easy generalization of an elementary result, also presented in the paper, stating that a subring \(A\) of a semisimple ring is semisimple if it admits a complement, which is left and right stable under \(A\). In fact, if \(\text{char }R\neq 2\), it suffices to assume that the complement is stable under commutation with \(A\).
This result is an easy generalization of an elementary result, also presented in the paper, stating that a subring \(A\) of a semisimple ring is semisimple if it admits a complement, which is left and right stable under \(A\). In fact, if \(\text{char }R\neq 2\), it suffices to assume that the complement is stable under commutation with \(A\).
Reviewer: Volodymyr Mazorchuk (Kyïv)
MSC:
16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
20G15 | Linear algebraic groups over arbitrary fields |