Indagationes Mathematicae-New Series
Invertibility in Banach algebras generated by two idempotents can be checked with the help of a theorem by Roch, Silbermann, Gohberg, and Krupnik. This theorem cannot be used to study generalized invertibility. The present paper is devoted to group invertibility in two types of finite-dimensional algebras which are generated by two idempotents, algebras generated by two tightly coupled idempotents on the one hand and algebras of dimension at most four on the other. As a side product, the paper gives the classification of all at most four-dimensional algebras which are generated by two idempotents. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Boettcher, A. and Spitkovsky, I. M., Group inversion in certain finite-dimensional algebras generated by two idempotents (2012). Indagationes Mathematicae-New Series, 23(4), 715-732.