On certain finite-dimensional algebras generated by two idempotents
Linear Algebra and Its Applications
This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)(m) = (QP)(m) and (PQ)(m-1) not equal (QP)(m-1). The main result is the classification of all these algebras, implying that for each m > = 2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given. (C) 2011 Elsevier Inc. All rights reserved.
Spitkovsky, I. M. and Boettcher, A., On certain finite-dimensional algebras generated by two idempotents (2011). Linear Algebra and Its Applications, 435(8), 1823-1836.