### Abstract

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^{m}=(QP)^{m} and (PQ)^{m-1}≠(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.

Original language | English (US) |
---|---|

Pages (from-to) | 1823-1836 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 435 |

Issue number | 8 |

DOIs | |

State | Published - Oct 15 2011 |

### Fingerprint

### Keywords

- Drazin inversion
- Finite-dimensional algebra
- Group inversion
- Idempotent
- Skew and oblique projection

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*435*(8), 1823-1836. https://doi.org/10.1016/j.laa.2011.03.046

**On certain finite-dimensional algebras generated by two idempotents.** / Böttcher, A.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 435, no. 8, pp. 1823-1836. https://doi.org/10.1016/j.laa.2011.03.046

}

TY - JOUR

T1 - On certain finite-dimensional algebras generated by two idempotents

AU - Böttcher, A.

AU - Spitkovsky, Ilya

PY - 2011/10/15

Y1 - 2011/10/15

N2 - This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)m=(QP)m and (PQ)m-1≠(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.

AB - This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)m=(QP)m and (PQ)m-1≠(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.

KW - Drazin inversion

KW - Finite-dimensional algebra

KW - Group inversion

KW - Idempotent

KW - Skew and oblique projection

UR - http://www.scopus.com/inward/record.url?scp=79958796229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79958796229&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2011.03.046

DO - 10.1016/j.laa.2011.03.046

M3 - Article

VL - 435

SP - 1823

EP - 1836

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 8

ER -