On certain finite-dimensional algebras generated by two idempotents

A. Böttcher, Ilya Spitkovsky

    Research output: Contribution to journalArticle

    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 languageEnglish (US)
    Pages (from-to)1823-1836
    Number of pages14
    JournalLinear Algebra and Its Applications
    Volume435
    Issue number8
    DOIs
    StatePublished - Oct 15 2011

    Fingerprint

    Finite Dimensional Algebra
    Idempotent
    Algebra
    Group Inverse
    Invertible
    Term

    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

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

    In: Linear Algebra and Its Applications, Vol. 435, No. 8, 15.10.2011, p. 1823-1836.

    Research output: Contribution to journalArticle

    Böttcher, A. ; Spitkovsky, Ilya. / On certain finite-dimensional algebras generated by two idempotents. In: Linear Algebra and Its Applications. 2011 ; Vol. 435, No. 8. pp. 1823-1836.
    @article{aaa7ab1cb002438e800fb2d12c9bec72,
    title = "On certain finite-dimensional algebras generated by two idempotents",
    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.",
    keywords = "Drazin inversion, Finite-dimensional algebra, Group inversion, Idempotent, Skew and oblique projection",
    author = "A. B{\"o}ttcher and Ilya Spitkovsky",
    year = "2011",
    month = "10",
    day = "15",
    doi = "10.1016/j.laa.2011.03.046",
    language = "English (US)",
    volume = "435",
    pages = "1823--1836",
    journal = "Linear Algebra and Its Applications",
    issn = "0024-3795",
    publisher = "Elsevier Inc.",
    number = "8",

    }

    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 -