Classification of the finite-dimensional algebras generated by two tightly coupled idempotents

A. Böttcher, Ilya Spitkovsky

    Research output: Contribution to journalArticle

    Abstract

    Let P and Q be idempotents in a real or complex associative algebra and consider the list of products P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,⋯ The number of factors is called the order of the product. We say that P and Q are tightly coupled if the list contains two products which take the same value and whose orders differ by at most 1. The main result of the paper is the classification of all algebras which are generated by two tightly coupled idempotents. In other words, we provide a list of algebras such that every algebra generated by two tightly coupled idempotents is isomorphic to exactly one algebra of the list. For example, it follows that up to isomorphisms there are exactly 16 copies of such algebras in which the equality PQP=PQPQ holds.

    Original languageEnglish (US)
    Pages (from-to)538-551
    Number of pages14
    JournalLinear Algebra and Its Applications
    Volume439
    Issue number3
    DOIs
    StatePublished - Aug 1 2013

    Fingerprint

    Finite Dimensional Algebra
    Idempotent
    Algebra
    Associative Algebra
    Isomorphism
    Equality
    Isomorphic

    Keywords

    • Finite-dimensional algebra
    • Idempotent
    • Two projections
    • Word problem

    ASJC Scopus subject areas

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

    Cite this

    Classification of the finite-dimensional algebras generated by two tightly coupled idempotents. / Böttcher, A.; Spitkovsky, Ilya.

    In: Linear Algebra and Its Applications, Vol. 439, No. 3, 01.08.2013, p. 538-551.

    Research output: Contribution to journalArticle

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