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

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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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