### 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 language | English (US) |
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Pages (from-to) | 538-551 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1 2013 |

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### 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

*Linear Algebra and Its Applications*,

*439*(3), 538-551. https://doi.org/10.1016/j.laa.2012.07.020

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 439, no. 3, pp. 538-551. https://doi.org/10.1016/j.laa.2012.07.020

}

TY - JOUR

T1 - Classification of the finite-dimensional algebras generated by two tightly coupled idempotents

AU - Böttcher, A.

AU - Spitkovsky, Ilya

PY - 2013/8/1

Y1 - 2013/8/1

N2 - 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.

AB - 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.

KW - Finite-dimensional algebra

KW - Idempotent

KW - Two projections

KW - Word problem

UR - http://www.scopus.com/inward/record.url?scp=84878293112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878293112&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.07.020

DO - 10.1016/j.laa.2012.07.020

M3 - Article

VL - 439

SP - 538

EP - 551

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 3

ER -