Polynomial Identities for Tangent Algebras of Monoassociative Loops

  1. Bremner, M.R. 2
  2. Madariaga, S. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 University of Saskatchewan
    info

    University of Saskatchewan

    Saskatoon, Canadá

    ROR https://ror.org/010x8gc63

Journal:
Communications in Algebra

ISSN: 0092-7872

Year of publication: 2014

Volume: 42

Issue: 1

Pages: 203-227

Type: Article

DOI: 10.1080/00927872.2012.709567 SCOPUS: 2-s2.0-84886384816 WoS: WOS:000325788500015 arXiv: 1111.6113v1 GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Communications in Algebra

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2011/00007/001

Abstract

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator, and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a 2 a ≡ aa 2. These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree. © 2014 Copyright Taylor and Francis Group, LLC.