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Formal multiplications, bialgebras of distributions and nonassociative Lie theory

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Abstract

We describe the general nonassociative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and nonassociative bialgebras.

Starting with a formal multiplication we construct a nonassociative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how identities in loops produce identities in bialgebras. While associativity in loops translates into associativity in algebras, other loop identities (such as the Moufang identity) produce new algebra identities. Finally, we define a class of unital formal multiplications for which Ado’s theorem holds and give examples of formal loops outside this class.

A by-product of the constructions of this paper is a new identity on Bernoulli numbers. We give two proofs: one coming from the formula for the nonassociative logarithm, and the other (due to D. Zagier) using generating functions.

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References

  1. E. Abe, Hopf Algebras, Cambridge Tracts in Mathematics, Vol. 74, Cambridge University Press, Cambridge, 1980.

    MATH  Google Scholar 

  2. L. Gerritzen, R. Holtkamp, Hopf co-addition for free magma algebras and the non-associative Hausdorff series, J. Algebra 265 (2003), 264–284.

    Article  MATH  MathSciNet  Google Scholar 

  3. А. И. Мальцев, Аналитические лупы, Матем. сб. 36(78) (1955), 569–576. (A. I. Mal’tsev, Analytic loops (in Russian), Mat. Sb. (N.S.) 36(78) (1955), 569–576.)

  4. P. O. Miheev, L. V. Sabinin, Quasigroups and differential geometry, in: Quasigroups and Loops: Theory and Applications, Sigma Series in Pure Mathematics, Vol. 8, Heldermann, Berlin, 1990, pp. 357–430.

    Google Scholar 

  5. J. M. Pérez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math. 208 (2007), 834–876.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. M. Pérez-Izquierdo, Right ideals in non-associative universal enveloping algebras of Lie triple systems, J. Lie Theory 18 (2008), no. 2, 375–382.

    MATH  MathSciNet  Google Scholar 

  7. J. M. Pérez-Izquierdo, I. P. Shestakov, An envelope for Malcev algebras, J. Algebra 272 (2004), 379–393.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. V. Sabinin, Smooth Quasigroups and Loops, Mathematics and its Applications, Vol. 492, Kluwer Academic, Dordrecht, 1999.

    MATH  Google Scholar 

  9. Л. В. Сабинин, П. О. Михеев, Локальные аналитические лупы с тождеством правой альтернативности, в сб., Проблемы теории сетей и квазигрупп, КГУ, Калинин, 1985, стр. 72–75, 158. (L. V. Sabinin, P. O. Mikheev, Local analytic loops with a right alternativity identity (Russian), in: Problems of the Theory of Webs and Quasigroups, Collect. sci. Works, Kalinin Univ., Kalinin, 1985, pp. 72–75, 158.)

  10. Л. В. Сабинин, П. О. Михеев, Об инфинитезимальной теории локальных аналитических луп, ДАН СССР 297 (1987), 801–804. Engl. transl.: L. V. Sabinin, P. O. Mikheev, On the infinitesimal theory of local analytic loops, Sov. Math., Dokl. 36 (1988), 545–548.

  11. I. P. Shestakov, U. U. Umirbaev, Free Akivis algebras, primitive elements, and hyperalgebras, J. Algebra 250 (2002), 533–548.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, CINVESTAV, A.P. 14-740, C.P. 07000, México, D.F., México

    J. Mostovoy

  2. Departamento Matemáticas y Computación, Universidad de La Rioja, Logroño, 26004, Spain

    J. M. Pérez–Izquierdo

Authors
  1. J. Mostovoy
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  2. J. M. Pérez–Izquierdo
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Correspondence to J. Mostovoy.

Additional information

Supported by SEP-CONACyT grant 44100. (J. Mostovoy)

Supported by the grants MTM2007-67884-C04-03, ANGI2005/05 and the SEPCONACyT grant 44100. (J. M. Pérez–Izquierdo)

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Mostovoy, J., Pérez–Izquierdo, J.M. Formal multiplications, bialgebras of distributions and nonassociative Lie theory. Transformation Groups 15, 625–653 (2010). https://doi.org/10.1007/s00031-010-9106-5

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  • DOI: https://doi.org/10.1007/s00031-010-9106-5

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