Non-Leibniz algebras with logarithms do not have the trigonometric identity

D. Przeworska-Rolewicz

Banach Center Publications (2000)

  • Volume: 53, Issue: 1, page 177-189
  • ISSN: 0137-6934

Abstract

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Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. [ C ( x ) ] 2 + [ S ( x ) ] 2 = e whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.

How to cite

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Przeworska-Rolewicz, D.. "Non-Leibniz algebras with logarithms do not have the trigonometric identity." Banach Center Publications 53.1 (2000): 177-189. <http://eudml.org/doc/209072>.

@article{Przeworska2000,
abstract = {Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.},
author = {Przeworska-Rolewicz, D.},
journal = {Banach Center Publications},
keywords = {sine mapping; algebra with unit; algebraic analysis; logarithmic mapping; Leibniz condition; cosine mapping; trigonometric identity},
language = {eng},
number = {1},
pages = {177-189},
title = {Non-Leibniz algebras with logarithms do not have the trigonometric identity},
url = {http://eudml.org/doc/209072},
volume = {53},
year = {2000},
}

TY - JOUR
AU - Przeworska-Rolewicz, D.
TI - Non-Leibniz algebras with logarithms do not have the trigonometric identity
JO - Banach Center Publications
PY - 2000
VL - 53
IS - 1
SP - 177
EP - 189
AB - Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.
LA - eng
KW - sine mapping; algebra with unit; algebraic analysis; logarithmic mapping; Leibniz condition; cosine mapping; trigonometric identity
UR - http://eudml.org/doc/209072
ER -

References

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  1. DB[1] A. di Bucchianico, Banach algebras, logarithms and polynomials of convolution type, J. Math. Anal. Appl. 156 (1991), 253-273. Zbl0765.46027
  2. PR[1] D. Przeworska-Rolewicz, Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988. 
  3. PR[2] D. Przeworska-Rolewicz, Consequences of the Leibniz condition, in: Different Aspects of Differentiability. Proc. Conf. Warsaw, September 1993. Ed. D. Przeworska-Rolewicz. Dissertationes Math. 340 (1995), 289-300. 
  4. PR[3] D. Przeworska-Rolewicz, Logarithms and Antilogarithms. An Algebraic Analysis Approach, With Appendix by Z. Binderman. Kluwer Academic Publishers, Dordrecht, 1998. 
  5. PR[4] D. Przeworska-Rolewicz, Linear combinations of right invertible operators in commutative algebras with logarithms, Demonstratio Math. 31 (1998), 887-898. 
  6. PR[5] D. Przeworska-Rolewicz, Postmodern logarithmo-technia, International Journal of Computers and Mathematics with Applications (to appear). Zbl0987.05014
  7. PR[6] D. Przeworska-Rolewicz, Some open questions in Algebraic Analysis, in: Unsolved Problems on Mathematics for the 21th Century - A Tribute to Kiyoshi Iséki's 80th Birthday, IOS Press, Amsterdam (to appear). 

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