Advertisement

European Journal of Mathematics

, Volume 5, Issue 3, pp 881–902 | Cite as

Nearly Frobenius algebras

  • Ana González
  • Ernesto LupercioEmail author
  • Carlos Segovia
  • Bernardo Uribe
Research/Review Article
  • 16 Downloads

Abstract

In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational results and some of the applications they encounter in geometry, topology and representation theory.

Keywords

Frobenius algebra TQFT String topology 

Mathematics Subject Classification

16G20 55N99 53D45 

Notes

Acknowledgements

We would like to thank the referee for very careful and useful remarks that improved this paper.

References

  1. 1.
    Abrams, L.: Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory Ramifications 5(5), 569–587 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Artenstein, D., González, A., Lanzilotta, M.: Constructing nearly Frobenius algebras. Algebr. Represent. Theory 18(2), 339–367 (2015)CrossRefGoogle Scholar
  3. 3.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. I. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Atiyah, M.F.: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brauer, R., Nesbitt, C.: On the regular representations of algebras. Proc. Natl. Acad. Sci. USA 23(4), 236–240 (1937)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cerf, J.: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Inst. Hautes Études Sci. Publ. Math. 39, 5–173 (1970)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chas, M., Sullivan, D.: String topology (1999). arXiv:math.GT/9911159
  8. 8.
    Cohen, R.L., Godin, V.: A polarized view of string topology. In: Tillmann, U. (ed.) Topology, Geometry and Quantum Field Theory. London Mathematical Society Lecture Note Series, vol. 308, pp. 127–154. Cambridge University Press, Cambridge (2004). arXiv:math.AT/0303003
  9. 9.
    Cohen, R.L., Jones, J.D.S.: A homotopy theoretic realization of string topology. Math. Ann. 324(4), 773–798 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohen, R.L., Klein, J.R., Sullivan, D.: The homotopy invariance of the string topology loop product and string bracket. J. Topol. 1(2), 391–408 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Moore, G.W., Segal, G.: D-branes and K-theory in 2D topological field theory (2006). arXiv:hep-th/0609042
  12. 12.
    Nakayama, T.: On Frobeniusean algebras. I. Ann. Math. 40, 611–633 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nakayama, T.: On Frobeniusean algebras. II. Ann. Math. 41, 1–21 (1941)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Facultad de IngenieríaInstituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”MontevideoUruguay
  2. 2.Centro de Investigaciones y Estudios AvanzadosMexico D.F.Mexico
  3. 3.Instituto de Matemáticas de la UNAMCampus OaxacaMexico
  4. 4.Universidad del NorteBarranquillaColombia

Personalised recommendations