Skip to main content
SpringerLink
Log in

PBW Deformations of a Fomin–Kirillov Algebra and Other Examples

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin–Kirillov algebra \(\mathcal {E}_{3}\). Another one appeared in a paper of García Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andruskiewitsch, N., Fantino, F., García, G.A., Vendramin, L.: On Nichols algebras associated to simple racks. In: Groups, algebras and applications, volume 537 of Contemp. Math., pp. 31–56. Amer. Math. Soc., Providence, RI (2011)

  2. Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J.: The Nichols algebra of a semisimple Yetter-Drinfeld module. Amer. J. Math. 132(6), 1493–1547 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Andruskiewitsch, N., Schneider, H.-J.: Pointed Hopf algebras. In: New directions in Hopf algebras, volume 43 of Math. Sci. Res. Inst. Publ., pp. 1–68. Cambridge University Press, Cambridge (2002)

  4. Bazlov, Y.: Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups. J. Algebra 297(2), 372–399 (2006)

    Article  MathSciNet  Google Scholar 

  5. Berger, R., Ginzburg, V.: Higher symplectic reflection algebras and non-homogeneous N-Koszul property. J. Algebra 304(1), 577–601 (2006)

    Article  MathSciNet  Google Scholar 

  6. Blasiak, J., Liu, R.I., Mészáros, K.: Subalgebras of the Fomin-Kirillov algebra. J Algebraic Combin. 44(3), 785–829 (2016)

    Article  MathSciNet  Google Scholar 

  7. Crawley-Boevey, W., Holland, M.P.: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92(3), 605–635 (1998)

    Article  MathSciNet  Google Scholar 

  8. Ştefan, D., Vay, C.: The cohomology ring of the 12-dimensional Fomin-Kirillov algebra. Adv. Math. 291, 584–620 (2016)

    Article  MathSciNet  Google Scholar 

  9. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147(2), 243–348 (2002)

    Article  MathSciNet  Google Scholar 

  10. Etingof, P., Rains, E.: New deformations of group algebras of Coxeter groups. Int. Math. Res. Not., (10):635–646 (2005)

  11. Etingof, P., Rains, E.: New deformations of group algebras of Coxeter groups. II. Geom. Funct. Anal. 17(6), 1851–1871 (2008)

    Article  MathSciNet  Google Scholar 

  12. Fløystad, G., Vatne, J.E.: PBW-deformations of N-Koszul algebras. J. Algebra 302(1), 116–155 (2006)

    Article  MathSciNet  Google Scholar 

  13. Fomin, S., Kirillov, A.N.: Quadratic algebras, Dunkl elements, and Schubert calculus. In: Advances in geometry, volume 172 of Progr. Math., pp. 147–182. Birkhäuser Boston, Boston (1999)

  14. Gan, W.L., Ginzburg, V.: Deformed preprojective algebras and symplectic reflection algebras for wreath products. J. Algebra 283(1), 350–363 (2005)

    Article  MathSciNet  Google Scholar 

  15. García Iglesias, A., Vay, C.: Finite-dimensional pointed or Hopf algebras over affine racks. J. Algebra 397, 379–406 (2014)

    Article  MathSciNet  Google Scholar 

  16. Graña, M.: On Nichols algebras of low dimension. In: New trends in Hopf algebra theory (La Falda, 1999), volume 267 of Contemp. Math., pp. 111–134. Amer. Math. Soc., Providence, RI (2000)

  17. Heckenberger, I., Vendramin, L.: Nichols algebras over groups with finite root system of rank two II. J. Group Theory 17(6), 1009–1034 (2014)

    Article  MathSciNet  Google Scholar 

  18. Lentner, S.: Quantum groups and Nichols algebras acting on conformal field theories. arXiv:1702.06431 (2017)

  19. Liu, R.I.: On the commutative quotient of Fomin-Kirillov algebras. European J. Combin. 54, 65–75 (2016)

    Article  MathSciNet  Google Scholar 

  20. Majid, S.: Noncommutative differentials and Yang-Mills on permutation groups S n. In: Hopf algebras in noncommutative geometry and physics, volume 239 of Lecture Notes in Pure and Appl. Math., pp. 189–213. Dekker, New York (2005)

  21. Majid, S., Raineri, E.: Electromagnetism and gauge theory on the permutation group S 3,. J. Geom. Phys. 44(2-3), 129–155 (2002)

    Article  MathSciNet  Google Scholar 

  22. Mészáros, K., Panova, G., Postnikov, A.: Schur Schubert via the Fomin-Kirillov algebra. Electron J. Combin. 21(1), Paper 1.39, 22 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Milinski, A., Schneider, H.-J.: Pointed indecomposable Hopf algebras over Coxeter groups. In: New trends in Hopf algebra theory (La Falda, 1999), volume 267 of Contemp. Math., pp. 215–236. Amer. Math. Soc., Providence, RI (2000)

  24. Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theoret. Comput. Sci. 134(1), 131–173 (1994). Second International Colloquium on Words, Languages and Combinatorics, Kyoto (1992)

    Article  MathSciNet  Google Scholar 

  25. Ngakeu, F., Majid, S., Lambert, D.: Noncommutative Riemannian geometry of the alternating group A 4. J. Geom. Phys. 42(3), 259–282 (2002)

    Article  MathSciNet  Google Scholar 

  26. Polishchuk, A., Positselski, L.: Quadratic algebras, volume 37 of University Lecture Series. American Mathematical Society, Providence, RI (2005)

  27. Roehrig, B.: Deformed Fomin-Kirillov algebras and applications. Phd Thesis, Philipps-Universität, Marburg (2016). https://doi.org/10.17192/z2017.0049

    Google Scholar 

  28. Scharlau, W.: Quadratic and Hermitian forms, volume 270 of Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1985)

    Google Scholar 

  29. Shepler, A.V., Witherspoon, S.: PBW deformations of skew group algebras in positive characteristic. Algebr. Represent. Theory 18(1), 257–280 (2015)

    Article  MathSciNet  Google Scholar 

  30. Walton, C., Witherspoon, S.: PBW deformations of braided products. J Algebra 504, 536–567 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank Pavel Etingof for pointing out the references [10, 11].

Author information

Authors and Affiliations

  1. Philipps-Universität Marburg, FB Mathematik und Informatik, Hans-Meerwein-Straße, 35032, Marburg, Germany

    I. Heckenberger

  2. IMAS–CONICET and Depto. de Matemática, FCEN, Universidad de Buenos Aires, Pabellón 1, Ciudad Universitaria (1428), Buenos Aires, Argentina

    L. Vendramin

Authors
  1. I. Heckenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Vendramin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Vendramin.

Additional information

Presented by: Sarah Witherspoon

The second-named author is partially supported by PICT-2014-1376, MATH-AmSud 17MATH-01, ICTP, ERC advanced grant 320974 and the Alexander von Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heckenberger, I., Vendramin, L. PBW Deformations of a Fomin–Kirillov Algebra and Other Examples. Algebr Represent Theor 22, 1513–1532 (2019). https://doi.org/10.1007/s10468-018-9830-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-018-9830-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation