Skip to main content
SpringerLink
Log in

Semi-simplicity of Temperley-Lieb Algebras of type D

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

Abstract

We provide a necessary and sufficient condition for a type D Temperley-Lieb algebra TLDn(δ) being semi-simple by studying branching rule for cell modules. As a byproduct, our result is used to study the so-called forked Temperley-Lieb algebra, which is a quotient algebra of TLDn(δ).

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. Abramsky, S.: Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics. arxiv: 0910.2737

  2. Andersen, H.H.: Simple modules for Temperley-Lieb algebras and related algebras. J. Algebra 520, 276–308 (2019)

    Article  MathSciNet  Google Scholar 

  3. Batchelor, M.T., Kuniba, A.: Temperley-Lieb lattice models arising from quantum group. J. Phys. A: Math. Gen. 24, 2599–2614 (1991)

    Article  MathSciNet  Google Scholar 

  4. Benkarta, G., Halverson, T.: Motzkin algebras. Eur. J. Comb. 36, 473–502 (2014)

    Article  MathSciNet  Google Scholar 

  5. Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)

    MathSciNet  MATH  Google Scholar 

  6. Dieck, T.: Bridges with pillars: a graphical calculus of knot algebra. Topology Appl. 78, 21–38 (1997)

    Article  MathSciNet  Google Scholar 

  7. Ehrig, M., Stroppel, C.: 2-row Springer fibres and Khovanov diagram algebras for Type D. Can. J. Math. 68, 1285–1333 (2016)

    Article  MathSciNet  Google Scholar 

  8. Enyang, J.: Representations of Temperley–Lieb algebras, arXiv:0710.3218

  9. Fan, C.K.: A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. Thesis, MIT (1995)

  10. Fan, C.K.: Structure of a Hecke algebra quotient. J. Amer. Math. Soc. 10, 139–167 (1997)

    Article  MathSciNet  Google Scholar 

  11. Geck, M.: Hecke algebras of finite type are cellular. Invent. Math. 169, 501–517 (2007)

    Article  MathSciNet  Google Scholar 

  12. Goodman, F., Wenzl, H.: The Temperley-Lieb algebra at roots of unity. Pac. J. Math. 161, 307–334 (1993)

    Article  MathSciNet  Google Scholar 

  13. Graham, J.J.: Modular representations of Hecke algebras and related algebras, Ph.D. Thesis, University of Sydney (1995)

  14. Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 34, 1–34 (1996)

    Article  MathSciNet  Google Scholar 

  15. Green, R.M.: Generalized Temperley-Lieb algebras and decorated tangles. J. Knot Theory Ram. 7, 155–171 (1998)

    Article  MathSciNet  Google Scholar 

  16. Grossman, P.: Forked Temperley-Lieb algebras and intermediate subfactors. J. Funct. Anal. 247, 477–491 (2007)

    Article  MathSciNet  Google Scholar 

  17. Hu, J.: A Morita equivalence theorem for Hecke algebra hq(dn) when n is even. Manuscripta Math. 108, 409–430 (2002)

    Article  MathSciNet  Google Scholar 

  18. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  MathSciNet  Google Scholar 

  19. Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull Amer. Math. Soc. 12, 103–111 (1985)

    Article  MathSciNet  Google Scholar 

  20. Jones, V.F.R.: Subfactors and knots, C.B.M.S. 80 Amer. Math. Soc Providence RI (1991)

  21. Kauffman, L.: An invariant of regular isotopy. Trans. Amer. Math. Soc. 318, 417–471 (1990)

    Article  MathSciNet  Google Scholar 

  22. Lejczyk, T., Stroppel, C.: A graphical description of (dn,An− 1) Kazhdan-Lusztig polynomials. Glasg. Math. J. 55, 313–340 (2013)

    Article  MathSciNet  Google Scholar 

  23. Losonczy, J.: The Kazhdan-Lusztig basis and the Temperley-Lieb quotient in type D. J. Algebra 233, 1–15 (2000)

    Article  MathSciNet  Google Scholar 

  24. Martin, P.P.: Temperley-lieb algebras for non-planar statistical mechanics-The partition algebra construction. J. Knot Theory Ram. 3, 51–82 (1994)

    Article  Google Scholar 

  25. Martin, P.P.: Potts models and related problems in statistical mechanics. World Scientific, Singapore (1991)

    Book  Google Scholar 

  26. Pallikaros, C.: Representations of Hecke algebras of type dn. J. Algebra 169, 20–48 (1994)

    Article  MathSciNet  Google Scholar 

  27. Ridout, D., Saint-Aubin, Y.: Standard modules, induction and the structure of the Temperley-Lieb algebra. Adv. Theor. Math. Phys. 18, 957–1041 (2012)

    Article  MathSciNet  Google Scholar 

  28. Rui, H.B., Xi, C.C.: The representation theory of cyclotomic Temperley-Lieb algebra. Comment. Math. Helv. 79, 427–450 (2004)

    Article  MathSciNet  Google Scholar 

  29. Rui, H.B., Xi, C.C., Yu, W.H.: On the semi-simplicity of the cyclotomic Temperley-Lieb algebras. Michigan Math. J. 53, 83–96 (2005)

    Article  MathSciNet  Google Scholar 

  30. Stroppel, C.: Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126, 547–596 (2005)

    Article  MathSciNet  Google Scholar 

  31. Stroppel, C., Wilbert, A.: Two-block Springer fibers of types C and d: a diagrammatic approach to Springer theory. Math. Z. 292, 1387–1430 (2019)

    Article  MathSciNet  Google Scholar 

  32. Temperley, H., Lieb, E.: Relations between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proc. Roy. Soc. London 322, 251–273 (1971)

    MathSciNet  MATH  Google Scholar 

  33. Wang, P.: The Grothendieck group of a tower of the Temperley-Lieb algebras. J. Algeba. App. 18, 1950136 (2019)

    Article  MathSciNet  Google Scholar 

  34. Westbury, B.W.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219, 539–565 (1995)

    Article  MathSciNet  Google Scholar 

  35. Xi, C.C.: Partition algebras are cellular. Compos. Math. 119, 99–109 (1999)

    Article  MathSciNet  Google Scholar 

  36. Xi, C.C.: On the quasi-heredity of Birman-Wenzl algebras. Adv. Math. 154, 280–298 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referee for her/his numerous helpful comments and corrections. The authors are grateful to Professor Shoumin Liu and Doctor Pei Wang for some discussions about this topic. Part of this work was done when Li visited School of Mathematics Sciences at Hebei Normal University in the summer of 2019. He takes this opportunity to express his sincere thanks to the School of Mathematics Sciences for the hospitality during his visit.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, People’s Republic of China

    Yanbo Li

  2. Department of mathematics, School of Science, Northeastern University, Shenyang, 110819, People’s Republic of China

    Xiaolin Shi

Authors
  1. Yanbo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaolin Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaolin Shi.

Additional information

Presented by: Alistair Savage

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Li is supported by the Natural Science Foundation of Hebei Province, China (A2017501003) and NSFC 11871107.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Shi, X. Semi-simplicity of Temperley-Lieb Algebras of type D. Algebr Represent Theor 25, 1133–1158 (2022). https://doi.org/10.1007/s10468-021-10062-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-021-10062-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation