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).
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References
Abramsky, S.: Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics. arxiv: 0910.2737
Andersen, H.H.: Simple modules for Temperley-Lieb algebras and related algebras. J. Algebra 520, 276–308 (2019)
Batchelor, M.T., Kuniba, A.: Temperley-Lieb lattice models arising from quantum group. J. Phys. A: Math. Gen. 24, 2599–2614 (1991)
Benkarta, G., Halverson, T.: Motzkin algebras. Eur. J. Comb. 36, 473–502 (2014)
Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
Dieck, T.: Bridges with pillars: a graphical calculus of knot algebra. Topology Appl. 78, 21–38 (1997)
Ehrig, M., Stroppel, C.: 2-row Springer fibres and Khovanov diagram algebras for Type D. Can. J. Math. 68, 1285–1333 (2016)
Enyang, J.: Representations of Temperley–Lieb algebras, arXiv:0710.3218
Fan, C.K.: A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. Thesis, MIT (1995)
Fan, C.K.: Structure of a Hecke algebra quotient. J. Amer. Math. Soc. 10, 139–167 (1997)
Geck, M.: Hecke algebras of finite type are cellular. Invent. Math. 169, 501–517 (2007)
Goodman, F., Wenzl, H.: The Temperley-Lieb algebra at roots of unity. Pac. J. Math. 161, 307–334 (1993)
Graham, J.J.: Modular representations of Hecke algebras and related algebras, Ph.D. Thesis, University of Sydney (1995)
Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 34, 1–34 (1996)
Green, R.M.: Generalized Temperley-Lieb algebras and decorated tangles. J. Knot Theory Ram. 7, 155–171 (1998)
Grossman, P.: Forked Temperley-Lieb algebras and intermediate subfactors. J. Funct. Anal. 247, 477–491 (2007)
Hu, J.: A Morita equivalence theorem for Hecke algebra hq(dn) when n is even. Manuscripta Math. 108, 409–430 (2002)
Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull Amer. Math. Soc. 12, 103–111 (1985)
Jones, V.F.R.: Subfactors and knots, C.B.M.S. 80 Amer. Math. Soc Providence RI (1991)
Kauffman, L.: An invariant of regular isotopy. Trans. Amer. Math. Soc. 318, 417–471 (1990)
Lejczyk, T., Stroppel, C.: A graphical description of (dn,An− 1) Kazhdan-Lusztig polynomials. Glasg. Math. J. 55, 313–340 (2013)
Losonczy, J.: The Kazhdan-Lusztig basis and the Temperley-Lieb quotient in type D. J. Algebra 233, 1–15 (2000)
Martin, P.P.: Temperley-lieb algebras for non-planar statistical mechanics-The partition algebra construction. J. Knot Theory Ram. 3, 51–82 (1994)
Martin, P.P.: Potts models and related problems in statistical mechanics. World Scientific, Singapore (1991)
Pallikaros, C.: Representations of Hecke algebras of type dn. J. Algebra 169, 20–48 (1994)
Ridout, D., Saint-Aubin, Y.: Standard modules, induction and the structure of the Temperley-Lieb algebra. Adv. Theor. Math. Phys. 18, 957–1041 (2012)
Rui, H.B., Xi, C.C.: The representation theory of cyclotomic Temperley-Lieb algebra. Comment. Math. Helv. 79, 427–450 (2004)
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)
Stroppel, C.: Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126, 547–596 (2005)
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)
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)
Wang, P.: The Grothendieck group of a tower of the Temperley-Lieb algebras. J. Algeba. App. 18, 1950136 (2019)
Westbury, B.W.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219, 539–565 (1995)
Xi, C.C.: Partition algebras are cellular. Compos. Math. 119, 99–109 (1999)
Xi, C.C.: On the quasi-heredity of Birman-Wenzl algebras. Adv. Math. 154, 280–298 (2000)
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.
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Presented by: Alistair Savage
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Li is supported by the Natural Science Foundation of Hebei Province, China (A2017501003) and NSFC 11871107.
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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
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DOI: https://doi.org/10.1007/s10468-021-10062-w