## Duke Mathematical Journal

### Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors

Catharina Stroppel

#### Abstract

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.

#### Article information

Source
Duke Math. J., Volume 126, Number 3 (2005), 547-596.

Dates
First available in Project Euclid: 11 February 2005

https://projecteuclid.org/euclid.dmj/1108155761

Digital Object Identifier
doi:10.1215/S0012-7094-04-12634-X

Mathematical Reviews number (MathSciNet)
MR2120117

Zentralblatt MATH identifier
1112.17010

#### Citation

Stroppel, Catharina. Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126 (2005), no. 3, 547--596. doi:10.1215/S0012-7094-04-12634-X. https://projecteuclid.org/euclid.dmj/1108155761

#### References

• \lccL. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5, (1996), 569-587.
• \lccH. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of Quantum Groups at a $p$th Root of Unity and of Semisimple Groups in Characteristic $p$: Independence of $p$, Astérisque 220, Soc. Math. France, Montrouge, 1994.
• \lccH. H. Andersen and C. Stroppel, Twisting functors on $\mathcal{O}_0$, Represent. Theory 7 (2003), 681-699.
• \lccE. Backelin, Koszul duality for parabolic and singular category $\mathcal{O}$, Represent. Theory 3 (1999), 139-152.
• ––––, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267-283.
• \lccH. Bass, Algebraic $K$-Theory, Benjamin, New York, 1968.
• \lccA. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527.
• \lccJ. Bernstein, I. Frenkel, and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), 199-241.
• \lccJ. N. Bernstein and S. I. Gel$'$fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245-285.
• \lccI. N. Bernstein, I. M. Gel$'$fand, and S. I. Gel$'$fand, A certain category of ${\mathfrak g}$-modules (in Russian), Funkcional. Anal. i Priložen. 10, no. 2 (1976), 1-8.
• \lccS. C. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin. 13 (2001), 111-136.
• \lccT. Brzeziński and J. Katriel, Representation-theoretic derivation of the Temperley-Lieb-Martin algebras, J. Phys. A 28 (1995), 5305-5312.
• \lccV. V. Deodhar, On some geometric aspects of Bruhat orderings, II: The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), 483-506.
• \lccT. tom Dieck, Temperley-Lieb algebras associated to the root system $D$, Arch. Math. (Basel) 71 (1998), 407-416.
• \lccT. J. Enright and B. Shelton, Categories of highest weight modules: Applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 67 (1987), no. 367.
• \lccI. B. Frenkel and M. G. Khovanov, Canonical bases in tensor products and graphical calculus for $U_q({\mathfrak{sl}_2})$, Duke Math. J. 87 (1997), 409-480.
• \lccR. M. Green, Generalized Temperley-Lieb algebras and decorated tangles, J. Knot Theory Ramifications 7 (1998), 155-171.
• \lccM. Härterich, Murphy bases of generalized Temperley-Lieb algebras, Arch. Math. (Basel) 72 (1999), 337-345.
• \lccR. Irving, Projective modules in the category $\mathcal{O}_{S}$: Self-duality, Trans. Amer. Math. Soc. 291 (1985), 701-732.
• ––––, Shuffled Verma modules and principal series modules over complex semisimple Lie algebras, J. London Math. Soc. (2) 48 (1993), 263-277.
• \lccJ. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math. 750, Springer, Berlin, 1979.
• ––––, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergeb. Math. Grenzgeb. (3) 3, Springer, Berlin, 1983.
• \lccL. H. Kauffman, Knots and Physics, 3rd ed., Ser. Knots Everything 1, World Sci., River Edge, N.J., 2001.
• \lccD. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.
• \lccM. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359-426.
• ––––, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665-741.
• \lccT. Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer, New York, 1991.
• \lccJ. Lipp, Vergleich von Basen von Hecke-Algebren vom Typ A, Diplomarbeit, Universität Stuttgart, Stuttgart, Germany, 1994.
• \lccP. Martin, “On Schur-Weyl duality, $A\sb n$ Hecke algebras and quantum ${\rm sl}(N)$ on $\bigotimes\sp {n+1}{\bf C}\sp N$” in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., World Sci., River Edge, N.J., 1992, 645-673.
• \lccV. Mazorchuk and C. Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, to appear in Trans. Amer. Math. Soc., http://www.ams.org/tran/0000-000-00/S0002-9947-04-03650-5/home.html; PII: S 0002-9947(04)03650-5
• \lccJ. Rickard, “Translation functors and equivalences of derived categories for blocks of algebraic groups” in Finite-Dimensional Algebras and Related Topics (Ottawa, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 424, Kluwer, Dordrecht, Netherlands, 1994, 255-264.
• \lccA. Rocha-Caridi, Splitting criteria for ${\mathfrak g}$-modules induced from a parabolic and the Ber\v nsteĭ n-Gel$'$fand-Gel$'$fand resolution of a finite-dimensional, irreducible ${\mathfrak g}$-module, Trans. Amer. Math. Soc. 262 (1980), 335-366.
• \lccS. Ryom-Hansen, Koszul duality of translation and Zuckerman functors, J. Lie Theory 14 (2004), 151-163.
• \lccW. Soergel, Kategorie $\mathcal{O}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445.
• ––––, Kazhdan-Lusztig polynomials and a combinatorics for tilting modules, Represent. Theory 1 (1997), 83-114.
• ––––, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (2000), 311-335.
• \lccC. Stroppel, Category $\mathcal{O}$: Gradings and translation functors, J. Algebra 268 (2003), 301-326.
• \lccH. N. V. Temperley and E. H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251-280.
• \lccV. G. Turaev, Quantum Invariants of Knots and $3$-Manifolds, de Gruyter Stud. Math. 18, de Gruyter, Berlin, 1994.
• \lccB. W. Westbury, The representation theory of the Temperley-Lieb algebras, Math. Z. 219 (1995), 539-565.