Webs and $q$-Howe dualities in types $\mathbf {BCD}$
HTML articles powered by AMS MathViewer
- by Antonio Sartori and Daniel Tubbenhauer PDF
- Trans. Amer. Math. Soc. 371 (2019), 7387-7431 Request permission
Abstract:
We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category and allow us to prove quantum versions of some classical type $\mathbf {BCD}$ Howe dualities.References
- H.H. Andersen, C. Stroppel and D. Tubbenhauer, Semisimplicity of Hecke and (walled) Brauer algebras, J. Aust. Math. Soc. 103 (2017), no. 1, 1–44, doi:10.1017/S1446788716000392, arXiv:1507.07676.
- D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499, doi:10.2140/gt.2005.9.1443, arXiv:math/0410495.
- C. Blanchet, An oriented model for Khovanov homology, J. Knot Theory Ramifications 19 (2010), no. 2, 291–312, doi:10.1142/S0218216510007863, arXiv:1405.7246.
- R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857–872, doi:10.2307/1968843.
- Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273. MR 992598, DOI 10.1090/S0002-9947-1989-0992598-X
- A. Berenstein and S. Zwicknagl, Braided symmetric and exterior algebras, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3429–3472, doi:10.1090/S0002-9947-08-04373-0, arXiv:math/0504155.
- S. Cautis, J. Kamnitzer and S. Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014), no. 1-2, 351–390, doi:10.1007/s00208-013-0984-4, arXiv:1210.6437.
- Shun-Jen Cheng and Weiqiang Wang, Dualities and representations of Lie superalgebras, Graduate Studies in Mathematics, vol. 144, American Mathematical Society, Providence, RI, 2012. MR 3012224, DOI 10.1090/gsm/144
- S.-J. Cheng and R. Zhang, Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras, Adv. Math. 182 (2004), no. 1, 124–172, doi:10.1016/S0001-8708(03)00076-8, arXiv:math/0206036.
- Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, DOI 10.1016/j.aim.2018.01.013
- M. Ehrig, C. Stroppel and D. Tubbenhauer, The Blanchet–Khovanov algebras, Categorification and higher representation theory, 183–226, Contemp. Math., 683, Amer. Math. Soc., Providence, RI, 2017, doi:10.1090/conm/683, arXiv:1510.04884.
- M. Ehrig, C. Stroppel and D. Tubbenhauer, Generic $\mathfrak {gl}_2$-foams, web and arc algebras, arXiv:1601.08010, (2016).
- M. Ehrig, D. Tubbenhauer, and A. Wilbert, Singular TQFTs, foams and type D arc algebras, arXiv:1611.07444, (2016).
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. MR 3242743, DOI 10.1090/surv/205
- Cheryl Grood, Brauer algebras and centralizer algebras for $\textrm {SO}(2n,\textbf {C})$, J. Algebra 222 (1999), no. 2, 678–707. MR 1734225, DOI 10.1006/jabr.1999.8069
- Reinhard Häring-Oldenburg, Actions of tensor categories, cylinder braids and their Kauffman polynomial, Topology Appl. 112 (2001), no. 3, 297–314. MR 1824165, DOI 10.1016/S0166-8641(00)00006-7
- Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
- Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
- Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532, DOI 10.1090/gsm/006
- Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
- M. Khovanov, $\mathfrak {sl}(3)$ link homology, Algebr. Geom. Topol. 4 (2004), 1045–1081, doi:10.2140/agt.2004.4.1045, arXiv:math/0304375.
- S. Kolb and J. Pellegrini, Braid group actions on coideal subalgebras of quantized enveloping algebras, J. Algebra 336 (2011), 395–416, doi:10.1016/j.jalgebra.2011.04.001, arXiv:1102.4185.
- Greg Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109–151. MR 1403861
- Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999), no. 2, 729–767. MR 1717368, DOI 10.1006/jabr.1999.8015
- G. Letzter, Cartan subalgebras for quantum symmetric pair coideals, arXiv:1705.05958, (2017).
- A.D. Lauda, H. Queffelec, and D.E.V. Rose, Khovanov homology is a skew Howe $2$-representation of categorified quantum $\mathfrak {sl}(m)$, Algebr. Geom. Topol. 15 (2015), no. 5, 2517–2608, doi:10.2140/agt.2015.15.2517, arXiv:1212.6076.
- George Lusztig, Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010. Reprint of the 1994 edition. MR 2759715, DOI 10.1007/978-0-8176-4717-9
- G. I. Lehrer and R. B. Zhang, Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra 306 (2006), no. 1, 138–174. MR 2271576, DOI 10.1016/j.jalgebra.2006.03.043
- G.I. Lehrer and R.B. Zhang, Invariants of the special orthogonal group and an enhanced Brauer category, Enseign. Math. 63 (2017), no. 1, 181–200, doi:10.4171/LEM/63-1/2-6 arXiv:1612.03998.
- G.I. Lehrer, H. Zhang, and R.B. Zhang, A quantum analogue of the first fundamental theorem of classical invariant theory, Comm. Math. Phys. 301 (2011), no. 1, 131–174, doi:10.1007/s00220-010-1143-3, arXiv:0908.1425.
- A.I. Molev, A new quantum analogue of the Brauer algebra, Czechoslovak J. Phys. 53 (2003), no. 11, 1073–1078, Quantum groups and integrable systems, doi:10.1023/B:CJOP.0000010536.64174.8e, arXiv:math/0211082.
- Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059
- H. Queffelec and A. Sartori, Mixed quantum skew Howe duality and link invariants of type A, arXiv:1504.01225, (2015).
- D.E.V. Rose and D. Tubbenhauer, Symmetric webs, Jones–Wenzl recursions, and $q$-Howe duality, Int. Math. Res. Not. IMRN (2016), no. 17, 5249–5290, doi:10.1093/imrn/rnv302, arXiv:1501.00915.
- G. Rumer, E. Teller and H. Weyl, Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten, Nachrichten von der Ges. der Wiss. Zu Göttingen. Math.-Phys. Klasse (1932), 498–504. (In German.)
- A. Sartori, Categorification of tensor powers of the vector representation of $U_q(\mathfrak {gl}(1\vert 1))$, Selecta Math. (N.S.) 22 (2016), no. 2, 669–734, doi:10.1007/s00029-015-0202-1, arXiv:1305.6162.
- A. Sartori, A diagram algebra for Soergel modules corresponding to smooth Schubert varieties, Trans. Amer. Math. Soc. 368 (2016), no. 2, 889–938, doi:10.1090/tran/6346, arXiv:1311.6968.
- H. 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), no. 1549, 251–280. MR 498284, DOI 10.1098/rspa.1971.0067
- D. Tubbenhauer, P. Vaz and P. Wedrich, Super $q$-Howe duality and web categories, Algebr. Geom. Topol. 17 (2017), no. 6, 3703–3749, doi:10.2140/agt.2017.17.3703, arXiv:1504.05069.
- Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/gsm/145
- Hans Wenzl, A $q$-Brauer algebra, J. Algebra 358 (2012), 102–127. MR 2905021, DOI 10.1016/j.jalgebra.2012.02.017
- S. Zwicknagl, $R$-matrix Poisson algebras and their deformations, Adv. Math. 220 (2009), no. 1, 1–58, doi:10.1016/j.aim.2008.08.006, arXiv:0706.0351.
Additional Information
- Antonio Sartori
- Affiliation: Mathematisches Institut, Albert–Ludwigs-Universität Freiburg, Eckerstra${\ss }$e 1, 79104 Freiburg im Breisgau, Germany
- Email: anton.sartori@gmail.com
- Daniel Tubbenhauer
- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, Switzerland
- MR Author ID: 1067860
- ORCID: 0000-0001-7265-5047
- Email: daniel.tubbenhauer@math.uzh.ch
- Received by editor(s): January 12, 2017
- Received by editor(s) in revised form: January 10, 2018, and March 21, 2018
- Published electronically: October 22, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7387-7431
- MSC (2010): Primary 17B37, 17B10, 17B20; Secondary 20G42, 81R50
- DOI: https://doi.org/10.1090/tran/7583
- MathSciNet review: 3939581