Webs and \(q\)-Howe dualities in types BCD. (English) Zbl 1472.17061
Summary: 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 BCD Howe dualities.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
Keywords:
diagrammatic presentation; webs; (quantum) Brauer algebras; (quantum) Howe duality; classical Lie type; coideal subalgebrasReferences:
| [1] | 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, http://dx.doi.org/10.1017/S1446788716000392doi:10.1017/S1446788716000392, http://arxiv.org/abs/1507.07676arXiv:1507.07676. · Zbl 1403.17016 |
| [2] | D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499, https://doi.org/10.2140/gt.2005.9.1443doi:10.2140/gt.2005.9.1443, https://arxiv.org/abs/math/0410495arXiv:math/0410495. · Zbl 1084.57011 |
| [3] | C. Blanchet, An oriented model for Khovanov homology, J. Knot Theory Ramifications 19 (2010), no. 2, 291-312, http://dx.doi.org/10.1142/S0218216510007863doi:10.1142/S0218216510007863, http://arxiv.org/abs/1405.7246arXiv:1405.7246. · Zbl 1195.57024 |
| [4] | R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857-872, http://dx.doi.org/10.2307/1968843doi:10.2307/1968843. · JFM 63.0873.02 |
| [5] | Birman, Joan S.; Wenzl, Hans, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc., 313, 1, 249-273 (1989) · Zbl 0684.57004 · doi:10.2307/2001074 |
| [6] | A. Berenstein and S. Zwicknagl, Braided symmetric and exterior algebras, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3429-3472, http://dx.doi.org/10.1090/S0002-9947-08-04373-0doi:10.1090/S0002-9947-08-04373-0, http://arxiv.org/abs/math/0504155arXiv:math/0504155. · Zbl 1220.17004 |
| [7] | S. Cautis, J. Kamnitzer and S. Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014), no. 1-2, 351-390, http://dx.doi.org/10.1007/s00208-013-0984-4doi:10.1007/s00208-013-0984-4, http://arxiv.org/abs/1210.6437arXiv:1210.6437. · Zbl 1387.17027 |
| [8] | Cheng, Shun-Jen; Wang, Weiqiang, Dualities and representations of Lie superalgebras, Graduate Studies in Mathematics 144, xviii+302 pp. (2012), American Mathematical Society, Providence, RI · Zbl 1271.17001 · doi:10.1090/gsm/144 |
| [9] | S.-J. Cheng and R. Zhang, Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras, Adv. Math. 182 (2004), no. 1, 124-172, http://dx.doi.org/10.1016/S0001-8708(03)00076-8doi:10.1016/S0001-8708(03)00076-8, http://arxiv.org/abs/math/0206036arXiv:math/0206036. · Zbl 1056.17004 |
| [10] | Ehrig, Michael; Stroppel, Catharina, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math., 331, 58-142 (2018) · Zbl 1432.16022 · doi:10.1016/j.aim.2018.01.013 |
| [11] | 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, https://doi.org/10.1090/conm/683doi:10.1090/conm/683, http://arxiv.org/abs/1510.04884arXiv:1510.04884. · Zbl 1419.57024 |
| [12] | M. Ehrig, C. Stroppel and D. Tubbenhauer, Generic \(\mathfrakgl_2\)-foams, web and arc algebras, http://arxiv.org/abs/1601.08010arXiv:1601.08010, (2016). |
| [13] | M. Ehrig, D. Tubbenhauer, and A. Wilbert, Singular TQFTs, foams and type D arc algebras, http://arxiv.org/abs/1611.07444arXiv:1611.07444, (2016). · Zbl 1473.57035 |
| [14] | Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205, xvi+343 pp. (2015), American Mathematical Society, Providence, RI · Zbl 1365.18001 · doi:10.1090/surv/205 |
| [15] | Grood, Cheryl, Brauer algebras and centralizer algebras for \({\rm SO}(2n,{\bf C})\), J. Algebra, 222, 2, 678-707 (1999) · Zbl 0959.20042 · doi:10.1006/jabr.1999.8069 |
| [16] | H\"aring-Oldenburg, Reinhard, Actions of tensor categories, cylinder braids and their Kauffman polynomial, Topology Appl., 112, 3, 297-314 (2001) · Zbl 0983.57005 · doi:10.1016/S0166-8641(00)00006-7 |
| [17] | Howe, Roger, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313, 2, 539-570 (1989) · Zbl 0674.15021 · doi:10.2307/2001418 |
| [18] | Howe, Roger, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc. 8, 1-182 (1995), Bar-Ilan Univ., Ramat Gan · Zbl 0844.20027 |
| [19] | Jantzen, Jens Carsten, Lectures on quantum groups, Graduate Studies in Mathematics 6, viii+266 pp. (1996), American Mathematical Society, Providence, RI · Zbl 0842.17012 |
| [20] | Kassel, Christian, Quantum groups, Graduate Texts in Mathematics 155, xii+531 pp. (1995), Springer-Verlag, New York · Zbl 0808.17003 · doi:10.1007/978-1-4612-0783-2 |
| [21] | M. Khovanov, \( \mathfraksl(3)\) link homology, Algebr. Geom. Topol. 4 (2004), 1045-1081, http://dx.doi.org/10.2140/agt.2004.4.1045doi:10.2140/agt.2004.4.1045, http://arxiv.org/abs/math/0304375arXiv:math/0304375. · Zbl 1159.57300 |
| [22] | S. Kolb and J. Pellegrini, Braid group actions on coideal subalgebras of quantized enveloping algebras, J. Algebra 336 (2011), 395-416, http://dx.doi.org/10.1016/j.jalgebra.2011.04.001doi:10.1016/j.jalgebra.2011.04.001, http://arxiv.org/abs/1102.4185arXiv:1102.4185. · Zbl 1266.17011 |
| [23] | Kuperberg, Greg, Spiders for rank \(2\) Lie algebras, Comm. Math. Phys., 180, 1, 109-151 (1996) · Zbl 0870.17005 |
| [24] | Letzter, Gail, Symmetric pairs for quantized enveloping algebras, J. Algebra, 220, 2, 729-767 (1999) · Zbl 0956.17007 · doi:10.1006/jabr.1999.8015 |
| [25] | G. Letzter, Cartan subalgebras for quantum symmetric pair coideals, https://arxiv.org/abs/1705.05958arXiv:1705.05958, (2017). · Zbl 1472.17057 |
| [26] | A.D. Lauda, H. Queffelec, and D.E.V. Rose, Khovanov homology is a skew Howe \(2\)-representation of categorified quantum \(sl(m)\), Algebr. Geom. Topol. 15 (2015), no. 5, 2517-2608, http://dx.doi.org/10.2140/agt.2015.15.2517doi:10.2140/agt.2015.15.2517, http://arxiv.org/abs/1212.6076arXiv:1212.6076. · Zbl 1330.81128 |
| [27] | Lusztig, George, Introduction to quantum groups, Modern Birkh\`“auser Classics, xiv+346 pp. (2010), Birkh\'”auser/Springer, New York · Zbl 1246.17018 · doi:10.1007/978-0-8176-4717-9 |
| [28] | Lehrer, G. I.; Zhang, R. B., Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra, 306, 1, 138-174 (2006) · Zbl 1169.17003 · doi:10.1016/j.jalgebra.2006.03.043 |
| [29] | 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, https://doi.org/10.4171/LEM/63-1/2-6doi:10.4171/LEM/63-1/2-6 http://arxiv.org/abs/1612.03998arXiv:1612.03998. · Zbl 1385.16040 |
| [30] | 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, http://dx.doi.org/10.1007/s00220-010-1143-3doi:10.1007/s00220-010-1143-3, http://arxiv.org/abs/0908.1425arXiv:0908.1425. · Zbl 1262.17008 |
| [31] | 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, http://dx.doi.org/10.1023/B:CJOP.0000010536.64174.8edoi:10.1023/B:CJOP.0000010536.64174.8e, http://arxiv.org/abs/math/0211082arXiv:math/0211082. |
| [32] | Murakami, Jun, The Kauffman polynomial of links and representation theory, Osaka J. Math., 24, 4, 745-758 (1987) · Zbl 0666.57006 |
| [33] | H. Queffelec and A. Sartori, Mixed quantum skew Howe duality and link invariants of type A, http://arxiv.org/abs/1504.01225arXiv:1504.01225, (2015). · Zbl 1469.57013 |
| [34] | D.E.V. Rose and D. Tubbenhauer, Symmetric webs, Jones-Wenzl recursions, and \(q-H\) owe duality, Int. Math. Res. Not. IMRN (2016), no. 17, 5249-5290, http://dx.doi.org/10.1093/imrn/rnv302doi:10.1093/imrn/rnv302, http://arxiv.org/abs/1501.00915arXiv:1501.00915. · Zbl 1404.17025 |
| [35] | G. Rumer, E. Teller and H. Weyl, Eine f\`“ur die Valenztheorie geeignete Basis der bin\'”aren Vektorinvarianten, Nachrichten von der Ges. der Wiss. Zu G\"ottingen. Math.-Phys. Klasse (1932), 498-504. (In German.) · Zbl 0006.14901 |
| [36] | A. Sartori, Categorification of tensor powers of the vector representation of \(U_q(\mathfrakgl(1\vert 1))\), Selecta Math. (N.S.) 22 (2016), no. 2, 669-734, http://dx.doi.org/10.1007/s00029-015-0202-1doi:10.1007/s00029-015-0202-1, http://arxiv.org/abs/1305.6162arXiv:1305.6162. · Zbl 1407.17012 |
| [37] | A. Sartori, A diagram algebra for Soergel modules corresponding to smooth Schubert varieties, Trans. Amer. Math. Soc. 368 (2016), no. 2, 889-938, http://dx.doi.org/10.1090/tran/6346doi:10.1090/tran/6346, http://arxiv.org/abs/1311.6968arXiv:1311.6968. · Zbl 1377.17011 |
| [38] | Temperley, H. N. V.; Lieb, E. H., 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, 1549, 251-280 (1971) · Zbl 0211.56703 · doi:10.1098/rspa.1971.0067 |
| [39] | D. Tubbenhauer, P. Vaz and P. Wedrich, Super \(q\)-Howe duality and web categories, Algebr. Geom. Topol. 17 (2017), no. 6, 3703-3749, https://doi.org/10.2140/agt.2017.17.3703doi:10.2140/agt.2017.17.3703, http://arxiv.org/abs/1504.05069arXiv:1504.05069. · Zbl 1422.57030 |
| [40] | Weibel, Charles A., The \(K\)-book, Graduate Studies in Mathematics 145, xii+618 pp. (2013), American Mathematical Society, Providence, RI · Zbl 1273.19001 |
| [41] | Wenzl, Hans, A \(q\)-Brauer algebra, J. Algebra, 358, 102-127 (2012) · Zbl 1288.20007 · doi:10.1016/j.jalgebra.2012.02.017 |
| [42] | S. Zwicknagl, \(R\)-matrix Poisson algebras and their deformations, Adv. Math. 220 (2009), no. 1, 1-58, http://dx.doi.org/10.1016/j.aim.2008.08.006doi:10.1016/j.aim.2008.08.006, http://arxiv.org/abs/0706.0351arXiv:0706.0351. · Zbl 1174.17019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
