Canonical bases arising from quantum symmetric pairs. (English) Zbl 1416.17001
Authors’ abstract: We develop a general theory of canonical bases for quantum symmetric pairs \((\mathbf{U}, \mathbf{U}^l )\) with parameters of arbitrary finite type. We construct new canonical bases for the finite-dimensional simple \(\mathbf{U}\)-modules and their tensor products regarded as \(\mathbf{U}^l\) -modules. We also construct a canonical basis for the modified form of the \(\iota\)-quantum group \(\mathbf{U}^l\). To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.
Reviewer: Sorin Dascalescu (Bucureşti)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
References:
| [1] | Araki, S, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ., 13, 1-34, (1962) · Zbl 0123.03002 |
| [2] | Bao, H, Kazhdan-Lusztig theory of super type \(D\) and quantum symmetric pairs, Represent. Theory, 21, 247-276, (2017) · Zbl 1411.17015 · doi:10.1090/ert/505 |
| [3] | Beilinson, A; Lusztig, G; MacPherson, R, A geometric setting for the quantum deformation of \(GL_n\), Duke Math. J., 61, 655-677, (1990) · Zbl 0713.17012 · doi:10.1215/S0012-7094-90-06124-1 |
| [4] | Balagovic, M; Kolb, S, The bar involution for quantum symmetric pairs, Represent. Theory, 19, 186-210, (2015) · Zbl 1376.17018 · doi:10.1090/ert/469 |
| [5] | Balagovic, M; Kolb, S, Universal \(K\)-matrix for quantum symmetric pairs, J. Reine Angew. Math., (2016) · Zbl 1376.17018 · doi:10.1515/crelle-2016-0012 |
| [6] | Bao, H., Kujawa, J., Li, Y., Wang, W.: Geometric Schur duality of classical type. Transform. Groups. arXiv:1404.4000v3(to appear) · Zbl 1440.17009 |
| [7] | Bao, H., Wang, W.: A new approach to Kazhdan-Lusztig theory of type \(B\) via quantum symmetric pairs, Astérisque. arXiv:1310.0103(to appear) · Zbl 1411.17001 |
| [8] | Bao, H; Wang, W, Canonical bases in tensor products revisited, Am. J. Math., 138, 1731-1738, (2016) · Zbl 1418.17031 · doi:10.1353/ajm.2016.0051 |
| [9] | Ehrig, M., Stroppel, C.: Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality. arXiv:1310.1972 · Zbl 1432.16022 |
| [10] | Fan, Z., Li, Y.: Positivity of canonical basis under comultiplication. arXiv:1511.02434 · Zbl 1127.17018 |
| [11] | Jantzen, J.C.: Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6. American Mathematical Society, Providence (1996) · Zbl 0842.17012 |
| [12] | Kashiwara, M, On crystal bases of the \(Q\)-analogue of universal enveloping algebras, Duke Math. J., 63, 456-516, (1991) · Zbl 0739.17005 · doi:10.1215/S0012-7094-91-06321-0 |
| [13] | Kashiwara, M, Crystal bases of modified quantized enveloping algebra, Duke Math. J., 73, 383-413, (1994) · Zbl 0794.17009 · doi:10.1215/S0012-7094-94-07317-1 |
| [14] | Kolb, S, Quantum symmetric Kac-Moody pairs, Adv. Math., 267, 395-469, (2014) · Zbl 1300.17011 · doi:10.1016/j.aim.2014.08.010 |
| [15] | Kolb, S; Pellegrini, J, Braid group actions on coideal subalgebras of quantized enveloping algebras, J. Algebra, 336, 395-416, (2011) · Zbl 1266.17011 · doi:10.1016/j.jalgebra.2011.04.001 |
| [16] | Letzter, G, Symmetric pairs for quantized enveloping algebras, J. Algebra, 220, 729-767, (1999) · Zbl 0956.17007 · doi:10.1006/jabr.1999.8015 |
| [17] | Letzter, G.: Coideal subalgebras and quantum symmetric pairs. In: Montgomery, S., Schneider, H.-J. (eds.) New directions in Hopf algebras (Cambridge), vol. 43, pp. 117-166. MSRI Publications, Cambridge University Press, Cambridge (2002) · Zbl 1025.17005 |
| [18] | Letzter, G, Quantum symmetric pairs and their zonal spherical functions, Transform. Groups, 8, 261-292, (2003) · Zbl 1107.17010 · doi:10.1007/s00031-003-0719-9 |
| [19] | Letzter, G, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math., 189, 88-147, (2004) · Zbl 1127.17018 · doi:10.1016/j.aim.2003.11.007 |
| [20] | Lusztig, G, Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., 3, 447-498, (1990) · Zbl 0703.17008 · doi:10.1090/S0894-0347-1990-1035415-6 |
| [21] | Lusztig, G, Quivers, perverse sheaves, and quantized enveloping algebras, J. Am. Math. Soc., 4, 365-421, (1991) · Zbl 0738.17011 · doi:10.1090/S0894-0347-1991-1088333-2 |
| [22] | Lusztig, G, Canonical bases in tensor products, Proc. Natl. Acad. Sci., 89, 8177-8179, (1992) · Zbl 0760.17011 · doi:10.1073/pnas.89.17.8177 |
| [23] | Lusztig, G.: Introduction to Quantum Groups, Modern Birkhäuser Classics, Reprint of the 1993 Edition, Birkhäuser, Boston (2010) · Zbl 1376.17018 |
| [24] | Li, Y., Wang, W.: Positivity vs negativity of canonical bases. In: Proceedings for Lusztig’s 70th Birthday Conference, Bulletin of Institute of Mathematics Academia Sinica (N.S.), vol. 13, pp. 143-198 (2018). arXiv:1501.00688 · Zbl 0713.17012 |
| [25] | Onishchik, A., Vinberg, E.: Lie groups and algebraic groups. Translated from the Russian by D.A. Leites. Springer Series in Soviet Mathematics. Springer, Berlin (1990) · Zbl 0722.22004 |
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.
