Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory
  • Cited by 27
Publisher:
Cambridge University Press
Online publication date:
January 2013
Print publication year:
2012
Online ISBN:
9781139226660
DOI:
https://doi.org/10.1017/CBO9781139226660
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
London Mathematical Society Lecture Note Series (401)
Information

Book description

The theory of Schur–Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur–Weyl theory. To begin, various algebraic structures are discussed, including double Ringel–Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum Schur–Weyl duality on three levels. This includes the affine quantum Schur–Weyl reciprocity, the bridging role of affine q-Schur algebras between representations of the quantum loop algebras and those of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel–Hall algebra with a proof of the classical case. This text is ideal for researchers in algebra and graduate students who want to master Ringel–Hall algebras and Schur–Weyl duality.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
Bibliography
Bibliography
[1] M., Auslander, I., Reiten and S. O., Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, No. 36, Cambridge University Press, Cambridge, 1995.
[2] J., Beck, Braid group action and quantum affine algebras, Commun. Math. Phys. 165 (1994), 555–568.
[3] J., Beck, V., Chari and A., Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455–487.
[4] A. A., Beilinson, G., Lusztig and R., MacPherson, A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), 655–677.
[5] R., Borcherds, Generalized Kac–Moody algebras, J. Algebra 115 (1988), 501–512.
[6] V., Chari and A., Pressley, Quantum affine algebras, Commun. Math. Phys. 142 (1991), 261–283.
[7] V., Chari and A., Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994.
[8] V., Chari and A., Pressley, Quantum affine algebras and their representations, Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., 16, American Mathematical Society, Providence, RI, 1995, 59–78.
[9] V., Chari and A., Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), 295–326.
[10] V., Chari and A., Pressley, Quantum affine algebras at roots of unity, Represen. Theory 1 (1997), 280–328.
[11] B., Deng and J., DuMonomial bases for quantum affine sln, Adv. Math. 191 (2005), 276–304.
[12] B., Deng, J., Du, B., Parshall and J., Wang, Finite Dimensional Algebras and Quantum Groups, Mathematical Surveys and Monographs Volume 150, American Mathematical Society, Providence, RI, 2008.
[13] B., Deng, J., Du and J., Xiao, Generic extensions and canonical bases for cyclic quivers, Can. J. Math. 59 (2007), 1260–1283.
[14] B., Deng and J., Xiao, On double Ringel–Hall algebras, J. Algebra 251 (2002), 110–149.
[15] R., Dipper and G., James, The q-Schur algebra, Proc. London Math. Soc. 59 (1989), 23–50.
[16] R., Dipper and G., James, q-Tensor spaces and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282.
[17] S., Donkin, The q-Schur algebra, LMS Lecture Note Series 253, Cambridge University Press, Cambridge, 1998.
[18] S., Doty and A., Giaquinto, Presenting Schur algebras, Internat. Math. Res. Not. 36 (2002), 1907–1944.
[19] S., Doty and R. M., Green, Presenting affine q-Schur algebras, Math. Z. 256 (2007), 311–345.
[20] V. G., Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 32 (1988), 212–216.
[21] J., Du, Kahzdan–Lusztig bases and isomorphism theorems for q-Schur algebras, Contemp. Math. 139 (1992), 121–140.
[22] J., Du, A note on the quantized Weyl reciprocity at roots of unity, Alg. Colloq. 2 (1995), 363–372.
[23] J., Du and Q., Fu, Quantum gl∞, infinite q-Schur algebras and their representations, J. Algebra 322 (2009), 1516–1547.
[24] J., Du and Q., Fu, A modified BLM approach to quantum affine gln, Math. Z. 266 (2010), 747–781.
[25] J., Du, Q., Fu and J., Wang, Infinitesimal quantum gln and little q-Schur algebras, J. Algebra 287 (2005), 199–233.
[26] J., Du and B., Parshall, Monomial bases for q-Schur algebras, Trans. Amer. Math. Soc. 355 (2003), 1593–1620.
[27] J., Du, B., Parshall and L., Scott, Quantum Weyl reciprocity and tilting modules, Commun. Math. Phys. 195 (1998), 321–352.
[28] E., Frenkel and E., Mukhin, The Hopf algebra Rep Uq (gl∞), Sel. Math., New Ser. 8 (2002), 537–635.
[29] E., Frenkel and N., Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras, Contemp. Math., AMS 248 (1998), 163–205.
[30] Q., Fu, On Schur algebras and little Schur algebras, J. Algebra 322 (2009), 1637–1652.
[31] V., Ginzburg, N., Reshetikhin and E., Vasserot, Quantum groups and flag varieties, Contemp. Math. 175 (1994), 101–130.
[32] V., Ginzburg and E., Vasserot, Langlands reciprocity for affine quantum groups of type An, Internat. Math. Res. Not. 3 (1993), 67–85.
[33] J. A., Green, Polynomial Representations of GLn, 2nd edn, with an appendix on Schensted correspondence and Littelmann paths by K., Erdmann, J. A., Green and M., Schocker, Lecture Notes in Mathematics, No. 830, Springer-Verlag, Berlin, 2007.
[34] J. A., Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361–377.
[35] R. M., Green, The affine q-Schur algebra, J. Algebra 215 (1999), 379–411.
[36] J. Y., Guo, The Hall polynomials of a cyclic serial algebra, Commun. Algebra 23 (1995) 743–751.
[37] J. Y., Guo and L., Peng, Universal PBW-basis of Hall–Ringel algebras and Hall polynomials, J. Algebra 198 (1997) 339–351.
[38] J., Hua and J., Xiao, On Ringel–Hall algebras of tame hereditary algebras, Algebra Represent. Theory 5 (2002), 527–550.
[39] A., Hubery, Symmetric functions and the center of the Ringel–Hall algebra of a cyclic quiver, Math. Z. 251 (2005), 705–719.
[40] A., Hubery, Three presentations of the Hopf algebra Uv(gln), preprint, 2009 (available at http://www1.maths.leeds.ac.uk/~ahubery).
[41] N., Iwahori and H., Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48.
[42] K., Jeong, S.-J., Kang and M., Kashiwara, Crystal bases for quantum generalized Kac–Moody algebras, Proc. London Math. Soc. 90 (2005), 395–438.
[43] M., Jimbo, A q-analogue of U(gl(N + 1), Hecke algebra, and the Yang–Baxter equation, Lett. Math. Physics 11 (1986), 247–252.
[44] N., Jing, On Drinfeld realization of quantum affine algebras, in: The Monster and Lie algebras (Columbus, OH, 1996), Ohio State University Mathematical Research Institute Publications, 7, de Gruyter, Berlin, 1998, 195–206.
[45] V. F. R., Jones, A quotient of the affine Hecke algebra in the Brauer algebra, Enseign. Math. 40 (1994), 313–344.
[46] A., Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 29, Springer-Verlag, Berlin, 1995.
[47] V., Kac, Infinite Dimensional Lie Algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.
[48] S.-J., Kang, Quantum deformations of generalized Kac–Moody algebras and their modules, J. Algebra 175 (1995), 1041–1066.
[49] S.-J., Kang and O., Schiffmann, Canonical bases for quantum generalized Kac–Moody algebras, Adv. Math. 200 (2006), 445–478.
[50] D., Kazhdan and G., Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.
[51] G., Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623–653.
[52] G., Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296.
[53] G., Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365–421.
[54] G., Lusztig, Introduction to Quantum Groups, Progress in Math. 110, Birkhäuser,Boston, 1993.
[55] G., Lusztig, Canonical bases and Hall algebras, in Representation Theories and Algebraic Geometry, A., Braer and A., Daigneault (eds.), Kluwer, Dordrecht, 1998, 365–399.
[56] G., Lusztig, Aperiodicity in quantum affine gln, Asian J. Math. 3 (1999), 147–177.
[57] G., Lusztig, Transfer maps for quantum affine sln, in: Representations and quantizations (Shanghai, 1998), China Higher Education Press, Beijing, 2000, 341–356.
[58] K., McGerty, Generalized q-Schur algebras and quantum Frobenius, Adv. Math. 214 (2007), 116–131.
[59] L., Peng, Some Hall polynomials for representation-finite trivial extension algebras, J. Algebra 197 (1997), 1–13.
[60] L., Peng and J., Xiao, Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (2000), 563–603.
[61] G., Pouchin, A geometric Schur–Weyl duality for quotients of affine Hecke algebras, J. Algebra 321 (2009), 230–247.
[62] M., Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147–163.
[63] C. M., Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–592.
[64] C. M., Ringel, Hall algebras revisited, Israel Math. Conf. Proc. 7 (1993), 171–176.
[65] C. M., Ringel, The composition algebra of a cyclic quiver, Proc. London Math. Soc. 66 (1993), 507–537.
[66] J. D., Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985), 443–465.
[67] O., Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Not. 8 (2000), 413–440.
[68] O., Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004) 113–168.
[69] I., Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Dissertation, Berlin 1902.
[70] I., Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe, Sitzber Königl. Preuß. Ak. Wiss., Physikal.-Math. Klasse, pages 58–75, 1927.
[71] B., Sevenhant and M., Van den Bergh, On the double of the Hall algebra of a quiver, J. Algebra 221 (1999), 135–160
[72] M. E., Sweedler, Hopf Algebras, Benjamin, New York, 1969.
[73] M., Varagnolo and E., Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297.
[74] M., Varagnolo and E., Vasserot, From double affine Hecke algebras to quantized affine Schur algebras, Internat. Math. Res. Not. 26 (2004), 1299–1333.
[75] E., Vasserot, Affine quantum groups and equivariant K-theory, Transf. Groups 3 (1998), 269–299.
[76] M.-F., Vignéras, Schur algebras of reductive p-adic groups. I, Duke Math. J. 116 (2003), 35–75.
[77] H., Weyl, The Classical Groups, Princeton University Press, Princeton, NJ, 1946.
[78] J., Xiao, Drinfeld double and Ringel–Green theory of Hall algebras, J. Algebra 190 (1997), 100–144.
[79] D., Yang, On the affine Schur algebra of type A, Commun. Algebra 37 (2009), 1389–1419.
[80] D., Yang, On the affine Schur algebra of type A. II, Algebr. Represent. Theory 12 (2009), 63–75.
[81] A. V., Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible representations of GLn, Ann. Sci. Ec. Norm. Sup. 4e Sér. 13 (1980), 165–210.
[82] G., Zwara, Degenerations for modules over representation-finite biserial algebras, J. Algebra 198 (1997), 563–581.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.