Majid conjecture: quantum Kac-Moody algebras version

Abstract

Based on the n-fold tensor product version of the generalized double-bosonization construction, we prove the Majid conjecture of the quantum Kac-Moody algebras version. Particularly, we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types G_2^{(1)} , E_2^{(1)},and Tp,q,r, and in this way, we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category. This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.

Keywords

Double-bosonization / quantum Kac-Moody algebras / R-matrix

References

[1] Benkart G, Kang S J, Misra K C. Graded Lie algebras of Kac-Moody type. Adv Math, 1993, 97(2): 154–190

[2] Benkart G, Kang S J, Misra K C. Indefinite Kac-Moody algebras of classical type. Adv Math, 1994, 105(1): 76–110

[3] Benkart G, Kang S J, Misra K C. Indefinite Kac-Moody algebras of special linear type. Pacific J Math, 1995, 170(2): 379–404

[4] Drinfeld V G. Quantum groups. Proceedings of the International Congress of Mathematicians, 1986: August 3-11, 1986, Berkeley, California, USA. Providence: Amer Math Soc, 1987, 798–820

[5] Faddeev L D, Reshetikhin N Yu, Takhtajan L A. Quantization of Lie groups and Lie algebras. Leningrad Math J, 1990, 1: 193–225

[6] Gu H X, Hu N H. Loewy filtration and quantum de Rham cohomology over quantum divided power algebra. J Algebra, 2015, 435: 1–32

[7] Hu H M, Hu N H. Double-bosonization and Majid's conjecture, (I): rank-inductive of A，B，C，D. J Math Phys, 2015, 56(11): 111702 (16 pp)

[8] Hu H M, Hu N H. Double-bosonization and Majid's conjecture, (IV): type-crossings from A to BCD. Sci China Math, 2016, 59(6): 1061–1080

[9] Hu H M, Hu N H. Double-bosonization and Majid's conjecture, (II): irregular R-matrices and type-crossing to G₂. Israel J Math (to appear)

[10] Hu H M, Hu N H. Grafting of quantum groups and Majid conjecture. Preprint

[11] Hu N H. Quantum divided power algebra, q-derivatives, and some new quantum groups. J Algebra, 2000, 232(2): 507–540

[12] Jimbo M. A q-difference analog of U(g) and the Yang-Baxter equation. Lett Math Phys, 1985, 10(1): 63–69

[13] Majid S. Braided momentum in the q-Poincaré group. J Math Phys, 1993, 34: 2045–2058

[14] Majid S. Double-bosonization of braided groups and the construction of U_q(g): Math Proc Cambridge Philos Soc, 1999, 125: 151–192

[15] Ringel C M. Hall algebras and quantum groups. Invent Math, 1990, 101(3): 583–591

[16] Rosso M. Quantum groups and quantum shuffles. Invent Math, 1998, 133(2): 399–416

[17] Sommerhäuser Y. Deformed enveloping algebras. New York J Math, 1996, 2: 35–58

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Higher Education Press