A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions

Atsuo Kuniba ; Masato Okado ; Yasuhiko Yamada

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U^+_q(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.

Détail

Keywords: quantized enveloping algebra; PBW bases; quantized algebra of functions; tetrahedron equation.

Bibliographie

[1] Bazhanov V. V., Mangazeev V. V., Sergeev S. M., “Quantum geometry of 3-dimensional lattices”, J. Stat. Mech. Theory Exp., 2008 (2008), P07004, 27 pp., arXiv: 0801.0129 | DOI | MR

[2] Bazhanov V. V., Sergeev S. M., “Zamolodchikov's tetrahedron equation and hidden structure of quantum groups”, J. Phys. A: Math. Gen., 39 (2006), 3295–3310, arXiv: hep-th/0509181 | DOI | MR | Zbl

[3] Berenstein A., Zelevinsky A., “Total positivity in {S}chubert varieties”, Comment. Math. Helv., 72 (1997), 128–166 | DOI | MR | Zbl

[4] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[5] Drinfeld V. G., “Quantum groups”, Proceedings of the {I}nternational {C}ongress of {M}athematicians ({B}erkeley, {C}alif., 1986), v. 1, 2, Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[6] Geiß C., Leclerc B., Schröer J., “Cluster structures on quantum coordinate rings”, Selecta Math. (N.S.), 19 (2013), 337–397, arXiv: 1104.0531 | DOI | Zbl

[7] Ip I. C. H., Positive representations of split real quantum groups of type $B_n$, $C_n$, $F_4$, and $G_2$, arXiv: 1205.2940

[8] Isaev A. P., Kulish P. P., “Tetrahedron reflection equations”, Modern Phys. Lett. A, 12 (1997), 427–437, arXiv: hep-th/9702013 | DOI | MR | Zbl

[9] Joseph A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 29, Springer-Verlag, Berlin, 1995 | MR

[10] Kapranov M. M., Voevodsky V. A., “{$2$}-categories and {Z}amolodchikov tetrahedra equations”, Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods ({U}niversity {P}ark, {PA}, 1991), Proc. Sympos. Pure Math., 56, Amer. Math. Soc., Providence, RI, 1994, 177–259 | DOI | MR | Zbl

[11] Kashiwara M., “Global crystal bases of quantum groups”, Duke Math. J., 69 (1993), 455–485 | DOI | MR | Zbl

[12] Kuniba A., Okado M., A solution of the 3D reflection equation from quantized algebra of functions of type $B$, arXiv: 1210.6430

[13] Kuniba A., Okado M., “Tetrahedron and 3{D} reflection equations from quantized algebra of functions”, J. Phys. A: Math. Theor., 45 (2012), 465206, 27 pp., arXiv: 1208.1586 | DOI | MR | Zbl

[14] Lusztig G., “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc., 3 (1990), 447–498 | DOI | MR | Zbl

[15] Lusztig G., Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser Boston Inc., Boston, MA, 1993 | MR | Zbl

[16] McConnell J. C., Robson J. C., Noncommutative {N}oetherian rings, Pure and Applied Mathematics (New York), John Wiley Sons Ltd., Chichester, 1987 | Zbl

[17] Noumi M., Yamada H., Mimachi K., “Finite-dimensional representations of the quantum group {${\rm GL}_q(n;C)$} and the zonal spherical functions on {${\rm U}_q(n-1)\backslash{\rm U}_q(n)$}”, Japan. J. Math., 19 (1993), 31–80 | MR | Zbl

[18] Reshetikhin N. Yu., Takhtadzhyan L. A., Faddeev L. D., “Quantization of {L}ie groups and {L}ie algebras”, Leningrad Math. J., 1 (1990), 193–225 | MR | Zbl

[19] Sasaki N., “Quantization of {L}ie group and algebra of {$G_2$} type in the {F}addeev–{R}eshetikhin–{T}akhtajan approach”, J. Math. Phys., 36 (1995), 4476–4488 | DOI | MR | Zbl

[20] Sergeev S. M., “Tetrahedron equations and nilpotent subalgebras of {${\mathcal U}_q({\rm sl}_n)$}”, Lett. Math. Phys., 83 (2008), 231–235, arXiv: 0707.4029 | DOI | MR | Zbl

[21] Soibelman Y. S., “Algebra of functions on a compact quantum group and its representations”, Leningrad Math. J., 2 (1991), 161–178 | MR | Zbl

[22] Soibelman Y. S., “Selected topics in quantum groups”, Internat. J. Modern Phys. A, 7, suppl. 1B (1992), 859–887 | DOI | MR | Zbl

[23] Vaksman L. L., Soibelman Y. S., “An algebra of functions on the quantum group {${\rm SU}(2)$}”, Funct. Anal. Appl., 22 (1988), 170–181 | DOI | MR | Zbl

[24] Xi N. H., “Canonical basis for type {$B_2$}”, J. Algebra, 214 (1999), 8–21 | DOI | MR | Zbl

[25] Xi N. H., “On the {PBW} bases of the quantum group {$U_v(G_2)$}”, Algebra Colloq., 2 (1995), 355–362 | MR

[26] Yakimov M., Proc. Lond. Math. Soc. (3), 101 (2010), Invariant prime ideals in quantizations of nilpotent {L}ie algebras, arXiv: 0905.0852 | DOI | MR

[27] Zamolodchikov A. B., “Tetrahedra equations and integrable systems in three-dimensional space”, Soviet Phys. JETP, 52 (1980), 325–336 | MR