Abstract
We construct quantization of semisimple conjugacy classes of the exceptional group G = G2 along with and by means of their representations on highest weight modules over the quantum group \(U_{q}(\mathfrak {g})\). With every point t of a fixed maximal torus we associate a highest weight module Mt over \(U_{q}(\mathfrak {g})\) and realize the quantized polynomial algebra of the class of t by linear operators on Mt. Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.
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Ashton, T., Mudrov A.: Quantization of borderline Levi conjugacy classes of orthogonal groups. J. Math. Phys. 55, 121702 (2014)
Ashton, T., Mudrov A.: Representations of quantum conjugacy classes of orthosymplectic groups. J. Math. Sci. 213, 637–650 (2016)
Bernstein, I. N., Gelfand, I. M., Gelfand, S.I.: Structure of representations that are generated by vectors of highest weight. Functional. Anal. Appl. 5, 1–8 (1971)
Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1995)
de Concini, C., Kac, V.G.: Representations of quantum groups at roots of 1, Operator algebras, unitary representations, enveloping algebras, and invariant theory. (Paris, 1989), Progress in Mathematics, 92. Birkhäuser, pp. 471–506 (1990)
Donin, J., Kulish, P., Mudrov, A.: On a universal solution to reflection equation. Lett. Math. Phys. 63(3), 179–194 (2003)
Drinfeld, V.: Quantum groups. In: Gleason, A. V. (ed.) Proceedings of the International Congress of Mathematicians, Berkeley 1986, pp 798–820. AMS, Providence (1987)
Drinfeld, V.: Almost cocommutative Hopf algebras. Leningrad Math. J. 1(2), 321–342 (1990)
Etingof, P., Varchenko, A.: Dynamical Weyl groups and applications. Adv. Math. 167, 74–127 (2002)
Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–226 (1990)
Hohm, O., Samtleben, H.: Exceptional Form of D = 11 Supergravity. Phys. Rev. Lett. 111, 231601 (2013)
Khoroshkin, S. M., Ogievetsky, O.: Mickelsson algebras and Zhelobenko operators. J. Algebra 319, 2113–2165 (2008)
Kulish, P. P., Sklyanin, E.K.: Algebraic structure related to the reflection equation. J. Phys. A 25, 5963–5975 (1992)
Kuniba, A.: Quantum R-matrix for G2 and a solvable 175-vertex model. J. Phys. A.: Math. Gen. 23, 1349–1362 (1990)
Mudrov, A.: On quantization of Semenov-Tian-Shansky Poisson bracket on simple algebraic groups. Algebra Analyz 5(5), 156–172 (2006)
Mudrov, A.: Quantum conjugacy classes of simple matrix groups. Commun. Math. Phys. 272, 635–660 (2007)
Mudrov, A.: Non-Levi closed conjugacy classes of SPq(2n). Commun. Math. Phys. 317, 317–345 (2013)
Mudrov, A.: R-matrix and inverse Shapovalov form. J. Math. Phys. 57, 051706 (2016)
Mudrov, A.: Contravariant form on tensor product of highest weight modules. SIGMA 15, 026, 10 pp (2019)
Ramond P.: Exceptional Groups and Physics, arXiv:hep-th/0301050
Reshetikhin, N.Y.: Quantized Universal Enveloping Algebras, The Yang-Baxter Equation and Invariant of Links 1,11. LOMI preprints (1988)
Sergeev, S.M.: Spectral decomposition of R-matrices for exceptional Lie algebras. Mod. Phys. Lett. A 06, 923–927 (1991)
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We are grateful to the anonymous referee for careful reading and valuable remarks and suggestions.
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Presented by: Iain Gordon
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In memoriam Petr Kulish
This study (Andrey Mudrov) is supported in part by the RFBR grant 15-01-03148.
Appendix A
Appendix A
1.1 A.1 Formulas for ηij and ξij
Below we present the explicit expressions for ηij and ξij, \(i\leqslant j\), arranging them into matrices.
1.2 A.2 Entries of the Matrix F
Here we present explicit expressions of the entries fij, i < j, participating in the reduced Shapovalov inverse form.
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Baranov, A., Mudrov, A. & Ostapenko, V. Quantum Exceptional Group G2 and its Semisimple Conjugacy Classes. Algebr Represent Theor 23, 1827–1848 (2020). https://doi.org/10.1007/s10468-019-09913-4
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DOI: https://doi.org/10.1007/s10468-019-09913-4