Geodesic
Representations of quantum conjugacy classes of orthosymplectic groups
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 23, Tome 433 (2015), pp. 20-40 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Let $G$ be the complex symplectic or special orthogonal group and $\mathfrak g$ its Lie algebra. With every point $x$ of the maximal torus $T\subset G$ we associate a highest weight module $M_x$ over the Drinfeld–Jimbo quantum group $U_q(\mathfrak g)$ and a quantization of the conjugacy class of $x$ by operators in $\mathrm{End}(M_x)$. These quantizations are isomorphic for $x$ lying on the same orbit of the Weyl group, and $M_x$ support different representations of the same quantum conjugacy class. 
@article{ZNSL_2015_433_a1,
     author = {Th. Ashton and A. Mudrov},
     title = {Representations of quantum conjugacy classes of orthosymplectic groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {20--40},
     year = {2015},
     volume = {433},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_433_a1/}
}
TY  - JOUR
AU  - Th. Ashton
AU  - A. Mudrov
TI  - Representations of quantum conjugacy classes of orthosymplectic groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 20
EP  - 40
VL  - 433
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_433_a1/
LA  - en
ID  - ZNSL_2015_433_a1
ER  - 
%0 Journal Article
%A Th. Ashton
%A A. Mudrov
%T Representations of quantum conjugacy classes of orthosymplectic groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 20-40
%V 433
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_433_a1/
%G en
%F ZNSL_2015_433_a1
Th. Ashton; A. Mudrov. Representations of quantum conjugacy classes of orthosymplectic groups. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 23, Tome 433 (2015), pp. 20-40. http://geodesic.mathdoc.fr/item/ZNSL_2015_433_a1/

[1] J. Donin, A. Mudrov, “Explicit equivariant quantization on coadjoint orbits of $GL(n,\mathbb C)$”, Lett. Math. Phys., 62 (2002), 17–32 | DOI | MR | Zbl

[2] A. Mudrov, “Quantum conjugacy classes of simple matrix groups”, Commun. Math. Phys., 272 (2007), 635–660 | DOI | MR | Zbl

[3] A. Mudrov, “Non-Levi closed conjugacy classes of $SP_q(2n)$”, Commun. Math. Phys., 317 (2013), 317–345 | DOI | MR | Zbl

[4] A. Mudrov, “Non-Levi closed conjugacy classes of $SO_q(N)$”, J. Math. Phys., 54 (2013), 081701 | DOI | MR | Zbl

[5] T. Ashton, A. Mudrov, “Quantization of borderline Levi conjugacy classes of orthogonal groups”, J. Math. Phys., 55 (2014), 121702 | DOI | MR | Zbl

[6] T. Ashton, A. Mudrov, “On representations of quantum conjugacy classes of $GL(n)$”, Lett. Math. Phys., 103 (2013), 1029–1045 | DOI | MR | Zbl

[7] J. Donin, D. Gurevich, S. Shnider, Quantization of function algebras on semisimple orbits in $\mathfrak g^*$, arXiv: q-alg/9607008

[8] T. Oshima, “Annihilators of generalized Verma modules of the scalar type for classical Lie algebras”, Harmonic Analysis, Group Representations, Automorphic forms and Invariant Theory, Lecture Notes Series, 12, IMS, National University of Singapore, 2007, 277–319 | MR

[9] M. Semenov-Tian-Shansky, “Poisson-Lie Groups, Quantum Duality Principle, and the Quantum Double”, Contemp. Math., 175 (1994), 219–248 | DOI | MR | Zbl

[10] J. Donin, A. Mudrov, “Dynamical Yang–Baxter equation and quantum vector bundles”, Commun. Math. Phys., 254 (2005), 719–760 | DOI | MR | Zbl

[11] B. Enriquez, P. Etingof, “Quantization of classical dynamical r-matrices with nonabelian base”, Commun. Math. Phys., 254 (2005), 603–650 | DOI | MR | Zbl

[12] B. Enriquez, P. Etingof, I. Marshall, “Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces”, Contemp. Math., 433 (2007), 135–176 | DOI | MR

[13] A. Alekseev, A. Lachowska, “Invariant $*$-product on coadjoint orbits and the Shapovalov pairing”, Comment. Math. Helv., 80 (2005), 795–810 | DOI | MR | Zbl

[14] E. Karolinsky, A. Stolin, V. Tarasov, “Irreducible highest weight modules and equivariant quantization”, Adv. Math., 211 (2007), 266–283 | DOI | MR | Zbl

[15] V. Drinfeld, “Quantum Groups”, Proc. Int. Congress of Mathematicians (Berkeley, 1986), ed. Gleason A. V., AMS, Providence, 1987, 798–820 | MR

[16] T. Ashton, A. Mudrov, $R$-matrix and Mickelsson algebras for orthosymplectic quantum groups, arXiv: 1410.6493

[17] A. Mudrov, $R$-matrix and inverse Shapovalov form, arXiv: 1412.3384

[18] V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge 1995 | MR | Zbl

[19] J. C. Jantzen, Lectures on quantum groups, Grad. Stud. in Math., 6, AMS, Providence, RI, 1996 | MR | Zbl

[20] C. de Concini, V. G. Kac, “Representations of quantum groups at roots of 1”, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progress in Mathematics, 92, Birkhäuser, 1990, 471–506 | MR

[21] S. Khoroshkin, O. Ogievetsky, “Mickelsson algebras and Zhelobenko operators”, Journal of Algebra, 319 (2008), 2113–2165 | DOI | MR | Zbl

[22] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, “Structure of Representations that are generated by vectors of highest weight”, Functional. Anal. Appl., 5:1 (1971), 1–8 | DOI | MR | Zbl

[23] V. Drinfeld, “Almost cocommutative Hopf algebras”, Leningrad Math. J., 1:2 (1990), 321–342 | MR | Zbl

[24] P. Pyatov, P. Saponov, “Characteristic relations for quantum matrices”, J. Phys. A, 28 (1995), 4415–4421 | DOI | MR | Zbl

[25] P. Pyatov, O. Ogievetsky, Orthogonal and symplectic quantum matrix algebras and Cayley–Hamilton theorem for them, arXiv: math/0511618[math.QA]