Geodesic
Shapovalov elements and Hasse diagrams
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 3, pp. 405-416 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Shapovalov elements of quantum groups are special polynomials in negative simple root vectors with coefficients in the rational Cartan subalgebra that relate singular vectors in reducible Verma modules with their highest vectors. We give explicit expressions for Shapovalov elements of nonexceptional quantum groups in terms of matrix elements of quantum $L$-operators using calculations on Hasse diagrams associated with auxiliary representations.
Keywords: Shapovalov elements, Hasse diagrams.
Mots-clés : $R$-matrix, Verma modules 
@article{TMF_2023_216_3_a2,
     author = {D. Algethami and A. I. Mudrov},
     title = {Shapovalov elements and {Hasse} diagrams},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {405--416},
     year = {2023},
     volume = {216},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_3_a2/}
}
TY  - JOUR
AU  - D. Algethami
AU  - A. I. Mudrov
TI  - Shapovalov elements and Hasse diagrams
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 405
EP  - 416
VL  - 216
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_216_3_a2/
LA  - ru
ID  - TMF_2023_216_3_a2
ER  - 
%0 Journal Article
%A D. Algethami
%A A. I. Mudrov
%T Shapovalov elements and Hasse diagrams
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 405-416
%V 216
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2023_216_3_a2/
%G ru
%F TMF_2023_216_3_a2
D. Algethami; A. I. Mudrov. Shapovalov elements and Hasse diagrams. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 3, pp. 405-416. http://geodesic.mathdoc.fr/item/TMF_2023_216_3_a2/

[1] N. N. Shapovalov, “Ob odnoi bilineinoi forme na universalnoi obertyvayuschei algebre kompleksnoi poluprostoi algebry Li”, Funkts. analiz i ego prilozh., 6:4 (1972), 65–70 | DOI | MR | Zbl

[2] I. N. Bernshtein, I. M. Gelfand, S. I. Gelfand, “Struktura predstavlenii, porozhdennykh vektorami starshego vesa”, Funkts. analiz i ego prilozh., 5:1 (1971), 1–9 | DOI | MR | Zbl

[3] 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, eds. J. Dixmier, A. Connes, Birkhäuser, Boston, MA, 1990, 471–506 | MR

[4] A. Mudrov, “Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n+1))$”, Rev. Math. Phys., 27:2 (2015), 1550004, 23 pp. | DOI | MR

[5] I. M. Musson, Šhapovalov elements and the Jantzen sum formula for contragradient Lie superalgebras, arXiv: 1710.10528

[6] A. Mudrov, Shapovalov elements for classical and quantum groups, arXiv: 2301.02624

[7] A. Mudrov, “$R$-matrix and inverse Shapovalov form”, J. Math. Phys., 57:5 (2016), 051706, 10 pp. | DOI | MR

[8] J. G. Nagel, M. Moshinsky, “Operators that lower or raise the irreducible vector spaces of $U_{n-1}$ contained in an irreducible vector space of $U_n$”, J. Math. Phys., 6:5 (1965), 682–694 | DOI | MR

[9] F. G. Malikov, B. D. Feigin, D. B. Fuks, “Osobye vektory v modulyakh Verma nad algebrami Katsa–Mudi”, Funkts. analiz i ego prilozh., 20:2 (1986), 25–37 | DOI | MR | Zbl

[10] S. Kumar, G. Letzter, “Shapovalov determinant for restricted and quantized restricted enveloping algebras”, Pacific J. Math., 179:1 (1997), 123–161 | DOI | MR

[11] D. P. Zhelobenko, Predstavlenie reduktivnykh algebr Li, Nauka, M., 1994

[12] V. G. Drinfeld, “Kvantovye gruppy”, Differentsialnaya geometriya, gruppy Li i mekhanika. VIII, Zap. nauch. sem. LOMI, 155, Nauka, L., 1986, 18–49 | DOI | Zbl

[13] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1995 | MR

[14] T. Ashton, A. Mudrov, “R-matrix and Mickelsson algebras for orthosymplectic quantum groups”, J. Math. Sci., 56:8 (2015), 081701, 8 pp. | DOI | MR

[15] A. Baranov, A. Mudrov, V. Ostapenko, Algebr. Represent. Theory, 23:4 (2020), Quantum exceptional group $G_2$ and its semisimple conjugacy classes, arXiv: 1609.02483 | DOI | MR