Split Casimir operator and solutions of the Yang–Baxter equation for the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

A. P. Isaev   ; A. A. Provorov

Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 259-301

Résumé

We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\otimes 2}$ of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the cases where $T$ is the defining or adjoint representation. For the defining representation, the $osp(M|N)$- and $s\ell(M|N)$-invariant solutions of the Yang–Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint representation, the characteristic identities and invariant projectors obtained are considered from the standpoint of a universal description of Lie superalgebras by means of the Vogel parameterization. We also construct a universal generating function for higher Casimir operators of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the adjoint representation.

Détail

Keywords: invariant subspace, projector, simple Lie superalgebra, split Casimir operator, Vogel parameters, generating function.

Bibliographie

[1] S. Okubo, “Casimir invariants and vector operators in simple and classical Lie algebras”, J. Math. Phys., 18:12 (1977), 2382–2394 | DOI

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

[3] Z. Ma, Yang–Baxter Equation and Quantum Enveloping Algebras, Advanced Series on Theoretical Physical Science, 1, World Sci., Singapore, 1993 | MR

[4] A. N. Sergeev, “Tenzornaya algebra tozhdestvennogo predstavleniya kak modul nad superalgebrami Li $\mathfrak Gl(n,m)$ i $Q(n)$”, Matem. sb., 123(165):3 (1984), 422–430 | DOI | MR | Zbl

[5] A. Berele, A. Regev, “Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras”, Adv. Math., 62:2 (1987), 118–175 | DOI | MR

[6] M. Ehrig, C. Stroppel, “Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra”, Math. Z., 284:1–2 (2016), 595–613 | DOI | MR

[7] P. Vogel, The universal Lie algebra, Preprint, Université Paris, 1999,\par http://www.math.jussieu.fr/<nobr>$\sim$</nobr>vogel/A299.ps.gz

[8] P. Deligne, “La serie exceptionnelle des groupes de Lie”, C. R. Acad. Sci. Paris Sér. I, 322:4 (1996), 321–326 | MR | Zbl

[9] J. M. Landsberg, L. Manivel, “Triality, exceptional Lie algebras and Deligne dimension formulas”, Adv. Math., 171:1 (2002), 59–85 | DOI | MR

[10] R. L. Mkrtchyan, A. N. Sergeev, A. P. Veselov, “Casimir eigenvalues for universal Lie algebra”, J. Math. Phys., 53:10 (2012), 102106, 7 pp. | DOI | MR

[11] J. M. Landsberg, L. Manivel, “A universal dimension formula for complex simple Lie algebras”, Adv. Math., 201:2 (2006), 379–407 | DOI | MR

[12] M. Y. Avetisyan, R. L. Mkrtchyan, “$X_2$ series of universal quantum dimensions”, J. Phys. A: Math. Theor., 53:4 (2020), 045202, 28 pp. | DOI | MR

[13] A. Mironov, R. Mkrtchyan, A. Morozov, “On universal knot polynomials”, JHEP, 02 (2016), 078, 34 pp., arXiv: 1510.05884 | DOI | MR

[14] A. Mironov, A. Morozov, “Universal Racah matrices and adjoint knot polynomials: Arborescent knots”, Phys. Lett. B, 755 (2016), 47–57, arXiv: 1511.09077 | DOI | MR

[15] A. P. Isaev, S. O. Krivonos, “Split Casimir operator for simple Lie algebras, solutions of Yang–Baxter equations and Vogel parameters”, J. Math. Phys., 62:8 (2021), 083503, 33 pp., arXiv: 2102.08258 | DOI | MR

[16] A. P. Isaev, A. A. Provorov, “Proektory na invariantnye podprostranstva predstavlenii $\operatorname{ad}^{\otimes 2}$ algebr Li $so(N)$ i $sp(2r)$ i parametrizatsiya Vozhelya”, TMF, 206:1 (2021), 3–22 | DOI | DOI | MR

[17] V. G. Kac, “Lie superalgebras”, Adv. Math., 26:1 (1977), 8–96 | DOI | MR

[18] F. A. Berezin, Vvedenie v algebru i analiz ot antikommutiruyuschikh peremennykh, Izd-vo Mosk. un-ta, M., 1983 | MR

[19] J. Fuksa, A. P. Isaev, D. Karakhanyan, R. Kirschner, “Yangians and Yang–Baxter R-operators for ortho-symplectic superalgebras”, Nucl. Phys. B, 917 (2017), 44–85, arXiv: 1612.04713 | DOI | MR

[20] A. P. Isaev, D. Karakhanyan, R. Kirschner, “Yang–Baxter $R$-operators for $osp$ superalgebras”, Nucl. Phys. B, 965 (2021), 115355, 28 pp., arXiv: 2009.08143 | DOI | MR

[21] V. G. Kac, “A sketch of Lie superalgebra theory”, Commun. Math. Phys., 53:1 (1977), 31–64 | DOI | MR

[22] F. A. Berezin, Introduction to Superanalysis, Mathematical Physics and Applied Mathematics, 9, Springer, Dordrecht, 1987 | DOI | MR

[23] L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press, London, 2000 | MR

[24] L. Frappat, A. Sciarrino, P. Sorba, “Structure of basic Lie superalgebras and of their affine extensions”, Commun. Math. Phys., 121:3 (1989), 457–500 | DOI | MR

[25] P. Cvitanović, Group Theory. Birdtracks, Lie's, and Exceptional Groups, Princeton Univ. Press, Princeton, 2008 | DOI

[26] A. P. Isaev, V. A. Rubakov, Teoriya grupp i simmetrii, Kniga 2: Predstavleniya grupp Li i algebr Li. Prilozheniya, URSS (Krasand), M., 2020 | MR

[27] P. P. Kulish, E. K. Sklyanin, “O resheniyakh uravneniya Yanga–Bakstera”, Zap. nauchn. sem. LOMI, 95 (1980), 129–160 | DOI | MR | Zbl

[28] A. P. Isaev, Quantum groups and Yang–Baxter equations, Preprint MPI 2004-132, Max-Plank-Institut für Mathematik, Bonn, 2004, \par http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2004/132.pdf

[29] V. G. Kac, M. Wakimoto, “Integrable highest weight modules over affine superalgebras and number theory”, Lie Theory and Geometry, Progress in Mathematics, 123, eds. J.-L. Brylinski, R. Brylinski, V. Guillemin, V. Kac, Birkhäuser, Boston, MA, 1994, 415–456 | MR

[30] A. P. Isaev, V. A. Rubakov, Teoriya grupp i simmetrii, Kniga I: Konechnye gruppy. Gruppy i algebry Li, URSS (Krasand), M., 2018 | MR