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
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 259-301
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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.
Keywords:
invariant subspace, projector, simple Lie superalgebra, split Casimir operator, Vogel parameters, generating function.
@article{TMF_2022_210_2_a5,
author = {A. P. Isaev and A. A. Provorov},
title = {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},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {259--301},
publisher = {mathdoc},
volume = {210},
number = {2},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a5/}
}
TY - JOUR AU - A. P. Isaev AU - A. A. Provorov TI - 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 JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2022 SP - 259 EP - 301 VL - 210 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a5/ LA - ru ID - TMF_2022_210_2_a5 ER -
%0 Journal Article %A A. P. Isaev %A A. A. Provorov %T 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 %J Teoretičeskaâ i matematičeskaâ fizika %D 2022 %P 259-301 %V 210 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a5/ %G ru %F TMF_2022_210_2_a5
A. P. Isaev; A. A. Provorov. 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. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 259-301. http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a5/