Abstract
Using the split Casimir operator, we find explicit formulas for the projectors onto invariant subspaces of the \( \mathrm{ad} ^{ \otimes 2}\) representation of the algebras \(so(N)\) and \(sp(2r)\). We also consider these projectors from the standpoint of the universal description of complex simple Lie algebras using the Vogel parameterization.
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Notes
The first of those equalities is a consequence of the fact that for each \(U\in \mathcal{U} ( \mathcal{A} )\), \((T_1 \otimes T_2)\Delta(U)\) commutes with the projectors \(P_\lambda\), which distinguish the irreducible subrepresentations of \((T_1 \otimes T_2)\) and are hence ad-invariant operators.
References
J. B. McGuire, “Study of exactly soluble one-dimensional \(N\)-body problems,” J. Math. Phys., 5, 622–636 (1964).
C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” Phys. Rev. Lett., 19, 1312–1315 (1967).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, New York (1982).
A. B. Zamolodchikov and Al. B. Zamolodchikov, “Factorized \(S\)-matrices in two dimensions as the exact solutions of certain relativistic quantum field models,” Ann. Phys. (N. Y.), 120, 253–291 (1979).
E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantum inverse problem method: I,” Theor. Math. Phys., 40, 688–706 (1979).
V. G. Drinfeld, “Quantum groups,” in: Proc. International Congress of Mathematicians (Berkeley, California, USA, 3–11 August 1986, A. M. Gleason, ed.), Vol. 1, Amer. Math. Soc., Providence, R. I. (1986), pp. 798–820.
M. Jimbo, “A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang–Baxter equation,” Lett. Math. Phys., 11, 247–252 (1986); “Introduction to the Yang–Baxter equation,” Internat. J. Modern Phys. A, 4, 3759–3777 (1989).
V. Pasquier and H. Saleur, “Common structures between finite systems and conformal field theories through quantum groups,” Nucl. Phys. B, 330, 523–556 (1990).
M. Karowski and A. Zapletal, “Quantum-group-invariant integrable \(n\)-state vertex models with periodic boundary conditions,” Nucl. Phys. B, 419, 567–588 (1994); arXiv:hep-th/9312008v1 (1996).
P. P. Kulish, “Quantum groups and quantum algebras as symmetries of dynamical systems,” Preprint YITP/K-959, Yukawa Institute for Theoretical Physics, Kyoto (1991).
L. Faddeev, N. Reshetikhin, and L. Takhtajan, “Quantization of Lie groups and Lie algebras,” Leningrad Math. J., 1, 193–225 (1990).
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge (1994).
A. P. Isaev, “Quantum groups and Yang–Baxter equations,” Preprint MPI 2004-132, http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2004/132.pdf, MPI, Bonn (2004).
P. Vogel, “The universal Lie algebra,” Preprint, Université Paris VII, UFR de mathématiques, Paris (1999).
P. Deligne, “La série exceptionnelle des groupes de Lie,” C. R. Acad. Sci. Paris Sér. I Math., 322, 321–326 (1996).
J. M. Landsberg and L. Manivel, “A universal dimension formula for complex simple Lie algebras,” Adv. Math., 201, 379–407 (2006).
R. L. Mkrtchyan, A. N. Sergeev, and A. P. Veselov, “Casimir eigenvalues for universal Lie algebra,” J. Math. Phys., 53, 102106 (2012).
J. M. Landsberg and L. Manivel, “Triality, exceptional Lie algebras, and Deligne dimension formulas,” Adv. Math., 171, 59–85 (2002).
A. Mironov, R. Mkrtchyan, and A. Morozov, “On universal knot polynomials,” JHEP, 1602, 78 (2016).
A. P. Isaev and V. A. Rubakov, Theory of Groups and Symmetries: Finite Groups, Lie Groups and Algebras [in Russian], KRASAND, Moscow (2018); English transl.: Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras, World Scientific, Singapore (2018).
R. Feger and Th. W. Kephart, “LieART – A Mathematica application for Lie algebras and representation theory,” Comput. Phys. Commun., 92, 166–195 (2015); arXiv:1206.6379v2 [math-ph] (2012).
N. Yamatsu, “Finite-dimensional Lie algebras and their representations for unified model building,” arXiv:1511.08771v2 [hep-ph] (2015).
P. Cvitanović, Group Theory: Birdtracks, Lie’s and Exceptional Groups, Princeton Univ. Press, Oxford (2008).
Acknowledgments
The authors thank S. O. Krivonos and O. V. Ogievetsky for the stimulating discussions.
Funding
The research of A. P. Isaev was supported by the Russian Foundation for Basic Research (Grant No. 19-01-00726).
The research of A. A. Provorov was supported by the Russian Foundation for Basic Research (Grant No. 20-52-12003\20).
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Isaev, A.P., Provorov, A.A. Projectors on invariant subspaces of representations \(\mathrm{ad}^{\otimes2}\) of Lie algebras \(so(N)\) and \(sp(2r)\) and Vogel parameterization. Theor Math Phys 206, 1–18 (2021). https://doi.org/10.1134/S0040577921010013
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DOI: https://doi.org/10.1134/S0040577921010013