Abstract
We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.
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References
Cvitanovic P.: Negative dimensions and E 7 symmetry. Nucl. Phys. B 188, 373 (1981)
Cvitanovic, P.: Group Theory. Princeton University Press, Princeton (2004). http://www.nbi.dk/grouptheory
Deligne P.: La série exceptionnelle des groupes de Lie. C. R. Acad. Sci. Paris Sér. I 322, 321–326 (1996)
Deligne, P.: Private communication (2013)
Deligne P., Man R.: La série exceptionnelle des groupes de Lie II. C. R. Acad. Sci. Paris. Sér. I 323, 577–582 (1996)
Di Francesco P., Mathieu P., Sénéchal D.: Conformal Field Theory. Springer, New York (1997)
’t Hooft G.: A planar diagram theory for strong interactions. Nucl.Phys. B 72, 461–473 (1974)
Jeurissen, R.H.: The automorphism groups of octave algebras, Doctoral dissertation, University of Utrecht, (1970)
King R.C.: The dimensions of irreducible tensor representations of the orthogonal and symplectic groups. Can. J. Math. 33, 176 (1972)
Kleinfeld E.: On extensions of quaternions. Indian J. Math. 9, 443–446 (1968)
Kneissler, J.: On spaces of connected graphs II: relations in the algebra Lambda. J. Knot Theory Ramif. 10(5), 667–674 (2001). arXiv:math/0301019
Landsberg J.M., Manivel L.: The sextonions and \({E_{7\frac{1}{2}}}\). Adv. Math. 201(1), 143–179 (2006)
Landsberg J.M., Manivel L.: A universal dimension formula for complex simple Lie algebras. Adv. Math. 201, 379–407 (2006)
Landsberg J.M., Manivel L.: Series of Lie groups. Mich. Math. J. 52, 453–479 (2004)
Mkrtchyan R.L.: The equivalence of Sp(2N) and SO(−2N) gauge theories. Phys. Lett. 105, 174–176 (1981)
Mkrtchyan, R.L.: Nonperturbative universal Chern–Simons theory. JHEP 09, 54 (2013). arXiv:1302.1507
Mkrtchyan, R.L.: Universal Chern–Simons partition functions as quadruple Barnes’ gamma-functions. JHEP 10, 190 (2013). arXiv:1309.2450
Mkrtchyan R.L., Veselov A.P.: On duality and negative dimensions in the theory of Lie groups and symmetric spaces. J. Math. Phys. 52, 083514 (2011)
Mkrtchyan, R.L., Veselov, A.P.: Universality in Chern–Simons theory. JHEP 08, 153 (2012). arXiv:1203.0766
Mkrtchyan, R.L., Sergeev, A.N., Veselov, A.P.: Casimir eigenvalues for universal Lie algebra. J. Math. Phys. 53, 102–106 (2012). arXiv:1105.0115
Proctor I.: Odd symplectic groups. Invent. Math. 92, 307–332 (1988)
Rumynin, D.: Lie algebras in symmetric monoidal categories. arXiv:1205.3705 [math.QA]
Vogel, P.: Algebraic structures on modules of diagrams. J. Pure Appl. Algebra 215(6), 1292–1339 (2011) [preprint (1995)]. http://www.math.jussieu.fr/~vogel/diagrams.pdf
Vogel, P.: The Universal Lie algebra (1999) (preprint). http://www.math.jussieu.fr/~vogel/A299.ps.gz
Westbury B.W.: R-matrices and the magic square. J. Phys. A. 36(7), 1947–1959 (2003)
Westbury, B.W.: Sextonions and the magic square. J. Lond. Math. Soc. 73(2), 455–474 (2006). arXiv:math/0411428
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Mkrtchyan, R.L. On the Road Map of Vogel’s Plane. Lett Math Phys 106, 57–79 (2016). https://doi.org/10.1007/s11005-015-0803-9
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DOI: https://doi.org/10.1007/s11005-015-0803-9