Lie algebra and Dynkin index

References

• American Mathematical Society MATHSCINET. Mathematical Review for Pierre Vogel’s paper Algebraic structures on modules of diagrams. Reviewed by Olga Kravchenko, https://mathscinet.ams.org/mathscinet-getitem?mr=2769234. [Online; accessed 25 June 2021].
   Google Scholar
• Adamović, D., Kac, V. G., Möseneder Frajria, P., Papi, P., and Perše, O. Conformal embeddings of affine vertex algebras in minimal W-algebras I: structural results. J. Algebra 500:117–152. DOI: 10.1016/j.jalgebra.2016.12.005.
   Web of Science ®Google Scholar
• Anderson, D. (2012). Degeneracy of triality-symmetric morphisms. Algebra Number Theory 6(4):689–706. DOI: 10.2140/ant.2012.6.689.
   Web of Science ®Google Scholar
• Arnold, V. I. (1994). The Vassiliev theory of discriminants and knots. In: First European Congress of Mathematics: Paris, July 6-10, 1992 Volume I Invited Lectures (Part 1), vol. 119 of Progress in Mathematics Basel: Birkhäuser, pp. 3–29.
   Google Scholar
• Bar-Natan, D. (1995). On the Vassiliev knot invariants. Topology 34(2):423–472. DOI: 10.1016/0040-9383(95)93237-2.
   Google Scholar
• Barton, C. H., Sudbery, A. (2003). Magic squares and matrix models of Lie algebras. Adv. Math. 180(2):596–647. DOI: 10.1016/S0001-8708(03)00015-X.
   Web of Science ®Google Scholar
• Birman, J. S., Lin, X.-S. (1993). Knot polynomials and Vassiliev’s invariants. Invent. Math. 111(2):225–270. DOI: 10.1007/BF01231287.
   Web of Science ®Google Scholar
• Borsten, L., Marrani, A. (2017). A kind of magic. Classical Quantum Gravity 34(23):235014, 40. DOI: 10.1088/1361-6382/aa8fe2.
   Web of Science ®Google Scholar
• Bourbaki, N. (2002). Lie Groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Berlin: Springer-Verlag. Translated from the 1968 French original by Andrew Pressley; first softcover printing of the 1st English edition of 2002, 2008.
   Google Scholar
• Cohen, A. M., de Man, R. (1996). Computational evidence for Deligne’s conjecture regarding exceptional Lie groups. C. R. Acad. Sci. Paris Sér. I Math. 322(5):427–432.
   Google Scholar
• Deligne, P. (1996). La série exceptionnelle de groupes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 322(4):321–326.
   Google Scholar
• Deligne, P., de Man, R. (1996). La série exceptionnelle de groupes de Lie. II. C. R. Acad. Sci. Paris Sér. I Math. 323(6):577–582.
   Google Scholar
• Deligne, P., Gross, B. H. (2002). On the exceptional series, and its descendants. C. R. Math. Acad. Sci. Paris 335(11):877–881. DOI: 10.1016/S1631-073X(02)02590-6.
   Web of Science ®Google Scholar
• Devchand, C. (2014). Oxidation of self-duality to 12 dimensions and beyond. Commun. Math. Phys. 329(2):461–482. DOI: 10.1007/s00220-014-1996-y.
   Web of Science ®Google Scholar
• Dynkin, E. B. (1952). Semisimple subalgebras of semisimple Lie algebras. Mat. Sbornik N.S. 30(72):349–462 (3 plates).
   Google Scholar
• Dynkin, E. B. (1957). Semisimple subalgebras of semisimple Lie algebras. In: Five Papers on Algebra and Group Theory, vol. 6 of American Mathematical Society Translations: Series 2. Providence, RI: American Mathematical Society, pp. 111–244. translated by Joel Brenner, original Russian paper published by Mat. Sbornik N.S. in 1952.
   Google Scholar
• Encyclopedia of Mathematics. (2021). Cayley-dickson algebra. http://encyclopediaofmath.org/index.php?title=Cayley-Dickson_algebra&oldid=46286. [Online; accessed 15 May 2021].
   Google Scholar
• Freudenthal, H. (1964). Lie groups in the foundations of geometry. Adv. Math. fasc. 1(2):145–190. DOI: 10.1016/0001-8708(65)90038-1.
   Google Scholar
• Gross, B. H., Wallach, N. R. (1996). On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math. 481:73–123.
   Web of Science ®Google Scholar
• Gürsey, F. (1975). Algebraic methods and quark structure. In: International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975). Lecture Notes in Physics, 39, pp. 189–195.
   Google Scholar
• Humphreys, J. What role does the “dual Coxeter number” play in Lie theory (and should it be called the “Kac number”)? MathOverflow. https://mathoverflow.net/q/25592 (version: 2016-06-09), Humphreys’ homepage in MathOverflow https://mathoverflow.net/users/4231/jim-humphreys. [Online; accessed 02 May 2021].
   Google Scholar
• Kantor, I. L., Solodovnikov, A. S. (1989). Hypercomplex Numbers. (trans. from the Russian by A. Shenitzer) New York: Springer-Verlag, 1989. An elementary introduction to algebras.
   Google Scholar
• Kawasetsu, K. (2018). W-algebras with non-admissible levels and the Deligne exceptional series. Int. Math. Res. Not. IMRN 3:641–676.
   Google Scholar
• Kirillov, Jr., A. An introduction to Lie groups and Lie algebras, vol. 113 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press.
   Google Scholar
• Kleinfeld, E. (1967). On extensions of quaternions. Indian J. Math. 9(2):443–446.
   Google Scholar
• Knapp, A. W. (2002). Lie Groups Beyond an Introduction, 2nd ed. vol. 140 of Progress in Mathematics. Boston, MA: Birkhäuser.
   Google Scholar
• Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P. (1998). The Book of Involutions, vol. 44 of American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society. With a preface in French by J. Tits.
   Google Scholar
• Kontsevich, M. (1993). Vassiliev’s knot invariants. In: I. M. Gelfand Seminar, vol. 16 of Advances in Soviet Mathematics. Providence, RI: American Mathematical Society, pp. 137–150.
   Google Scholar
• Kubo, T. (2014). The Dynkin index and conformally invariant systems associated to parabolic subalgebras of Heisenberg type. Osaka J. Math. 51(2):359–373.
   Web of Science ®Google Scholar
• Kumar, S., Narasimhan, M. S., Ramanathan, A. (1994). Infinite Grassmannians and moduli spaces of G-bundles. Math. Ann. 300(1):41–75. DOI: 10.1007/BF01450475.
   Web of Science ®Google Scholar
• Landsberg, J. M., Manivel, L. (2002). Triality, exceptional Lie algebras and Deligne dimension formulas. Adv. Math. 171(1):59–85. DOI: 10.1006/aima.2002.2071.
   Web of Science ®Google Scholar
• Landsberg, J. M., Manivel, L. (2004). Series of Lie groups. Michigan Math. J. 52(2):453–479. DOI: 10.1307/mmj/1091112085.
   Web of Science ®Google Scholar
• Landsberg, J. M., Manivel, L. The sextonions and E712. Adv. Math. 201(1):143–179.
   Web of Science ®Google Scholar
• Lieberum, J. (1999). On Vassiliev invariants not coming from semisimple Lie algebras. J. Knot Theory Ramifications 8, 5:659–666. DOI: 10.1142/S0218216599000420.
   Web of Science ®Google Scholar
• Luan, Y. (2021). Nilpotent orbits and Dynkin indices of Lie algebras. PhD thesis, The Hong Kong University of Science and Technology. https://lbdiscover.ust.hk/bib/991012980219803412.
   Google Scholar
• Marrani, A., Truini, P. Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations. J. Phys. A 47(26):265202, 30. DOI: 10.1088/1751-8113/47/26/265202.
   Google Scholar
• Marrani, A., Truini, P. (2017). Sextonions, Zorn matrices, and e712. Lett. Math. Phys. 107(10):1859–1875.
   Web of Science ®Google Scholar
• Mkrtchyan, R. L. (2016). On the road map of Vogel’s plane. Lett. Math. Phys. 106(1):57–79. DOI: 10.1007/s11005-015-0803-9.
   Web of Science ®Google Scholar
• Panyushev, D. I. (2015). The Dynkin index and sl2-subalgebras of simple Lie algebras. J. Algebra 430:15–25.
   Web of Science ®Google Scholar
• Piccinni, P. (2017). On the cohomology of some exceptional symmetric spaces. In: Chiossi, S. G., Fino, A., Musso, E., Podestà, F., Vezzoni, L., eds. Special Metrics and Group Actions in Geometry, vol. 23 of Springer INdAM Ser. Cham: Springer, pp. 291–305.
   Google Scholar
• SageMath PREP Tutorials v9.2. (2021). Maximal Subgroups and Branching Rules. https://doc.sagemath.org/html/en/thematic_tutorials/lie/branching_rules.html. [Online; accessed 21 March 2021].
   Google Scholar
• SageMath PREP Tutorials v9.2. (2021). The Weyl Character Ring. https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html. [Online; accessed 20 March 2021].
   Google Scholar
• Shafarevich, I. R. (2005). Basic Notions of Algebra, vol. 11 of Encyclopaedia of Mathematical Sciences. Berlin: Springer-Verlag. Translated from the 1986 Russian original by Miles Reid, Reprint of the 1997 English translation [MR1634541], Algebra, I.
   Google Scholar
• Slodowy, P. (1980). Simple Singularities and Simple Algebraic Groups, vol. 815 of Lecture Notes in Mathematics. Berlin: Springer.
   Google Scholar
• Tits, J. (1966). Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28:223–237.
   Google Scholar
• Truini, P. (2012). Exceptional Lie algebras, su(3), and Jordan pairs. Pacific J. Math. 260(1):227–243.
   Web of Science ®Google Scholar
• van Leeuwen, M. A. A., Cohen, A. M., Lisser, B. LiE. http://www-math.univ-poitiers.fr/∼maavl/LiE. distributed under the GNU Lesser General Public Licence, with a manual in http://www-math.univ-poitiers.fr/∼maavl/LiEman/manual.pdf.
   Google Scholar
• Vassiliev, V. A. (1990). Cohomology of knot spaces. In: Theory of Singularities and its Applications, vol. 1 of Advances in Soviet Mathematics Providence, RI: Amer. Math. Soc., pp. 23–69.
   Google Scholar
• Vinberg, E. B. (2005). Construction of the exceptional simple Lie algebras. In: Vinberg, E., ed. Lie Groups and Invariant Theory, vol. 213 of American Mathematical Society Translations: Series 2. Providence, RI: American Mathematical Society, pp. 241–242. Translated from Trudy Sem. Vekt. Tenz. Anal. 13:7–9 (1966).
   Google Scholar
• Vinberg, E. B. (2005). Short SO3-structures on simple Lie algebras and associated quasielliptic planes. In: Vinberg, E., ed. Lie Groups and Invariant Theory, vol. 213 of American Mathematical Society Translations: Series 2. Providence, RI: American Mathematical Society, pp. 243–270.
   Google Scholar
• Vogel, P. (2011). Algebraic structures on modules of diagrams. Tech. rep., UNIVERSITÉ PARIS VII, 1997. downloaded from CiteSeerX https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.5848&rep=rep1&type=pdf on 29 April 2021, an updated version of this paper is published in J. Pure Appl. Algebra.
   Google Scholar
• Vogel, P. (2011). Algebraic structures on modules of diagrams. J. Pure Appl. Algebra 215(6):1292–1339. DOI: 10.1016/j.jpaa.2010.08.013.
   Web of Science ®Google Scholar
• Westbury, B. W. (2006). Sextonions and the magic square. J. London Math. Soc. (2), 73(2):455–474. DOI: 10.1112/S0024610706022605.
   Web of Science ®Google Scholar
• Zhevlakov, K. A., Slin’ko, A. M., Shestakov, I. P., Shirshov, A. I. (1983). Rings that are nearly associative, vol. 104 of Pure and Applied Mathematics. New York-London: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. Translated from the Russian by Harry F. Smith.
   Google Scholar