dimensions of all homogeneous components dim Ji, i ∈ ℤ are uniformly bounded. Then either
1 1) J has finitely many negative (respectively, positive) non-zero homogeneous components;
2 2) or J is isomorphic to one of the superalgebras Jn, JCK(6), n ≥ 1 or a twisted version of it.
This theorem agrees with the Kac–van de Leur conjecture on classification of superconformal algebras.
1.9. References
Ademollo, M., Brink, L., d’Adda, A., d’Auria, R., Napolitano, E., Scinto, S., Del Giudice, E., Di Vecchia, P., Ferrera, S., Gliozzi, F., Musto, R., Pettorino, R. (1976). Supersymmetric strings and color confinement. Phys. Lett., 62B, 105–110.
Benkart, G., Elduque, A. (2002). A new construction of the Kac Jordan superalgebra. Proc. Amer. Math. Soc., 130(11), 3209–3217.
Cantarini, N., Kac, V.G. (2007). Classification of linearly compact simple Jordan and generalized Poisson superalgebras. J. Algebra, 313(1), 100–124.
Cheng, S.-J., Kac, V.G. (1997). A new N = 6 superconformal algebra. Comm. Math. Phys., 186(1), 219–231.
Grozman, P., Leites, D., Shchepochkina, I. (2001). Lie superalgebras of string theories. Acta Math. Vietnam., 26(1), 27–63.
Jacobson, N. (1968). Structure and Representations of Jordan Algebras. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence.
Kac, V.G. (1977a). Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra, 5(13), 1375–1400.
Kac, V.G. (1977b). Lie superalgebras. Adv. Math., 26(1), 8–96.
Kac, V.G., Martinez, C., Zelmanov, E. (2001). Graded simple Jordan superalgebras of growth one. Mem. Amer. Math. Soc., 150(711).
Kac, V.G., van de Leur, J.W. (1989). On classification of superconformal algebras. In Strings ‘88 (College Park, MD). World Science Publishing, Teaneck, 77–106.
Kantor, I.L. (1972). Certain generalizations of Jordan algebras. Trudy Sem. Vektor. Tenzor. Anal., 16, 407–499.
Kantor, I.L. (1990). Connection between Poisson brackets and Jordan and Lie superalgebras. In Lie Theory, Differential Equations and Representation Theory, Montreal, PQ, 1989. University of Montréal, Montreal, 213–225.
Kashuba, I., Serganova, V. (2020). Representations of simple Jordan superalgebras. Adv. Math., 370.
King, D., McCrimmon, K. (1992). The Kantor construction of Jordan superalgebras. Comm. Algebra, 20(1), 109–126.
Koecher, M. (1967). Imbedding of Jordan algebras into Lie algebras. I. Amer. J. Math., 89, 787–816.
Martin, C., Piard, A. (1992). Classification of the indecomposable bounded admissible modules over the virasoro Lie algebra with weightspaces of dimension not exceeding two. Comm. Math. Phys., 150(3), 465–493.
Martínez, C., Shestakov, I. (2020). Jordan bimodules over the superalgebra M1+1. Glasgow Math. J., 62(3). [Online].
Martínez, C., Shestakov, I., Zelmanov, E. (2001). Jordan superalgebras defined by brackets. J. London Math. Soc., 64(2), 357–368.
Martínez, C., Shestakov, I., Zelmanov, E. (2010). Jordan bimodules over the superalgebras P(n) and Q(n). Trans. Amer. Math. Soc., 362(4), 2037–2051.
Martinez, C., Zelmanov, E. (2001). Simple finite-dimensional Jordan superalgebras of prime characteristic. J. Algebra. 236(2), 575–629.
Martínez, C., Zelmanov, E. (2006). Unital bimodules over the simple Jordan superalgebra D(t). Trans. Amer. Math. Soc., 358(8), 3637–3649.
Martínez, C., Zelmanov, E. (2009). Jordan superalgebras and their representations. In Algebras, Representations and Applications. Futorney, V., Kac, V., Kashuba, I., Zelmanov, E. (eds). American Mathematical Society, Providence, 179–194.
Martínez, C., Zelmanov, E. (2010). Representation theory of Jordan superalgebras. I. Trans. Amer. Math. Soc., 362(2), 815–846.
Martínez, C., Zelmanov, E. (2014). Irreducible representations of the exceptional Cheng-Kac superalgebra. Trans. Amer. Math. Soc., 366(11), 5853–5876.
Martínez, C., Zelmanov, E.I. (2003). Lie superalgebras graded by P(n) and Q(n). Proc. Natl. Acad. Sci. USA, 100(14), 8130–8137.
Medvedev, Y.A., Zelmanov, E.I. (1992). Some counterexamples in the theory of Jordan algebras. In Nonassociative Algebraic Models (Zaragoza, 1989). Nova Science Publishing, Commack, 1–16.
Neveu, A., Schwarz, J. (1971). Factorizable dual model of pions. Nuclear Physics B, 31(1), 86–112.
Racine, M.L., Zelmanov, E.I. (2003). Simple Jordan superalgebras with semisimple even part. J. Algebra, 270(2), 374–444.
Racine, M.L., Zelmanov, E.I. (2015). An octonionic construction of the Kac superalgebra K10. Proc. Amer. Math. Soc., 143(3), 1075–1083.
Ramond, P. (1971). Dual theory for free fermions. Phys. Rev. D, 3, 2415–2418.
Schoutens, K. (1987). A nonlinear representation of the d = 2 SO(4) extended superconformal algebra. Phys. Lett., B194, 75–80.
Schwimmer, A., Seiberg, N. (1987). Comments on the n = 2, 3, 4 superconformal algebras in two dimensions. Physics Letters B, 184(2), 191–196.
Shestakov, I.P. (1997). Prime alternative superalgebras of arbitrary characteristic. Algebra i Logika. 36(6), 675–716, 722.
Shtern, A.S. (1987). Representations of an exceptional Jordan superalgebra. Funktsional. Anal. i Prilozhen., 21(3), 93–94.
Solarte, O.F., Shestakov, I. (2016). Irreductible representations of the simple Jordan superalgebra of Grassmann Poisson bracket. J. Algebra, 455, 29–313.
Stern, A.S. (1995). Representations of finite-dimensional Jordan superalgebras of Poisson brackets. Comm. Algebra, 23(5), 1815–1823.
Tits, J. (1962). Une classe d’algèbres de Lie en relation avec les algèbres de Jordan. Nederl. Akad. Wetensch. Proc., A 65, 24, 530–535.
Tits, J. (1966). Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction. Nederl. Akad. Wetensch. Proc., A 69, 28, 223–237.
Trushina, M. (2008). Modular representations of the Jordan superalgebras D(t) and K3. J. Algebra, 320(4), 1327–1343.
Wall, C.T.C. (1963, 1964). Graded Brauer groups. J. Reine Angew. Math., 213, 187–199.
Zelmanov,
