Leites, Dimitry A.; Serganova, V. Metasymmetry and Volichenko algebras. (English) Zbl 0753.17006 Phys. Lett., B 252, No. 1, 91-96 (1990). Summary: We continue the study of a generalization of supermanifolds (called here metamanifolds) on which “functions” form a metabelian algebra (one for which \([[x, y], z] = 0\)). The usual superspaces considered as metaspaces and some conventional Lagrangians have a symmetry wider than supersymmetry. Infinitesimal transformations of these metaspaces constitute Volichenko algebras. The Volichenko algebras are natural generalizations of Lie superalgebras. Here we classify simple finite-dimensional complex Volichenko algebras (under a technical hypothesis). Their list is as discrete as the list of simple Lie superalgebras. The results may be significant for applications to physics in connection with parastatistics. Cited in 1 ReviewCited in 11 Documents MSC: 17A70 Superalgebras 17B70 Graded Lie (super)algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 58A50 Supermanifolds and graded manifolds Keywords:superspaces; metamanifolds; Volichenko algebras; nonhomogeneous subspaces of Lie superalgebras; superbracket × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Majid, S., Intern. J. Mod. Phys. A, 5, 1, 1990 · Zbl 0709.17009 [2] Manin, Yu., Quantum groups and noncommutative geometry, 1988, CRM: CRM Montreal · Zbl 0724.17006 [3] Smith, S. P., to appear [4] to appear; meanwhile see preprints 1–34 [5] Drinfeld, V. G., 798, Berkeley, CA [6] Manin, Yu., Topics in noncommutative geometry, 1989, Rice University: Rice University Houston, TX [7] Leites, D., Usp. Mat. Nauk, 29, No. 3, 157, 1974, [in Russian] · Zbl 0326.14001 [8] Berezin, F.; Leites, D., Sov. Math. Dokl., 16, 1218, 1976 · Zbl 0331.58005 [9] Leites, D., Duke Math. J., 54, No. 2, 649, 1987 · Zbl 0646.58012 [10] Fredenhagen, K.; Rehren, K. H.; Schroer, B., Commun. Math. Phys., 125, 201, 1989 · Zbl 0682.46051 [11] Doplicher, S.; Roberts, J. E., Commun. Math. Phys., 1989, to appear [12] N. Jing, M.-L. Ge and Y.-S. Wu, University of Utah preprint (1990). [13] Beckers, J.; Deberg, N., J. Phys. A, 23, L751, 1990 · Zbl 0721.17005 [14] Serganova, V.; Serganova, V., meanwhile see preprint 33 [15] Berezin, F., Mat. Zametki, 1, 269, 1967, [in Russian] · Zbl 0204.03602 [16] Berezin, F., Analysis with anticommuting variables, 1987, Kluwer: Kluwer Dordrecht [17] Leites, D., Russ. Math. Surv., 35, 1, 1980; Leites, D., Supermanifold theory, 1983, Karelia Branch of the USSR Academy of Sciences: Karelia Branch of the USSR Academy of Sciences Petrozavodsk, [in Russian] [18] Corwin, L.; Ne’eman Y, Y.; Sternberg, S., Rev. Mod. Phys., 47, 573, 1975 · Zbl 0557.17004 [19] R.-O. Buchweitz, University of Toronto, Department of Mathematics preprint (1980). [20] Leites, D.; Poletaeva, E., to appear; Leites, D.; Serganova, V.; Vinel, G. [21] Kac, V., Adv. Math., 26, 8, 1977 · Zbl 0366.17012 [22] Scheunert, M., The theory of Lie superalgebras (an introduction) · Zbl 0407.17001 [23] Kac, V., Comm. Alg., 5, 1375, 1977 · Zbl 0367.17007 [24] Leites, D., J. Sov. Math. (JOSMAR), 30, 2056, 1985 [25] Serganova, V., Sov. J. Funct. Anal. Appl., 19, No. 3, 75, 1985 · Zbl 0578.17002 [26] Onishchik, A.; Vinberg, E., Seminar on algebraic groups and Lie groups, 1988, Nauka: Nauka Moscow, [in Russian, English translation: Springer, Berlin, 1990] · Zbl 0648.22009 [27] Serganova, V., Sov. J. Funct. Anal. Appl., 17, No. 3, 46, 1983 · Zbl 0545.17001 [28] Kochetkov, Yu.; Leites, D., to appear This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.