# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Variational calculus of supervariables and related algebraic structures. (English) Zbl 1012.37048
The author establishes a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel’fand-Dikii and the fermionic theory in his earlier work [J. Phys. A, Math. Gen. 28, No. 6, 1681-1698 (1995; Zbl 0852.58043)]. In terms of his theory he finds certain interesting new algebraic structures in connection with Hamiltonian superoperators. In particular, he finds connections between Hamiltonian superoperators and the Novikov-Poisson algebras that he introduced in [J. Algebra 185, No. 3, 905-934 (1996; Zbl 0863.17003)] in order to establish a tensor theory of Novikov algebras. He also proves that an odd linear Hamiltonian superoperator in his variational calculus induces a Lie superalgebra, which is a natural generalization of the super-Virasoro algebra under certain conditions.

##### MSC:
 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws 37K30 Relations of infinite-dimensional systems with algebraic structures 17B80 Applications of Lie algebras to integrable systems
Full Text:
##### References:
 [1] Balinskii, A. A.; Novikov, S. P.: Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras. Soviet math. Dokl. 32, 228-231 (1985) · Zbl 0606.58018 [2] Dolan, L.; Goddard, P.; Montague, P.: Conformal field theory of twisted vertex operators. Nuclear phys. B 338, 529-601 (1990) · Zbl 0745.17011 [3] Daletsky, Yu.L.: Lie superalgebras in Hamiltonian operator theory. (1984) [4] Daletsky, Yu.L.: Hamiltonian operators in graded formal calculus of variables. Funct. anal. Appl. 20, 136-138 (1986) [5] Dewitt, B.: Supermanifolds. (1992) · Zbl 0874.53055 [6] Feingold, A. J.; Frenkel, I. B.; Ries, J. F.: Spinor construction of vertex operator algebras, triality and $E(1)$8. Contemp. math. 121 (1991) · Zbl 0743.17029 [7] Frenkel, I. B.; Lepowsky, J.; Meurman, A.: Vertex operator algebras and the monster. Pure and applied mathematics (1988) · Zbl 0674.17001 [8] Gel’fand, I. M.; Dikii, L. A.: Asymptotic behavior of the resolvent of Sturm--Liouville equations and the algebra of the Korteweg--de Vries equations. Russian math. Surveys 30, 77-113 (1975) · Zbl 0334.58007 [9] Gel’fand, I. M.; Dikii, L. A.: A Lie algebra structure in a formal variational calculation. Funct. anal. Appl. 10, 16-22 (1976) · Zbl 0347.49023 [10] Gel’fand, I. M.; Dorfman, I. Ya.: Hamiltonian operators and algebraic structures related to them. Funct. anal. Appl. 13, 248-262 (1979) [11] Mathieu, P.: Supersymmetry extension of the Korteweg--de Vries equation. J. math. Phys. 29, 2499-2507 (1988) · Zbl 0665.35076 [12] Negele, J. W.; Orland, H.: Quantum many-particle systems. (1988) · Zbl 0984.82503 [13] Osborn, J. M.: Novikov algebras. Nova J. Algebra geom. 1, 1-14 (1992) · Zbl 0876.17005 [14] Osborn, J. M.: Simple Novikov algebras with an idempotent. Comm. algebra 20, 2729-2753 (1992) · Zbl 0772.17001 [15] Osborn, J. M.: Infinite dimensional Novikov algebras of characteristic 0. J. algebra 167, 146-167 (1994) · Zbl 0814.17002 [16] J. M. Osborn, Modules for Novikov algebras, in, Proceedings of the II International Congress on Algebra, Barnaul, 1991. · Zbl 0842.17002 [17] J. M. Osborn, Modules for Novikov algebras of characteristic 0, preprint. · Zbl 0846.17002 [18] Tsukada, H.: Vertex operator superalgebras. Comm. math. Phys. 18, 2249-2274 (1990) · Zbl 0704.17001 [19] X. Xu, On spinor vertex operator algebras and their modules, J. Algebra191, 427--460. · Zbl 0883.17027 [20] Xu, X.: Hamiltonian operators and associative algebras with a derivation. Lett. math. Phys. 33, 1-6 (1995) · Zbl 0837.16034 [21] Xu, X.: Hamiltonian superoperators. J. phys. A 28, 1681-1698 (1995) · Zbl 0852.58043 [22] Xu, X.: On simple Novikov algebras and their irreducible modules. J. algebra 185, 905-934 (1996) · Zbl 0863.17003 [23] Xu, X.: Novikov--Poisson algebras. J. algebra 190, 253-279 (1997) · Zbl 0872.17030 [24] Xu, X.: Skew-symmetric differential operators and combinatorial identities. Mh. math. 127, 243-258 (1999) · Zbl 0924.05002 [25] Zel’manov, E. I.: On a class of local translation invariant Lie algebras. Soviet math. Dokl. 35, 216-218 (1987)