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Classification of nearly associative function algebras by the method of functional equations. (English. Russian original) Zbl 0629.17017

Math. Notes 39, 438-444 (1986); translation from Mat. Zametki 39, No. 6, 806-818 (1986).
The classification of function algebras close to associative ones by means of the method of functional equations is usually divided into two steps. At first, in the given class of function algebras the general solution of functional equations, generated by identities of the algebra, is described. Then this part of the general solution, which on the given class of elements of the carrier of the function algebra defines the multiplication in sensible way, is selected. But in the paper under review only the first step of classification is considered for Jordan and alternative function algebras \(\{\phi\),\(\rho\}\) on \({\mathbb{R}}^ n\), which are invariant with respect to the group of translations, in the case when \(\rho \in C^ 2(\Omega)\), (\(\Omega\) is a neighbourhood of zero in \({\mathbb{R}}^{2n}\) and \(n\in {\mathbb{N}})\).
Reviewer: M.Abel

MSC:

17D99 Other nonassociative rings and algebras
17C99 Jordan algebras (algebras, triples and pairs)
46J10 Banach algebras of continuous functions, function algebras
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[1] G. K. Tolokonnikov, ?On associative Hamiltonian algebras,? Teor. Mat. Fiz.,31, No. 2, 250-255 (1977). · Zbl 0386.70022
[2] G. K. Tolokonnikov, ?On Hamiltonian algebras,? Teor. Mat. Fiz.,37, No. 3, 336-346 (1978). · Zbl 0401.17005
[3] G. K. Tolokonnikov, ?On algebras of observables of physical theories, close to canonical ones,? Teor. Mat. Fiz.,60, No. 1, 87-92 (1984).
[4] G. K. Tolokonnikov, ?On alternative function algebras,? Izv. Vyssh. Uchebn. Zaved., Mat. No. 9, 47-51 (1985).
[5] G. K. Tolokonnikov, ?On algebras of observables of a certain class of associative mechanics,? Teor, Mat, Fiz.,63, No. 2, 164-174 (1985).
[6] H. Weyl, The Theory of Groups and Quantum Mechanics, New York (1937).
[7] J. Dixmier, Universal Enveloping Algebras [Russian translation]. Mir, Moscow (1978).
[8] F. Trev, Introduction to the Theory of Pseudodifferential Operators and Fourier Integral Operators [Russian translation], Vol. 1, Mir, Moscow (1984).
[9] G. S. Agarwal and E. Wolf, ?Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics,? Phys. Rev. D.,2, No, 4, 2161-2226 (1970). · Zbl 1227.81196 · doi:10.1103/PhysRevD.2.2161
[10] Yu. M. Shirokov, ?On admissible canonical mechanics,? Teor. Mat. Fiz.,30, No. 1, 6-11 (1977). · Zbl 0399.70004
[11] F. A. Berezin and F. I. Karpelevich, ?On associative function algebras,? Vestn. Mosk. Univ. Ser. I Mat. Mekh., No. 1, 33-38 (1976). · Zbl 0317.46045
[12] E. Grgin and A. Peterson, ?Classical quantum mechanics in auxiliary algebras,? Phys. Rev. D.,5, No. 2, 300-306 (1972). · doi:10.1103/PhysRevD.5.300
[13] J. F. Ritt, ?Associative differential operators,? Ann. Math.,51, No. 3, 756-765 (1950). · Zbl 0037.18501 · doi:10.2307/1969379
[14] J. F. Ritt, ?Differential groups of order two,? Ann. Math.,53, No. 2, 491-519 (1951). · Zbl 0042.25801 · doi:10.2307/1969568
[15] J. F. Ritt, ?Subgroups of differential groups,? Ann. Math.,54, No. 2, 110-146 (1951). · Zbl 0043.02905 · doi:10.2307/1969315
[16] J. F. Ritt, ?Differential groups and formal Lie theory for an infinite number of parameters,? Ann. Math.,52, No. 3, 708-726 (1950). · Zbl 0038.16801 · doi:10.2307/1969444
[17] B. Weisfeiler, ?On Lie algebras of differential formal groups of Ritt,? Bull. Am. Math. Soc.,84, No. 1, 127-130 (1978). · Zbl 0382.14016 · doi:10.1090/S0002-9904-1978-14437-1
[18] A. Lichnerowicz, ?Existence and equivalence of twisted products on a symplectic manifold,? Lett. Math. Phys.,3, No. 6, 495-502 (1979). · Zbl 0433.53027 · doi:10.1007/BF00401931
[19] V. A. Ginzburg, ?Enveloping algebras and deformations,? Usp. Mat. Nauk,34, No. 2, 191-192 (1979). · Zbl 0405.17011
[20] W. Arveson, ?Quantization and the uniqueness of invariant structures,? Commun. Math. Phys.,89, 77-102 (1983). · Zbl 0522.58022 · doi:10.1007/BF01219527
[21] J. F. Jordan and E. C. G. Sudarshan, ?Lie group dynamical formalism and the relation between quantum mechanics and classical mechanics,? Rev. Mod. Phys.,33, No. 4, 515-524 (1961). · Zbl 0118.44003 · doi:10.1103/RevModPhys.33.515
[22] O. W. Greenberg, ?General free fields and models of local field theory,? Ann. Phys.,16, No. 2, 158-176 (1961). · Zbl 0099.23103 · doi:10.1016/0003-4916(61)90032-X
[23] Yu. M. Shirokov, ?Axiomatics of Hamiltonian theories of general form, including classical and quantum as special cases,? Teor. Mat. Fiz.,25, No. 3, 307-312 (1975).
[24] G. A. Zaitsev, Algebraic Problems of Mathematical and Theoretical Physics [in Russian], Nauka, Moscow (1974).
[25] E. Grgin and A. Petersen, ?Duality of observables and generators in classical and quantum mechanics,? J. Math. Phys.,15, No. 6, 764-769 (1974). · Zbl 0287.17003 · doi:10.1063/1.1666726
[26] C. L. Mehta, ?Phase space formalism of the dynamics of canonical variables,? J. Math. Phys.,5, No. 5, 677-686 (1964). · Zbl 0152.23502 · doi:10.1063/1.1704163
[27] A. V. Yagzhev, ?On a certain functional equation of theoretical physics,? Funkts. Anal. Prilozhen.,16, No. 1, 49-57 (1982). · Zbl 0492.47025 · doi:10.1007/BF01081810
[28] A. A. Kirillov, ?Local Lie algebras,? Usp. Mat. Nauk,31, No. 1, 57-76 (1976).
[29] P. Jordan, J. von Neumann, and E. Wigner, ?On the algebraic generalization of the quantum mechanical formalism,? Ann. Math.,35, 29-64 (1934). · JFM 60.0902.02 · doi:10.2307/1968117
[30] G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory [Russian translation], Mir, Moscow (1976).
[31] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings Close to Associative Ones [in Russian], Nauka, Moscow (1978).
[32] N. Jacobson, ?Structure theory for a class of Jordan algebras,? Proc. Nat. Acad. Sci, USA,55, No. 1, 243-251 (1966). · Zbl 0133.28903 · doi:10.1073/pnas.55.2.243
[33] E. M. Alfsen, F. W. Schultz, and E. A. Stormer, ?A Gelfand-Neumark theorem for Jordan algebras,? Adv. Math.,28, No. 1, 11-56 (1978). · Zbl 0397.46065 · doi:10.1016/0001-8708(78)90044-0
[34] E. I. Zel’manov, ?Jordan algebras with division,? Algebra Logika,18, No. 3, 286-310 (1979). · Zbl 0457.03040 · doi:10.1007/BF01673946
[35] A. I. Mal’tsev, Classical Algebra [in Russian], Nauka, Moscow (1976).
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