C-alkylation of indoles by 1,1-bis(trifluoromethyl)-2,2-di-cyanoethylene and 2-trifluoromethyl-3,3-dicyanoacrylic acid esters

Indole and its derivatives with various substituants in the 1 and 2 positions of the indole ring react with 1,1-bis(trifluoromethyl)-2,2-dicyanoethylene and 2-trifluoromethyl-3,3-dicyanoacrylic acid esters to give C³-alkylation products.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 5, pp. 1193–1195, May, 1991.     

Deformations of the central extension of the Poisson superalgebra

The Poisson superalgebra realized on smooth Grassmann-valued functions with compact support has a central extension at some values of the superdimension. We find formal deformations of these central extensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 250–269, August, 2008.     

Change in conductivity and optical absorption of fluorite after irradiation in a reactor and by a synchrocyclotron

Measurements have been made for pure crystals and for ones containing oxygen after exposure to neutrons and to a 660 MeV proton beam. The conductivity for a pure crystal is unaltered, whereas that of an oxygen-bearing crystal decreases. The two types of crystal differ only slightly in optical absorption. It is concluded that there is no single mechanism responsible for the coloring and conductivity reduction in oxygen-bearing fluorite crystals.     

Renormalization of gauge theories in nonlinear gauges

The usual rules for constructing the Lagrangian of fictitious particles in gauge theories is generalized. The gauge invariant and multiplicative renormalizability of the theory with generalized action in nonlinear gauges is proved.Translated from Izvestiya Vysshikh, Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–15, June, 1981.The author thanks I. A. Batalin and B. L. Voronov for discussing the problems considered here.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions

This paper completes the review of the theory of self-adjoint extensions of symmetric operators for physicists as a basis for constructing quantum-mechanical observables. It contains a comparative presentation of the well-known methods and a newly proposed method for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions in terms of self-adjoint boundary conditions. The new method has the advantage that it does not require explicitly evaluating deficient subspaces and deficiency indices (these latter are determined in passing) and that boundary conditions are of explicit character irrespective of the singularity of a differential expression. General assertions and constructions are illustrated by examples of well-known quantum-mechanical operators like momentum and Hamiltonian.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–36, August, 2007.     

Jacobi-type identities in algebras and superalgebras

We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of an arbitrary associative algebra. One is a consequence of the other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.