Deformations of the central extension of the Poisson superalgebra.pdf

a r X i v : h e p - t h / 0 5 0 1 0 2 7 v 1 5 J a n 2 0 0 5 Deformations of the central extension of the Poisson superalgebra. S.E.Konstein? and I.V.Tyutin?? I.E.Tamm Department of Theoretical Physics, P. N. Lebedev Physical Institute, 119991, Leninsky Prospect 53, Moscow, Russia. Abstract Poisson superalgebras realized on the smooth Grassmann valued functions with compact support in Rn have the central extensions. The deformations of these central extensions are found. 1 Introduction This paper continues the investigation started in [2], [3] and [4]. We consider the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support in Rn. As it is shown in [2] this superalgebra has central extensions for some dimensions. For these dimensions, we found the second adjoint cohomology space and the deformations of the Poisson superalgebra under consideration. 1.1 Poisson superalgebra Let K be either R or C. We denote by D(Rn) the space of smooth K-valued functions with compact support on Rn. This space is endowed with its standard topology. We set Dn?n+ = D(R n+)?Gn?, En?n+ = C ∞(Rn+)?Gn?, D′n?n+ = D ′(Rn+)?Gn? , where Gn? is the Grassmann algebra with n? generators and D ′(Rn+) is the space of con- tinuous linear functionals on D(Rn+). The generators of the Grassmann algebra (resp., the coordinates of the space Rn+) are denoted by ξα, α = 1, . . . , n? (resp., x i, i = 1, . . . , n+). We shall also use collective variables zA which are equal to xA for A = 1, . . . , n+ and are equal to ξA?n+ for A = n+ + 1, . . . , n+ + n?. The spaces D n? n+ , En?n+ , and D ′n? n+ possess a natural grading which is determined by that of the Grassmann algebra. The parity of an element ?E-mail: konstein@lpi.ru ?E-mail: tyutin@lpi.ru ?This work was supported by the RFBR (grants No. 02-02-16944 (I.T.) and No. 02-02-17067 (S.K.)), and by the grant LSS-1578.2003.2. 1 2f of these spaces is denoted by ε(f). We also set εA = 0 for A = 1, . . . , n+ and εA = 1 for A =