Poisson and Hamiltonian Superpairs over Polarized Associative Algebras

Poisson superpair is a pair of Poisson superalgebra structures on a super commutative associative algebra, whose any linear combination is also a Poisson superalgebra structure. In this paper, we first construct certain linear and quadratic Poisson superpairs over a finite-dimensional or semi-finitely-filtered polarized $\Bbb{Z}_2$-graded associative algebra. Then we give a construction of certain Hamiltonian superpairs in the formal variational calculus over any finite-dimensional $\Bbb{Z}_2$-graded associative algebra with a supersymmetric nondegenerate associative bilinear form. Our constructions are based on the Adler mapping in a general sense. Our works in this paper can be viewed as noncommutative generalizations of the Adler-Gel'fand-Dikii Hamiltonian pair. As a preparatory work, some structural properties of polarized associative algebras have been studied.Comment: 30 pages, Latex fil

Conformal Oscillator Representations of Orthogonal Lie Algebras

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a, we prove that the space forms an irreducible o(n+2)-module for any constant c if the vector a is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2) on the polynomial algebra C in n variables. Moreover, we prove that the algebra C forms an infinite-dimensional irreducible weight o(n+2)-module with finite-dimensional weight subspaces if the constant c is not a half integer.Comment: 13page