On invariants and equivalence of differential operators under Lie pseudogroups actions

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms

Invariant chiral differential operators and the W_3 algebra

Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected Lie group with Lie algebra g, and V is a linear G-representation, there is an action of the corresponding affine algebra on S(V). The invariant space S(V)^{g[t]} is a commutant subalgebra of S(V), and plays the role of the classical invariant ring D(V)^G. When G is an abelian Lie group acting diagonally on V, we find a finite set of generators for S(V)^{g[t]}, and show that S(V)^{g[t]} is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W_3 algebra with c=-2 plays a fundamental role in the structure of S(V)^{g[t]}.Comment: a few typos corrected, final versio

Supertransvectants and symplectic geometry

We consider the $osp(1|2)$-invariant bilinear operations on weighted densities on the supercircle $S^{1|1}$ called the supertransvectants. These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that these operations coincide with the iterated Poisson and ghost Poisson brackets on ${\mathbb R}^{2|1}$ and apply this result to construct star-products involving the supertransvectants

The ternary invariant differential operators acting on the spaces of weighted densities

Over n-dimensional manifolds, I classify ternary differential operators acting on the spaces of weighted densities and invariant with respect to the Lie algebra of vector fields. For n=1, some of these operators can be expressed in terms of the de Rham exterior differential, the Poisson bracket, the Grozman operator and the Feigin-Fuchs anti-symmetric operators; four of the operators are new, up to dualizations and permutations. For n>1, I list multidimensional conformal tranvectors, i.e.,operators acting on the spaces of weighted densities and invariant with respect to o(p+1,q+1), where p+q=n. Except for the scalar operator, these conformally invariant operators are not invariant with respect to the whole Lie algebra of vector fields.Comment: 13 pages, no figures, to appear in Theor. Math. Phy

A new approach to deformation equations of noncommutative KP hierarchies

Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly nonassociative' (WNA) algebra structure, from which we can conclude, refering to previous work, that any solution of the Riccati system also solves the potential KP hierarchy (in the corresponding matrix algebra). We then turn to the case where the components of the matrices are multiplied using a (generalized) star product. Associated with the deformation parameters, there are additional symmetries (flow equations) which enlarge the respective KP hierarchy. They have a compact formulation in terms of the WNA structure. We also present a formulation of the KP hierarchy equations themselves as deformation flow equations.Comment: 25 page

Lie Algebras of Formal Power Series

Pseudodifferential operators are formal Laurent series in the formal inverse -1 of the derivative operator whose coefficients are holomorphic functions. Given a pseudodifferential operator, the corresponding formal power series can be ob tained by using some constant multiples of its coefficients. The space of pseu dodifferential operators is a noncommutative algebra over C and therefore has a natural structure of a Lie algebra. We determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferen tial operators, Jacobi-like forms, and modular forms for a discrete subgroup of SL(2, R)

Building Abelian Functions with Generalised Baker-Hirota Operators

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus