×

Invariant operators on a symplectic supermanifold. (English. Russian original) Zbl 0539.58009

Math. USSR, Sb. 48, 521-533 (1984); translation from Mat. Sb., Nov. Ser. 120(162), No. 3, 528-539 (1983).
The author deals with the structure of formal differential operators on a (2n,m)-dimensional supermanifold \((m\geq 2,n>0)\). The exposition is based upon a paper of the same author [Serdica 8, 408-417 (1982; Zbl 0532.17009)]. The dimensional cases (0,m), (2n,1) were considered previously by other authors. One denotes by \(\Omega_ k\) the superspace of differential forms and by \(\Omega'_ k\) the superspace of the forms with constant coefficients. The Lie superalgebra H(2n,m) of Hamiltonian vector fields is usually defined with respect to the symplectic form \[ \omega =\sum^{n}_{i=1}dp_ idq_ i+(1/2)\sum^{m}_{j=1}d\xi_ jd\xi_{m-j+1}. \] Let \(L_ 0=osp(m,2n)\) be the Lie superalgebra of linear vector fields. The weights in \(L_ 0\) are taken relative to the elements \(E_ i=-\xi_ i\partial \xi_ i+\xi_{m-i}\partial \xi_{m- i}, H_ j=-p_ j\partial p_ j+q_ j\partial q_ j, i=1,...,[m/2],\quad j=1,...,n\). One defines the Jordan-Hölder graph for an \({\mathcal A}\)-module V, (\({\mathcal A}\) being an associative ring), by considering the sequence \(0\subset V_ n\subset...\subset V_ 0=V\) with the irreducible factors \(V_ i\) and \(\alpha_ i=V_{i-1}/V_ i\) as the edges of the graph. The set \(\{\alpha_ i\}\) is partially ordered by \(\alpha_ i>\alpha_ j\Leftrightarrow^{def}M\cap V_{i-1}/M\cap V_ i\neq \emptyset\) for every submodule \(M\subset V\). The obtained graph is denoted \(\Gamma\) (V). Proposition 1 claims a bijection between the inferior semi-intervals in \(\Gamma\) (V) and the submodules of V. By taking \({\mathcal A}=U(osp(m,2n))\), \(V=\Omega_ k\), Theorem 1 considers the graph \(\Gamma(\Omega'_ k)\). The other theorems give the decomposition of the exterior derivative operator \(d: \Omega_ k\to \Omega_{k+1}\) as well as the full description of the invariant operators \(d: T(V_ 1)\to T(V_ 2)\) on tensorial fields, for \(m>2\) and \(m=2\), by using the results about the existence of the singular vectors for H(2n,m).
Reviewer: M.Tarina

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B70 Graded Lie (super)algebras
17A70 Superalgebras

Citations:

Zbl 0532.17009
PDFBibTeX XMLCite
Full Text: DOI