Poisson structures on Weil bundles (Q1780386)

\loadeufm A Weil algebra is a real associative commutative algebra with unit which is of the form \(A={\mathbb R}\cdot 1\oplus N\), where \(N\) is a finite-dimensional ideal of nilpotent elements. Weil algebras can be equivalently characterized as finite-dimensional quotients of the algebras \(J^k_0({\mathbb R}^n,{\mathbb R})\) of jets. To each Weil algebra \(A\) there corresponds a functor \(T^A\) from the category of smooth manifolds into the category of fibered manifolds (\textit{Weil functor}). A Weil algebra \(A\) is said to be Frobenius if there exists a non-degenerate bilinear form \(Q:A\times A\rightarrow{\mathbb R}\) satisfying \(Q(XY,Z)=Q(X,YZ)\). For example, the algebra of dual numbers \(D={\mathbb R}[\epsilon]=\{ a+b\epsilon:a,b\in{\mathbb R},\;\epsilon^2=0\}\) is a Frobenius Weil algebra and, for any smooth manifold \(M\), the Weil bundle \(T^BM\) is just the tangent bundle \(TM\). In the paper, the author constructs complete lifts of covariant and contravariant tensor fields from the smooth manifold \(M\) to its Weil bundle \(T^AM\) for the case of Frobenius Weil algebra \(A\). They reduce to the standard complete lifts from \(M\) to \(TM\) in the case \(A=D\). For multivector fields, these lifts are proven to respect the Schouten bracket. In particular, the complete lift \(w^C\) of a Poisson tensor \(w\) is again a Poisson tensor. The modular class of the Poisson manifold \((T^AM,w^C)\) is computed for Frobenius Weil algebra \(A\) which is weakly symmetric, i.e. \(d_k(A)=d_{q-k}(A)\), where \(d_k(A)=\text{dim}(N^k/N^{k+1})\) and \(q\) is the height of \(A\).