Strict lifts and cohomological obstructions in symplectic Lie--Rinehart--Jacobi prequantization

Abstract

Let $(\G,\rho,\omega)$ be a symplectic Lie--Rinehart--Jacobi algebra over a commutative algebra $A$, equivalently a Lie--Rinehart algebra $(\G,\bar\rho)$ endowed with a $1$-cocycle $\chi$ and a nondegenerate $2$-form closed for the twisted differential $d_\rho=d_{\bar\rho}+\chi\wedge$. Starting from the standard extension by the $2$-cocycle $\omega$, we study the strict lifting problem for its canonical $1$-form $\eta$ in the LRJ setting. We prove that the strict symmetries of $\eta$ are precisely the elements $X_f+fe$ and that $f\mapsto X_f+fe$ identifies the Jacobi Lie algebra $(A,\{\,,\})$ with this Lie algebra of strict symmetries. For locally Hamiltonian elements, the obstruction to a strict lift is the class $[i_x\omega]\in H^1_\rho(\G,A)$, yielding the exact sequence $0\to\ham\to\hamloc\to H^1_\rho(\G,A)\to0$. We also relate Okassa's symplectic curl to the variation of $d\eta$ and establish a conformal triviality criterion. In the smooth case the framework reduces to locally conformally symplectic geometry, and on a closed surface of genus $g\geq2$ we obtain structures that are not conformally trivial and whose obstruction space has dimension $2g-2$.