The modular class of a Lie algebroid comorphism (Q2871270)

The author introduces the definition of modular class of a Lie algebroid comorphism and also some of its properties are investigated.NEWLINENEWLINENEWLINEFor two given Lie algebroids \(A\to M\) and \(B\to N\) a comorphism between \(A\) and \(B\) covering \(\phi:M\to N\) is a vector bundle map \(\Phi:\phi^{-1}B\to A\) from the pullback vector bundle \(\phi^{-1}B\) to \(A\) such that if \(\Phi^*:A^*\to B^*\) is a Poisson map for the natural linear Poisson structures on the dual Lie algebroids. In this case \(\phi^{-1}B\to M\) carries a natural Lie algebroid structure and the natural maps \(\Phi:\phi^{-1}B\to A\) and \(j:\phi^{-1}B\to B\) are Lie algebroid morphisms.NEWLINENEWLINENEWLINEThe modular class of a Lie algebroid comorphism \(\Phi:\phi^{-1}B\to A\) is defined as the cohomology class NEWLINE\[NEWLINE\mathrm{mod}(\Phi):=\Phi^*\mathrm{mod}(A)-j^*\mathrm{mod}(B)\in H^1(\phi^{-1}B),NEWLINE\]NEWLINE where \(\text{mod}\,(A)\) denotes the modular class of \(A\).NEWLINENEWLINEAfter introducing this definition the author gives an explicit description of a representative of the modular class of \(\Phi\) in terms of modular cocycles of \(A\) and \(B\), and also some of its properties in relation with corresponding notions concerning the modular class of Poisson maps are investigated.